# B/PS bulges and barlenses from a kinematic viewpoint. I

Daria Zakharova1,2, Iliya S. Tikhonenko3, Natalia Ya. Sotnikova4, and Anton A.
Smirnov5
1Dipartimento di Fisica e Astronomia "Galileo Galilei", Universita’ degli
studi di Padova, Vicolo dell’Osservatorio, 3, I-35122, Padova, Italy
2INAF - Osservatorio astronomico di Padova, Vicolo dell’Osservatorio, 5,
I-35122, Padova, Italy
3Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße, D-85748
Garching, Germany
4St. Petersburg State University, Universitetskij pr. 28, 198504 St.
Petersburg, Stary Peterhof, Russia
5Central (Pulkovo) Astronomical Observatory of RAS, Pulkovskoye Chaussee 65/1,
196140 St. Petersburg, Russia
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

A significant part of barred disc galaxies exhibits boxy/peanut-shaped
structures (B/PS bulges) at high inclinations. Another structure also
associated with the bar is a barlens, often observed in galaxies in a position
close to face-on. At this viewing angle, special kinematic tests are required
to detect a 3D extension of the bars in the vertical direction (B/PS bulges).
We use four pure $N$-body models of galaxies with B/PS bulges, which have
different bar morphology from bars with barlenses to the so-called face-on
peanut bars. We analyse the kinematics of our models to establish how the
structural features of B/PS bulges manifest themselves in the kinematics for
galaxies at intermediate inclinations and whether these features are related
to the barlenses. We apply the dissection of the bar into different orbital
groups to determine which of them are responsible for the features of the
LOSVD (line-of-sight velocity distribution), i.e., for the deep minima of the
$h_{4}$ parameter along the major axis of the bar. As a result, we claim that
for our models at the face-on position, the kinematic signatures of a ‘peanut’
indeed track the vertical density distribution features. We conclude that
orbits responsible for such kinematic signatures differ from model to model.
We pay special attention to the barlens model. We show that orbits assembled
into barlens are not responsible for the kinematic signatures of B/PS bulges.
The results presented in this work are applicable to the interpretation of IFU
observations of real galaxies.

###### keywords:

methods: numerical – galaxies: evolution – galaxies: kinematics and dynamics –
galaxies: structure

††pubyear: 2023††pagerange: B/PS bulges and barlenses from a kinematic
viewpoint. I–B

##  1 Introduction

The barlens component has been introduced only recently in morphological
classifications of nearly face-on galaxies. It is described as a new type of
lens embedded into the bar and covering about half its length (Laurikainen et
al., 2011). Its outline is circular-like, while its azimuth-averaged profile
is exponential, which distinguishes barlenses from classical bulges
(Laurikainen & Salo, 2017; Athanassoula et al., 2015). The barlenses are
typically four times larger than nuclear discs and rings and are also distinct
from standard lenses (Kormendy, 1979) because they are smaller than bars
(Laurikainen et al., 2018; Athanassoula et al., 2015).

The so-called B/PS (boxy/peanut-shaped) bulges are also associated with the
bars and stand out well against the edge-on stellar disc background (Lütticke
et al., 2000a). B/PS bulges are the thickest inner parts of the bars, which
have grown vertically as the bars have evolved (Combes & Sanders, 1981;
Athanassoula, 2016; Laurikainen & Salo, 2016). Although B/PS structures in
$N$-body simulations inevitably form as parts of bars (Combes & Sanders,
1981), the relationship between B/PS structures and bars is not obvious in
observations due to different viewing conditions. This connection is
established through the statistics of large samples of galaxies. Lütticke et
al. (2000a) (see also Li et al., 2017; Erwin & Debattista, 2017) found that up
to 45% of all S0-Sd galaxies have B/PS structures, which is close to the
fraction of barred galaxies ($\sim$ 50%-60%, Marinova & Jogee, 2007). Erwin &
Debattista (2013, 2017) found that B/PS features extend from 1/3 to 2/3 of the
bar radius. In addition to the statistical arguments, kinematic arguments also
work: in edge-on galaxies with B/PS structures, traces of a bar can be found
from the kinematic characteristics of elongated orbits (Bureau & Athanassoula,
2005; Iannuzzi & Athanassoula, 2015).

Thus, two structural features of galaxies, B/PS bulges and barlenses, are
closely associated with bars. B/PS bulges stand out when the galaxy is viewed
edge-on, and barlenses are well identified visually when the galaxy positioned
almost face-on. That is why the idea has recently spread that ‘barlens’ and
‘box/peanut’ are the same stellar structure, namely the inner part of the bar,
but seen at different viewing angles (Laurikainen et al., 2014; Athanassoula,
2016). The identification of the two structures was based on several
arguments, mostly indirect. The parent galaxies of barlenses and B/PS
structures have a similar total stellar mass distribution (Laurikainen et al.,
2014). Relative to the bar, barlenses have similar sizes (Athanassoula et al.,
2015) as obtained for the B/PS bulges by Lütticke et al. (2000b) and Erwin &
Debattista (2013). However, the main evidence came from analysing the results
of $N$-body simulations. By viewing their snapshots from many different
angles, Athanassoula et al. (2015) concluded that barlenses are simply the
vertically thick part of the bar, i.e., the barlenses in the edge-on position
appear as B/PS bulges. Stronger evidence comes from an analysis of the
kinematics of galaxies with barlenses. First, Méndez-Abreu et al. (2008) found
kinematic evidence of the B/PS bulge for the galaxy NGC 98 classified as
having a barlens (Laurikainen & Salo, 2017). Then, for NGC 1640, which
classified as a galaxy with a barlens (Laurikainen & Salo, 2017), Méndez-Abreu
et al. (2014) showed a kinematic hint of the presence of a B/PS bulge in the
central regions of the bar. Finally, the TIMER team (Gadotti et al., 2020)
claim to have found in the barlens region of several nearly face-on galaxies
kinematic signatures of both a bar and a box/peanut, as expected from
numerical simulations. But for almost face-on galaxies it is not yet clear
which structural component of the bar (the bar itself or the barlens) is
associated with the discovered kinematic signatures of a ‘peanut’. That is why
even the kinematic data needs a more accurate interpretation.

This question can be solved using the results of the orbital analysis of
numerical models. Recently Tikhonenko et al. (2021) have dissected a bar in
one typical $N$-body model with a barlens into separate orbital components
supported by completely different types of orbits. This work shows that the
part of the barlens with rounded isophotes in the face-on view is a rather
flat structure in the vertical direction without any significant off-centre
protrusions and signatures of a peanut. It was also concluded that the edge-on
B/PS bulge is a barlens-independent component of the bar and is supported by
various types of orbits. One can not separate the face-on image of the galaxy
into individual orbital components and find kinematic parameters for these
components, but by using the results of $N$-body simulations and orbital
analysis, one can understand which orbital subsystem contributes to the
kinematic features of a peanut.

Four parameters are often used in kinematic diagnostics: the mean velocity
$\overline{V}$, the velocity dispersion $\sigma$, and $h_{3}$, $h_{4}$
parameters, which measure the deviations of the LOSVD (line-of-sight velocity
distribution) from a Gaussian. The last two parameters are the corresponding
coefficients that arise when the velocity profile is expanded by orthogonal
Gauss-Hermite functions. The parameter $h_{3}$ measures asymmetric deviations
from the Gaussian, while the parameter $h_{4}$ measures symmetric ones. These
deviations occur when different velocity components on the line of sight do
not contribute equally to the LOSVD (van der Marel & Franx, 1993; Gerhard,
1993).

Debattista et al. (2005) used kinematic diagnostic to analyse the LOSVD of
barred face-on galaxies. In this case, there is a single velocity component, a
vertical one ($v_{z}$). Debattista et al. (2005) noted that the velocity
distribution (VD) for the vertical component $f(v_{z})$ may differ from the
Gaussian and lead to negative values of $h_{4}$ if the density distribution in
the vertical direction is flat-topped, i.e., not so strongly concentrated
towards the mid-plane compared to the surrounding disc. The flat-topped volume
density distribution in the vertical direction is expressed in terms of
negative values of the fourth coefficient of Gauss-Hermite expansions $d4$ (an
analogue of $h_{4}$). It opens up the possibility of confirming the
relationship between bars (structures that are well identified in a nearly
face-on view) and B/PS bulges, which are seen predominantly edge-on.
Debattista et al. (2005) have noticed that negative values of $h_{4}$ along
the major axis of the bar at inclinations less than $30^{\circ}$ indicate the
presence of the peanut if they form two deep minima on both sides of the
centre of the galaxy. Two shallow minima of $h_{4}$ along the major axis of
the bar indicate not a peanut, but a contamination of the LOSVD by other
velocity components, in addition to $v_{z}$.

The origin of the negative $h_{4}$ values is still not clearly understood
while being particularly important in the light of recent high resolution
integral field unit (IFU) spectroscopy data (for instance, the TIMER project,
Gadotti et al., 2019). The accurate interpretation of these features (negative
$h_{4}$ values) would require a set of synthetic 2D maps of the common
kinematic parameters ($\overline{V}$, $\sigma$ and $h_{3}$, $h_{4}$
parameters) based on a wide grid of numerical models viewed from different
inclinations and bar major axis viewing angles.

We intend to explore model kinematic maps for intermediate inclinations, thus
extending the data available from Debattista et al. (2005) for a face-on view
and from Iannuzzi & Athanassoula (2015) for mainly high inclined models. In
addition, since quite a lot of galaxies with barlenses fall into the sample by
Gadotti et al. (2020), we would like to conduct a comparative analysis of 2D
maps for models with gradually changing bar morphology from models with
barlenses to models without features in the face-on structure of the bar. We
also intend to collect the analysed model data into data cubes with an
appropriate spectral resolution for the most plausible comparison with
observational data cubes. Such maps will be useful both for interpreting
existing and future observational data. Since 2D kinematic maps for inclined
galaxies can differ from face-on maps, it is necessary to extend the analysis
of the LOSVD distortions due to i) density distribution features, ii)
contamination of the LOSVD by various velocity components. This will help to
better understand the relationship of kinematic parameters and B/PS bulges in
cases when the B/PS bulges themselves are not visible. Finally, having the
results of an orbital analysis for a number of numerical models, we have a
unique opportunity to study the influence of different orbital groups on the
kinematic picture and to understand which orbital groups (assembled into a
barlens or into the bar itself) raise distortions into the LOSVD and create
peanut signatures when the galaxy is viewed face-on.

In this article, we focus only on the features of the LOSVD, which are
measured by the parameter $h_{4}$. We leave the joint analysis of the
parameters $h_{3}$ and $h_{4}$ for future works.

The paper is organised as follows. In Section 2, we describe our $N$-body
models as well as our dissection of the models into various orbital groups
assembled into different sub-strictures of the bar, explain how we prepare
data cubes for all models and for a set of inclinations and position angles of
a bar and discuss how we extract the main kinematic parameters from LOSVDs in
each pixel. In Section 3, we present our results, compare 2D kinematic maps
for models with a barlens and with an ordinary bar at face-on view and other
positions of a bar. In Section 4, we differentiate the effects of a flat-
topped vertical density distribution and effects of inclinations associated
with contamination of the LOSVD by velocity components other than $v_{z}$. In
Section 5, we focus on the contributions of different orbital groups into
features of the LOSVD. In Section 6 we discuss the astrophysical applications
of our results. In Section 7, we give our conclusions.

##  2 Data preparation

###  2.1 N-body models

To study the kinematic properties of B/PS bulges, we use a set of self-
consistent pure $N$-body (i.e., without the gas component) models from Smirnov
et al. (2021). In the following description and most of the illustrations, we
adopt the unit system where 1 length unit is equal to $3.5$ kpc, $G=1$, and
unit mass is equal to $5\times 10^{10}M_{\odot}$. Assuming these values, a
time unit would be equal to $13.8$ Myr.

The initial conditions for each model were prepared with mkgalaxy script
(McMillan & Dehnen, 2007). At the start of simulations, all models have a
stellar disc embedded into a self-consistent (“live”) dark halo.

The stellar disc of each model has an exponential radial profile and is
isothermal in the vertical direction:

$\rho_{\mathrm{d}}(R,z)=\frac{M_{\mathrm{d}}}{4\pi
R_{\mathrm{d}}^{2}z_{\mathrm{d}}}\cdot\exp(-R/R_{\mathrm{d}})\cdot\mathrm{sech}^{2}(z/z_{\mathrm{d}})\,,$
(1)

where $R$ is the cylindrical radius, $M_{\mathrm{d}}=1$ is the total mass of
the disc, $R_{\mathrm{d}}=1$ and $z_{\mathrm{d}}=0.05$ are scale lengths in
radial and vertical directions, respectively.

The halos are spherical and have Navarro-Frenk-White-like (Navarro et al.,
1996) profile with dark matter mass for each model equal to
$M_{\mathrm{h}}(r<4R_{\mathrm{d}})/M_{\mathrm{d}}\approx 1.5$.

For three models, a spherical Hernquist bulge (Hernquist, 1990) in the central
area was added. All models in the course of their evolution develop bars with
B/PS bulges and X-structures (the brightest part of the B/PS bulges) visible
at the edge-on position. We distinguish models by their nicknames according to
the final morphology of their bars: BL (BarLens), BLx (BarLens + slight traces
of face-on X-shaped bar), Xb (face-on X-shaped bar + bulge), X (face-on
X-shaped bar). BL and BLx models are the galaxies with a barlens in a face-on
position (first and second columns in Fig. 1). X and Xb models represent
galaxies with X-shaped (or peanut-shaped) bars in a face-on position (third
and fourth columns in Fig. 1). Such a morphology difference is due to the
presence/absence of the bulge. The bulges in models differ in the total mass
$M_{\mathrm{b}}$ and radial scale lengths $r_{\mathrm{b}}$ (Table 1).
111blackOne should bear in mind that galaxies which show X-shape at low
inclinations are quite rare.

Each subsystem was represented by a specific number of “star”-particles:
black$4\times 10^{6}$ for the disc, black$4.5\times 10^{6}$ for the halo. The
total number of bulge particles is determined according to the value of the
ratio $M_{\mathrm{b}}/M_{\mathrm{d}}$, with a reference point for the Xb
model, where the number is equal to black$0.8\times 10^{6}$.

The disc radial dispersion profile was exponential with
$\sigma_{R}=\sigma_{0}\cdot\exp(-R/2R_{\mathrm{d}})$, where $\sigma_{0}$ is a
normalisation constant. It is inferred from the Toomre parameter value:
$Q(2\,R_{\mathrm{d}})=Q_{0}$ with $Q_{0}$ set to 1.2. Thus, we consider a
marginally stable stellar disc. The evolution of all models was traced by
gyfaclON integrator (Dehnen, 2002) from the NEMO suite (Teuben, 1995) for
approximately 8 Gyr with 1.7 Myr time step. Fig. 1 shows snapshots for our
models for $t=450$ ($\approx 6.2$ Gyr).

The key difference between the models is their initial bulge concentration,
which results in different gradient of the rotation curve in the centre. For
example, the BL model has the steepest rotation curve in the central regions
(see Smirnov et al., 2021, figure 1). The relation between the slope of the
rotation curve in the inner parts of the galaxy and different face-on bar
morphology was first noted by (Salo & Laurikainen, 2017). Our BL model does
fall into the range of parameters for which the model develops a clear
barlens. Thus, the chosen setup of models yields a gradual progression in
face-on bar morphology from a bar with a barlens to a face-on peanut, which
makes this model set optimal for understanding the nature of the barlens and
its connection with a B/PS bulge: either in terms of its vertical structure
(done in Tikhonenko et al. 2021) or its kinematics (this work).

Figure 1: The set of models (BL, BLx, Xb, and X) from Smirnov et al. (2021) used in this work. The top row shows face-on views of the corresponding model in $(x,y)=[-3;3]\times[-3;3]$ area, while the bottom row displays edge-on views in $(x,z)=[-3;3]\times[-1.5;1.5]$.  model | X | Xb | BLx | BL
---|---|---|---|---
$M_{\mathrm{b}}$ | $0$ | $0.2$ | $0.1$ | $0.1$
$r_{\mathrm{b}}$ | - | $0.2$ | $0.1$ | $0.05$
Table 1: The properties of classical bulges in our models, see text for the
adopted unit system description.

###  2.2 Orbital groups

In Parul et al. (2020) and Smirnov et al. (2021), the entire orbital structure
of several self-consistent $N$-body models with bars were analysed using
Fourier analysis of spatial coordinates of each star-particle. Four models
presented in Section 2.1 fall into the grid of models for which orbital
analysis was performed. Thus, we have unique data for studying the kinematics
of individual orbital groups and searching for kinematic signatures of a
peanut on face-on snapshots that we can attribute to one or another orbital
group.

The conducted orbital analysis is based on spectral dynamics methods (Binney &
Spergel, 1982). For all orbits in the disc, the Cartesian frequencies
($f_{x}$, $f_{y}$, and $f_{z}$) in the rotating coordinate frame with $x$-axis
aligned with the bar major axis222blackTo identify the bar major axis we
diagonalise the moment-of-inertia tensor of a subset of particles ($R<0.8$,
$|z|<0.3$) and find principal axes. have been computed by finding the position
of the highest peaks in Fourier spectra of the corresponding Cartesian
coordinates are $x$, $y$, $z$.

In addition to $f_{x}$, $f_{y}$, and $f_{z}$, we have determined the radial
oscillations frequency $f_{R}$. Thus, a set of four numbers for every orbit in
a disc for each model have been obtained. Using the ratios of these
frequencies $f_{x}/f_{\mathrm{R}}$ and $f_{y}/f_{\mathrm{R}}$, Smirnov et al.
(2021) found several orbital groups in their grid of models. These orbital
groups assemble in specific structures, and a dominant group or a combination
of groups are responsible for a particular morphological feature of a model
bar. Thus, the bar can be disassembled into the following orbital groups.

  * -

A so-called ‘classic’ bar, identified by the condition $f_{R}/f_{x}=2.0\pm
0.1$. This orbital group primarily consists of loop orbits with
$f_{y}/f_{x}=1\pm 0.1$ (in the sense of the orbital classification byValluri
et al. 2016) with x1-like and x2-like (prograde) or x4-like (retrograde)
orbits being the prominent members of it. An x1-like family gives a narrow and
elongated bar with an inner bar-like structure superimposed on an outer bar
and oriented perpendicular to it (x2-like or x4-like orbits).

  * -

Boxy orbits, selected via criterion $f_{R}/f_{x}=2.0\pm 0.1$ and
$f_{y}/f_{x}>1.1$. These orbits constitute a face-on peanut, or the so called
‘boxy bar’, and are part of the extended bar.

  * -

Two non-classical groups: $\text{bl}_{\text{o}}$ and $\text{bl}_{\text{u}}$ in
notation of Smirnov et al. (2021). blackOrbits from these two groups have
$f_{R}/f_{x}$ either larger than 2 ($\text{bl}_{\text{o}}$) or smaller than 2
($\text{bl}_{\text{u}}$). They are definitely not from the outer disc, but at
the same time they are not quite elongated with the bar and do not precess
synchronously with it. These orbits have rather complex frequency proportions.
Orbits from the $\text{bl}_{\text{u}}$ family have $f_{y}/f_{x}=1.0$. As to
the orbits from the $\text{bl}_{\text{o}}$ family, they split into two
branches, with $f_{y}/f_{x}=1.0$ (more extended structure) and with
$(f_{y}+f_{x})/f_{R}=1.0$ (more compact structure). All this gives them the
overall ‘roundish’ shape, which results in the formation of a barlens in the
$N$-body model that has it. The $\text{bl}_{\text{u}}$ orbital group is a key
component of a barlens and maintains the rounded shape in the central area of
the bar. The $\text{bl}_{\text{o}}$ orbits are assembled into a more extended
square-like structure, which is nevertheless a part of the barlens.

The percentage of orbits of each type in the models is compiled in Table 2.

One of the main conclusions made by Smirnov et al. (2021) is that the
dominance of one or another orbital group determines the face-on morphology of
a bar and its features. In Section 5, we use the data on the bar orbital
groups revealed in Smirnov et al. (2021) to determine their contribution to
pixels with peanut signatures at the face-on position of the galaxy.

orbital | X | Xb | BLx | BL |
---|---|---|---|---|---
group |  |  |  |  |
x1+x2+x4 | 3.79 | 6.52 | 9.51 | 11.35 |
boxy bar | 45.19 | 25.68 | 23.88 | 10.45 |
$\text{bl}_{\text{o}}$ | 5.36 | 7.82 | 9.98 | 13.49 |
$\text{bl}_{\text{u}}$ | 0.96 | 1.05 | 3.42 | 9.22 |
Table 2: The percentage of orbits of each type in the models. The fraction (%)
is given relative to the total number of particles in the disc ($4\times
10^{6}$).

###  2.3 Datacubes

To study kinematics, one needs to represent the velocities of model particles
in a way that can be faithfully compared with observational data. The most
common form of data used in kinematic studies is a datacube. Here, we
construct synthetic datacubes from our simulations.

The best stellar kinematics available to date were obtained by TIMER project
(Gadotti et al., 2019), conducted on the MUSE spectrograph on the VLT. The
high-level products of their processing pipeline are the maps with stellar
kinematics parameters, which are publicly available on the TIMER
website333https://www.muse-timer.org. We use these maps as a reference point
for the construction of our synthetic maps. We model our datacubes to have a
comparable spatial resolution to MUSE. Specifically, we assume the whole image
to be $30^{\prime}\times 30^{\prime}$ and the pixel size of each slice to be
$300\text{px}\times 300\text{px}$. The proper choice of the spectral axis was
more arbitrary. Normally, the velocities from integral-field spectroscopy
(IFS) data are inferred via advanced pipelines, such as GIST (Bittner et al.,
2019), which in turn uses pPXF (Cappellari & Emsellem, 2004) for stellar
kinematics. These pipelines estimate the LOSVD by fitting a linear combination
of template library spectra to the real spectrum of the galaxy. However, in
case of our $N$-body models, a realistic imitation of stellar spectra would be
unnecessary complex for the task at hand. Instead, we take the mean velocity
uncertainty from (Bittner et al., 2019), where is was estimated to be $\approx
5\text{km}\,\text{s}^{-1}$, and assume the same uncertainty for the positions
of the LOSVD bins. Thus, we simply assume the bin width to be equal to
$2\sigma\approx 10\text{km}/\text{s}$.

To improve signal-to-noise444In absence of real errors, we assume an
artificial noise with Poisson distribution, thus S/N scales as $N^{-1/2}$.
ratio (S/N) of the final datacubes, we stack 17 snapshots blackin the time
interval $t=448$ – $452$. During this period by $\Delta t=4$, the change in
the bar pattern speed is insignificant (see Figure 2 in Smirnov et al., 2021).
blackThus,74.8 million particles in total blackare reached. Contrary to the
similar procedure in Iannuzzi & Athanassoula (2015) we do not enforce any kind
of symmetry to the model along the principal axes of the bar, because the
total number of particles was already sufficient for the purposes of this
paper after stacking.

To obtain the line-of-sight velocities, we rotate the model to match its
$z$-axis, initially perpendicular to the disc plane, with the chosen direction
of the line-of-sight and project the full velocity ($v$) on the new $z$-axis.
Then, the velocity is binned from $\min v_{\text{LOS}}$ to $\max
v_{\text{LOS}}$ with the bin width equal to the velocity resolution assumed
above. In addition to that, we also compute the LOS projections of the
vertical ($v_{z}$), azimuthal ($v_{\varphi}$), and radial ($v_{R}$) velocity
components in the _original_ coordinate frame. Thus, for every spatial pixel
of the datacube, we effectively have four distributions: one of the total LOS
velocity and three distributions of the LOS velocity components. In practice,
we just made a dedicated datacube for each velocity component to simplify the
further processing.

In a similar fashion, for each model we prepare the set of datacubes excluding
each orbital family one-by-one. In Section 5, we would use these datacubes to
estimate the contributions of orbital groups in LOSVDs of datacubes. This
approach was chosen instead of considering each component alone, because it
results in datacubes with a larger number of particles that are easier to
analyse and to compare with observational data.

We deliberately do not apply the Voronoi-binning technique, which is quite
common in similar studies, because in this case it harder to compare our
datacubes for different models fiber-by-fiber (our Section 3) as well as
different velocity components for the same model (Section 4). In addition to
that, it is rather unclear how to study the effect of individual orbital
groups (our Section 5), because the tessellation should be again different for
every datacube and the additive property of LOSVD in a given fiber can no
longer be used. For the same reasons, we also do not smooth the layers of the
datacubes with the instrumental PSF.

###  2.4 Determination of kinematic parameters

To describe deviations from the Gaussian profile, line-of-sight velocity
distributions of galaxies are often represented by a product of its Gaussian
and non-Gaussian parts. The latter term is further expanded into Gauss-Hermit
series(van der Marel & Franx, 1993; Gerhard, 1993), resulting in the following
expression for the overall profile:

$\mathcal{L}(v)=I\tfrac{1}{\sqrt{2\pi}}e^{-w^{2}/2}\cdot\sum\limits_{j=0}^{N}h_{j}H_{j}(v),\
w=\frac{v-\overline{V}}{\sigma},\\\ $ (2)

where $I$ is a normalization constant (corresponding to the intensity),
$\overline{V}$ is a mean velocity, $\sigma$ is a velocity dispersion,
$H_{j}(v)$ are Hermite polynomials, and $h_{j}$ are coefficient (or factors)
of Gauss-Hermit series.

The most commonly used parameters in kinematic studies apart from the mean
velocity and its dispersion are $h_{3}$ and $h_{4}$, which tell about the
‘skewness’ and the ‘broadness’ of the distribution respectively. The values of
these parameters can be estimated in different ways. One approach would be to
use the orthogonality property of Hermit polynomials and compute the factors
directly via the corresponding scalar products (see Debattista et al., 2005
for an example of its application). However, in this work, we follow van der
Marel & Franx (1993); Brown et al. (2013) and Iannuzzi & Athanassoula (2015),
and simultaneously obtain the values for all free parameters ($I$,
$\overline{V}$, $\sigma$, $h_{3}$, and $h_{4}$). blackThis approach was chosen
to ease the comparison with previous studies and for perfomance reasons.
blackThus, we truncate the series up to the fourth order to avoid overfitting
to noise and effectively fit the profiles with the following function:

$\mathcal{L}(v)=I\tfrac{1}{\sqrt{2\pi}}e^{-w^{2}/2}\Bigr{(}1+h_{3}H_{3}(w)+h_{4}H_{4}(w)\Bigl{)},\
w=\frac{v-\overline{V}}{\sigma}.\\\ $ (3)

In the case of a nearly Gaussian distribution, $h_{3}$ and $h_{4}$ parameters
are around zero. However, if the system is dominated by radial orbits, the
distribution becomes narrower compared to the Gaussian and the fit would
produce positive values of $h_{4}$. If the system has predominantly circular
orbits, then the distribution broadens. In this case, one measures the
negative values of $h_{4}$ (Gerhard, 1993; van der Marel & Franx, 1993).
Negative values of $h_{4}$ may also appear when the LOSVD is a sum of two
Gaussians with clear double-peaked structure, which is typical for two
subsystems with different average rotation speeds (Gerhard, 1993).

Using the fitting procedure described above555We additionally checked the
fitting results with the MCMC parameter estimation and found no significant
deviations from the former method, which is much faster. (Eq. (3)), we obtain
an intensity, mean velocity and its dispersion, and $h_{3}$ and $h_{4}$ map
for each synthetic datacube. To characterise the density distribution, we
calculate the $d_{4}$ parameter in a similar way using Eq. (4)

$\mathcal{L}(z)=\mu_{0}\tfrac{1}{\sqrt{2\pi}}e^{-\psi^{2}/2}\Bigr{(}1+d_{3}H_{3}(\psi)+d_{4}H_{4}(\psi)\Bigl{)},\
\psi=\frac{\mu_{1}-z}{\mu_{2}}\\\ $ (4)

with $(\mu_{0},\mu_{1},\mu_{2},d_{3},d_{4})$ being the free parameters of the
fit.

##  3 LOSVD moment maps

For galaxies with a B/PS bulge, the density distribution in the vertical
direction in the area of the B/PS bulge differs from the usual exponential
distribution or the distribution characteristic of a self-gravitating
isothermal layer (see Eq. (1)). The density distribution is more vertically
extended than the surrounding disc, i.e., the thickest parts of the bars are
very loose, and the parameter $d_{4}$ in Eq. (4)) becomes negative. In this
section, we follow the approach of Debattista et al. (2005) and explore the
connection between vertical density features ($d_{4}$) and the thickness with
the kinematics ($h_{4}$) for different inclinations of the model and different
bar position angles.

###  3.1 Face-on maps

Figure 2: 2D maps of the $h_{4}$ parameter (upper row), the vertical density
distribution parameter $d_{4}$ (middle row) and the $\operatorname{med}|z|$
(bottom row) for all model galaxies at face-on position ($i=0^{\circ}$,
$\mathrm{PA}=0^{\circ}$). The isophotes for each model correspond to the
intensity maps shown in Fig. 1. The vertical lines define the position of the
$h_{4}$ and $d_{4}$ minima and the maximum of thickness (med|z|) along the
major axis of the bar (the exact values of the distance from the centre of the
model galaxy to the extremum are indicated at the bottom of the maps).

Fig. 2 (upper row) demonstrates 2D maps of the parameter $h_{4}$ for our
models (BL, BLx, Xb, and X). We obtained $h_{4}$ values for each pixel
constituting the face-on snapshots of models as described in Section 2.4. The
$h_{4}$ maps for prepared models have different morphology for the face-on
case due to the different morphology of the bar.

Regardless of the type of the B/PS bulge, for all modelled maps there are
negative values of $h_{4}$ within the bar radii (Fig. 2). The cuts along the
major axis also display the double-sided minima as discussed in Debattista et
al. (2005). Although, we note that the difference between these $h_{4}$ maps
depends on the exact morphology of the model. For the BL model, the minima of
the $h_{4}$ parameter lie outside the rounded central isophotes, and the
pixels with negative values of $h_{4}$ are grouped mainly on the major axis of
the bar. At the same time, for BLx, Xb, and X models, the negative values of
$h_{4}$ parameter draw an area that looks like a ‘bow tie’. In the particular
case of the model X, which has the largest B/PS bulge of all of the models,
the negative values of the parameter $h_{4}$ still lie close to the site
occupied by the face-on peanut-shaped bar, while the morphology of the $h_{4}$
maps reflects the X-shaped morphology of the its bar (compare Fig. 1 and Fig.
2).

Using cuts along the major axis of the bar (Fig. 3, solid lines), we show the
values of the $h_{4}$ parameter along the major axis for all models. In Fig. 2
(upper row) the position of $h_{4}$ minimum is shown by a vertical line and
its location is indicated in the dimensionless units of this work. In all
models, the minimum values of $h_{4}$ are similar and equal to $-0.05$.
However, the minimum of $h_{4}$ along the major axis of the bar is reached at
different distances from the centre. For the BL and BLx models, the $h_{4}$
minimum along the major axis coincides with the boundary of inner rounded
isophotes where the transition to a narrow extended bar occurs. Xb and X
models have their $h_{4}$ minima along the major axis closer to the centre of
the galaxy than the BL and BLx models, but they are wide (especially for the X
model). This is clearly seen on the profiles of the $h_{4}$ parameter along
the major axis of the bar (Fig. 3). If we compare the position of the negative
minima of the parameter $h_{4}$ on the major axis with the dimensions of the
B/PS bulge, as it is seen on the side-on snapshots (Fig. 1), then they do not
match. B/PS bulge is larger. At the same time, the maps in the Fig. 2 (upper
row) show that negative values of the $h_{4}$ parameter are present not only
along the major axis of the bar, but are also partially localized in the bar
itself away the minima on the major axis along four rays. The rays roughly
outline the dimensions of the B/PS bulge.

We also checked the relationship between the flat-topped distribution of the
LOSV (measured by $h_{4}$) and the features of the vertical density
distribution ($d_{4}$), as well as between $h_{4}$ and the thickness for all
our models. To measure the thickness, we calculate the median value of
absolute distance to the disc plane (hereinafter, $\operatorname{med}|z|$) for
all particles that create a signal in a given pixel. We also computed $d_{4}$
using Eq. (4). The middle and bottom rows in Fig. 2 show the $d_{4}$ and
$\operatorname{med}|z|$, respectively, for all our models. Regardless of a
specific model, all our models have a good correlation between $h_{4}$ and
$d_{4}$ parameters, as galaxies with B/PS bulges. We additionally checked the
Pearson correlation coefficient between $h_{4}$ and $d_{4}$ along the major
axis. For our models, it varies from 0.79 to 0.83.

The Xb model is a little out of the general range of models. It has the
loosest and most massive classical bulge. In the area of the bar and at the
face-on position, the bulge particles through which the line of sight passes
blur the features of the vertical density distribution of the B/PS bulge,
which, for this reason, hardly appear on the $d_{4}$ map (Fig. 2, middle row).
At the same time, B/PS bulge signatures are still visible on the kinematic
maps ($h_{4}$ maps, Fig. 2, upper row), and green pixels (negative values) are
located along the bar up to the boundaries of the B/PS bulge, as seen edge-on
(Fig. 1).

Thus, the flat-topped vertical density distribution can be traced from the
flat-topped LOSVD distribution for face-on galaxies. In the following
sections, we discuss the limitations of this approach.

The relationship between $h_{4}$ and the thickness ($\operatorname{med}|z|$)
of the B/PS bulge is shown in Fig. 2 (upper and bottom rows). We have obtained
a formal match between the minima of the $h_{4}$ parameter and the thickness
only for the BL model. For all other B/PS bulge models, the $h_{4}$ parameter
does not trace the thickness. Thus, the parameter $h_{4}$ is only an indicator
of features of the vertical density distribution, but not the thickness.

Figure 3: The $h_{4}$ (solid lines) and $d_{4}$ (dashed lines) profiles along
the major axis for the BL, BLx, Xb, and X models. The short vertical bars at
the bottom of the plots indicate the position of the $h_{4}$ minima and
correspond to the vertical lines in Fig. 2 (upper row).

###  3.2 Inclination effects for $h_{4}$ maps

In this subsection, we explore in detail how the $h_{4}$ maps change depending
on the galaxy inclination and the position angle (PA) of the bar major axis
relative to the line of nodes. We also discuss the absence of connection
between $h_{4}$ and $d_{4}$ when the model is inclined. We now focus only on
the BL model, since all other models behave in a similar way. The
corresponding figures for BLx, Xb, and X model can be found in Appendix A. In
Fig. 4 we present the changes of $h_{4}$ maps varying both inclination angles
($0^{\circ}$, $20^{\circ}$, $40^{\circ}$, $60^{\circ}$) and position angles
($0^{\circ}$, $45^{\circ}$, $90^{\circ}$) of the bar. This set of observer
positions broadly covers all relevant options. We additionally add inset plots
with $d_{4}$ maps to the upper right corner of each panel with $h_{4}$ maps.

The leftmost columns of Fig. 4 show $h_{4}$ (main panels) and $d_{4}$ (inset
panels) parameters for different inclinations with $\mathrm{PA}=0^{\circ}$.
The parameter $d_{4}$ stops being a reliable vertical density distribution
indicator with the inclination of a model galaxy even if the position angle is
close to zero. It happens because particles at different distances from the
center of the galaxy end up in the same line of sight. Respectively, the
connection between $h_{4}$ and $d_{4}$ is broken. The reason is the
contamination of $v_{z}$ by other velocity components (we discuss it in detail
in Sec. 4). This is especially well seen on the $h_{4}$ and $d_{4}$ profiles
along the major axis of a bar (Fig. 5), where the $h_{4}$ parameter along the
major axis behaves in a different way than the $d_{4}$. The inclination of the
galaxy moves the minima of $h_{4}$ away from the galaxy center. Also, for
inclined galaxies ($i\gtrsim 20^{\circ}$), the $h_{4}$ minima become less
prominent. We support this statement by the cuts along the major axis for the
BL model in Fig. 5.

The same tendency to move the minimum of $h_{4}$ to the periphery along the
major axis with increasing inclination is also observed when the inclined
galaxy has a non-zero position angle. The second and the third columns of Fig.
4 show the $h_{4}$ for $\mathrm{PA}=45^{\circ}$ and $\mathrm{PA}=90^{\circ}$
respectively. The prominent two-sided $h_{4}$ minima along the major axis
create a semi-closed ‘ring’ of negative values already at
$\mathrm{PA}=45^{\circ}$ and closed ‘ring’ at $\mathrm{PA}=90^{\circ}$ . Maps
with $i\approx$ 40∘-60∘ look very disturbed and the deepness of $h_{4}$ minima
on the line of nodes increases (Fig. 5). A change of the PA for an inclined
galaxy also increases the $h_{4}$ minima, and creates a prominent
‘ring’666Signatures of a ring from negative values of $h_{4}$ can be noticed
in figure 29 by Iannuzzi & Athanassoula (2015) (the case
$\mathrm{PA}=90^{\circ}$, $i=60^{\circ}$, models GTR101 and GTR116, which have
barlenses). of $h_{4}$ minima. Moreover, such a ring is typical not only for
the model with barlens (BL) but also for all the others (Fig. 16, 17, 18).

When $\mathrm{PA}=0^{\circ}$ and $i\geq 40^{\circ}$, the minima of $h_{4}$ lie
on the major axis of the bar, and the minima of $d_{4}$ form two arcs
symmetrically crossing the minor axis. At the same time, for
$\mathrm{PA}=90^{\circ}$ and $i\geq 40^{\circ}$, the deepest minima of the
parameter $h_{4}$ appear along the minor axis, which are not reflected in any
way in the minima of the parameter $d_{4}$, although the relationship between
the minima can be traced along the major axis. We return to the discussion of
these features in Sections 4 and 5.

To conclude this section, we note that the appearance of $h_{4}$ maps is
affected not only by the actual morphology of the bar but also by the position
of the barred galaxy with respect to the observer. We note that for all our
models with B/PS bulges, these effects manifest themselves in a similar way.
At the same time, for inclined galaxies ( $i\geq 20^{\circ}$), the features of
the LOSVD are no longer related to the features of the density distribution in
the vertical direction. In the next subsection, we compare all our models at
an intermediate inclination in an attempt to better characterise the features
of the maps. We also pay attention to the fact that when the galaxy is
inclined, the minima move away from the center of the model, and their
position can no longer reflect the size of the bar.

Figure 4: Effects of galaxy inclination ($i$) and PA in the $h_{4}$ maps for
the barlens model (BL). The isophotes correspond to the intensity maps for a
given inclination. Straight lines with arrows show the position of the $h_{4}$
minima along the major axis of the bar. The number at the bottom shows the
distance from the centre of the model galaxy to the $h_{4}$ minimum along the
major axis. For $i=60^{\circ}$ and $\mathrm{PA}=45^{\circ},\,90^{\circ}$ the
positions of the minima are not indicated since there are no global minima
along the major axis. In the upper right corner of each panel, the insets with
$d_{4}$ corresponding to the given position of the galaxy are added. The
colors are the same as in Fig. 2. blackThe minima of $h_{4}$ are broad.
Therefore, the precise location of exact global minima is not very well
defined and may slightly differ for different viewing angles even in case of
$i=0^{\circ}$. Figure 5: The $h_{4}$ (solid lines) and $d_{4}$ (dashed lines)
profiles along the major axis for the BL model at different values of $i$ and
PA.

###  3.3 The differences between models on $h_{4}$ maps

Many galaxies from the TIMER project (Gadotti et al., 2019) have inclinations
in the range $i=30^{\circ}-60^{\circ}$. So, in this section, we pay special
attention to the differences between our models with a significant
inclination. Fig. 6 shows the $h_{4}$ maps for all models at $i=40^{\circ}$
and different position angles
$\mathrm{PA}=0^{\circ}\,,45^{\circ}\,,90^{\circ}$. The first line corresponds
to $\mathrm{PA}=0^{\circ}$ and we do not notice a significant difference
between the models BL, BLx, and Xb. Although the map for the model without the
bulge (X) differs from the others and the position of its $h_{4}$ minimum
along the major axis of the bar lies as far as possible from the centre of the
galaxy (2.4 units), it does not make it possible to distinguish between the
vertical structure of bars of different morphology, i.e., to identify B/PS
bulges with certain parts of the bar.

The rotation of the kinematic axis by $45^{\circ}$ (second row in Fig. 6)
makes the BL, BLx and Xb models even less visually distinguishable. The main
difference between the $h_{4}$ maps for these models at such position angles
is the degree of compactness of the location of the features on the maps: the
minimum of $h_{4}$ is slightly further from the centre for the BLx model. The
differences between the BL and Xb models are completely insignificant,
although the Xb model does not contain a barlens. As for the X model, we note
that the $h_{4}$ maps corresponding to this model differ from the rest only in
that the minima are located almost one and a half times farther from the
centre of the galaxy (0.9 units versus 0.6), due to the fact that the bar of
this model is more extended, as well as its B/PS bulge (cf. Fig. 1).

In the case $\mathrm{PA}=90^{\circ}$, the map shows the ring of negative
values of $h_{4}$ for all models, regardless of the existence or absence of a
barlens. The main difference between the rings of negative values of $h_{4}$
for the BL and X models is their shape (for the X model the ring is more
squared) and the size of the ring (for the X model the ring is larger).
However, there is practically no difference between the BL, BLx, and Xb models
for this position of the galaxy.

Thus, the presence of the ring of negative values of $h_{4}$ is not related to
the exact bar morphology, but to the inclination effects, which can manifest
themselves in two ways. First, at intermediate inclinations, the LOSVD is
contaminated by other velocity components, in addition to $v_{z}$. Thus, the
negative values of $h_{4}$ are determined by the contribution of these other
components to the LOSVD and their features, and not by the looseness of the
density distribution. Second, stars with other kinematics, different from the
kinematics of the stars of the bar or any of its subsystems, may appear on the
line of sight. Even if the VD for individual subsystems is close to a
Gaussian, the sum of Gaussians with different mean velocities can lead to the
effect of negative values of $h_{4}$ (Gerhard, 1993).

Figure 6: Differences between $h_{4}$ maps for all models (BL, BLx, Xb, X) at
intermediate inclination $i=40^{\circ}$ and varying position angles. The
isophotes correspond to the intensity maps for a given position of the model
relative to the observer. The vertical lines in the top row indicate the
positions of $h_{4}$ minima along the major axis. The exact numbers at the
bottom right corner of each panel show the distance from the centre of the
model galaxy to the $h_{4}$ minima. The colors are the same as in Fig. 2.

##  4 Contamination of the LOSVD by different velocity components

In Section 3, we only discussed how the $h_{4}$ maps generally change for our
four models depending on the bar morphology, inclination, and position angle
of a bar major axis relative to the line of nodes. In this section, our goal
is to explain why the location of negative $h_{4}$ minima for our BL model
shifts on the maps depending on the inclination and position angle. To do
this, we consider only the BL model and separate the velocity components in
each cube since now. In contrast to the face-on position, for inclined galaxy,
the LOSVDs can contain contributions from both $v_{\varphi}$ and
$v_{\mathrm{R}}$ in addition to $v_{z}$. These velocity components distort the
LOSVD, although they have nothing to do with the flat-topped density
distribution in the vertical direction.

###  4.1 Contamination of LOSVD by azimuthal velocity component

Figure 7: Left: LOSVDs for the pixel (-1.7; 0) of the BL model at face-on
position (top) and for the same model but inclined at $i=60^{\circ}$ (bottom)
with respect to the bar major axis. The total LOSV (dark blue color) was
additionally decomposed into vertical ($v_{z}$ , green color) and azimuthal
($v_{\varphi}$, cyan) components. The selected pixel corresponds to the
$h_{4}$ minima along the major axis for the BL model at $i=60^{\circ}$.
Centre: the vertical density distribution for both inclinations. Right:
$h_{4}$ maps for $i=0^{\circ}$ (top) and $i=60^{\circ}$ (bottom). The
isophotes (black lines) correspond to the intensity maps for a given
inclination. The cross symbols show the position of selected pixels. Figure 8:
Left: a $(yz)$ projection of the region selected by $|x+1.7|<0.02$ (i.e., a
thin layer); LOS rays have angles $i=0^{\circ}$ (light gray) and
$i=60^{\circ}$ (orange) and pass through the zero point of this view. Right:
an $(xz)$ projection of the layer $|y|<0.02$; LOS rays have angles
$i=0^{\circ}$ (light gray) and $i=40^{\circ}$ (light green) and pass through
the point $(x,z)=(-1.57,0)$.

Our first example is a pixel on the bar major axis for the BL model, at the
distance of 1.7 units from the centre (Fig. 7, upper right plot, a cross). In
this pixel, particles with different $z$, but with the nearly same values of
$y=0$ (for $i=0^{\circ}$), fall on the line of sight (Fig. 8, left plot, light
grey line). In the face-on position, when only the vertical component of the
velocity $v_{z}$ contributes to the LOSVD, the value of $h_{4}$ in this pixel
is positive (Fig. 5, upper left cut, solid line and Fig. 7, upper right plot,
a cross in red region), and the distribution function $f(v_{z})$, in contrast
to the Gaussian, has a sharp peak (Fig. 7, upper left plot). This is in good
agreement with the density distribution in the vertical direction, which also
has a narrow peak (Fig. 7, upper middle plot).

When the stellar disc is inclined around the major axis of the bar at an angle
$i=60^{\circ}$, the distance from this pixel to the centre of the disc does
not change (Fig. 7, bottom right plot, a cross). At the same time, the line of
sight in this case captures particles not only with different $z$ but also
with different $y$ (Fig. 8, left plot, orange line). Now, not only the
vertical velocity component $v_{z}$ (green line on the bottom left panel of
Fig. 7), but also the azimuthal one (cyan line), contributes to the LOSVD in
this pixel, and the LOSVD (dark blue line) is characterised by a negative
value of $h_{4}$ (Fig. 7, bottom right plot, a cross in green area). Moreover,
in this pixel, there is a negative minimum of $h_{4}$, which was not in this
place for the model in the face-on position (Fig. 5, left cuts, solid lines).
The LOSVD in this pixel is flat-topped (Fig. 7, bottom left plot) but its
shape is determined mainly by $f(v_{\varphi})$ because $f(v_{z})$ in this
pixel is narrow peak around zero, which means that the projection of the
$v_{z}$ on the LOS is rather small . We know that it is the shape of the
distribution function $f(v_{z})$ that is related to the features of the
density distribution in the vertical direction, i.e., only a flat-topped
$f(v_{z})$, which gives a negative value of $h_{4}$, implies a flat-topped
density distribution. Thus, it is impossible to conclude that this pixel shows
the signatures of a peanut. This is confirmed by the density distribution
along the line of sight (Fig. 7, bottom middle plot), although formally the
value of $h_{4}$ in this pixel falls into negative minimum.

At smaller inclinations ($i=20^{\circ}-40^{\circ}$), pixels with negative
values of $h_{4}$ along the bar major axis are located approximately in the
same area as for the face-on view (Fig 4, left column), especially considering
that the negative minimum of $h_{4}$ for the face-on view is quite wide. This
is clearly seen in the cuts along the major axis for different inclinations
(Fig. 5, left column, solid lines). As mentioned above, the position of the
formal negative minimum of $h_{4}$ moves outward. At the same time, at
$i=20^{\circ}$, in the region of the negative minimum of $h_{4}$, there are
still pixels with negative values of $d_{4}$ along the major axis, and there
is a pronounced negative minimum of $d_{4}$, although it does not coincide
with the position of the $h_{4}$ minimum. At $i=40^{\circ}$, pixels with
negative values of $d_{4}$ disappear completely, and the negative minimum of
$h_{4}$ is strongly blurred in the B/PS bulge region. This means that for the
low inclinations ($i<40^{\circ}$) and $\mathrm{PA}=0$, negative values of
$h_{4}$ along the major axis carry the information about the presence of a
peanut despite the contamination of the LOSVD with the $v_{\varphi}$ velocity
component. However, this is no longer the case for larger inclinations of the
galaxy even on the major axis of the bar.

###  4.2 Contamination of LOSVD by radial velocity component

Figure 9: Left: LOSVDs for the pixel $(-1.2;0)$ of the BL model at face-on
position (top) and for the same model but inclined by $i=40^{\circ}$ and
rotated up to $\mathrm{PA}=90^{\circ}$ (bottom). The total LOSV (dark blue
color) was additionally decomposed into vertical ($v_{z}$ , green color) and
radial ($v_{R}$, red) components. The selected pixel corresponds to the
$h_{4}$ minima along the major axis for the BL model at $i=40^{\circ}$,
$\mathrm{PA}=90^{\circ}$. Centre: the vertical density distribution for both
cases. Right: $h_{4}$ maps for $i=0^{\circ}$ (top) and $i=40^{\circ}$,
$\mathrm{PA}=90^{\circ}$ (bottom). The isophotes (black lines) correspond to
the intensity maps for a given inclination. The cross symbols show the
position of selected pixels.

Our second example is the contamination of the LOSVD by the radial velocity
components. For the BL model, we choose the pixel at the bar major axis at the
distance of 1.57 units from the centre. Fig. 5 (upper right plot, solid line)
and Fig. 9 (upper right plot, a cross in a red area) demonstrate a small
positive value of $h_{4}$ in this pixel. The distribution function $f(v_{z})$
in this pixel has a narrow peak around zero (Fig. 9, upper left plot) and
corresponds to a narrow density distribution in the vertical direction (Fig.
9, upper middle plot) since the line of sight passes exclusively through the
peripheral rather thin parts of the bar (Fig. 8, right plot, light grey line).

The inclination of the stellar disc around the minor axis of the bar by
$i=40^{\circ}$ shifts the positions along the major axis of the bar and
reduces the distances due to projection effects. Thus, the selected pixel end
up at the distance of 1.2 units from the centre and falls into a deep negative
minimum of $h_{4}$ (Fig. 5, right column, solid line, $i=40^{\circ}$ and Fig.
9, bottom right plot, a cross in a dark green area). Now, two velocity
components, $v_{\mathrm{R}}$ and $v_{z}$ contribute to the LOSVD (Fig. 9,
bottom left plot), and the features of the LOSVD are determined by the
features of the radial velocity component distribution function
$f(v_{\mathrm{R}})$. At the same time, the distribution function $f(v_{z})$
remains narrow, although the line of sight in the upper half-plane of the B/PS
bulge passes partially through its loose parts (Fig. 8, right plot, light
green line). This is reflected in the skewed density distribution in the
vertical direction (Fig. 9, bottom middle plot). Such a distribution has a
negative value of the parameter $d_{4}$ (Fig. 5, right column, dashed line,
$i=40^{\circ}$). Additional particles from the loose region have almost no
effect on $f(v_{z})$, but they broaden the distribution $f(v_{\mathrm{R}})$
since particles located in a wide range of distances from the centre make an
additional contribution to it.

At $i=40^{\circ}$ and $\mathrm{PA}=90^{\circ}$, the B/PS bulge itself is
located closer to the centre from the negative minimum of $h_{4}$ along the
major axis. In this region, there are many negative pixels of $h_{4}$ and
$d_{4}$ for $i=0^{\circ}$ which implies that the kinematic diagnostics of
peanuts still works here, although the pixels themselves are not located at
the very minimum of $h_{4}$. Thus, at $\mathrm{PA}=90^{\circ}$ up to
inclination $i=40^{\circ}$, if we take a cut along the major axis of the bar
we can assume that the negative minimum of $h_{4}$ points to the upper
boundary of the B/PS bulge along the major axis (its tip).

For $i=60^{\circ}$ and $\mathrm{PA}=90^{\circ}$, there is no formal negative
minimum of $h_{4}$ along the major axis in the B/PS bulge region, and the
cuts, as well as $h_{4}$ maps, are difficult to interpret.

##  5 Orbital groups and $h_{4}$

In this section, we discuss in detail the features of $h_{4}$ maps for all our
models and contributions of various orbital components to the LOSVD (Section
2.2 contains information about the bar dissection into orbital groups). In
Smirnov et al. (2021) and Tikhonenko et al. (2021), each of our models was
disassembled into several orbital groups to analyse the face-on and edge-on
morphology of an overall bar. We have excluded each orbital group in turn from
the final model to determine how the $h_{4}$ maps change with the changes in
the orbital groups that constitute the bar. These results should be
interpreted with caution since such a simple removal of a particular orbital
group does not eliminate the gravitational potential of these points in the
rest of the model. This means that the influence of removed particles would be
still imprinted in the kinematics of remaining particles and will show up in
some way on the maps. 777As an example of such a pitfall, one may try to
naively remove the thickest layers of the B/PS bulge only to find that face-on
$h_{4}$ maps still look almost the same as they were for the original model.

For the face-on models, we analyse which orbital groups are responsible for
peanut signatures on the $h_{4}$ maps. We now focus only on the BL and X
models for a better example since the existence of the big bulges blurs the
picture for the BLx and Xb models, albeit the reasons for the appearance of
certain features on the $h_{4}$ maps are the same for all four models. The
corresponding maps for the BLx and Xb models can be found in Appendix B.

###  5.1 Exclusion of orbital groups

Fig. 10 shows how the $h_{4}$ maps change when the orbits of each orbital
group are excluded from the face-on snapshots for the BL and X models.

The BL model (upper row) consists of 55.49% particles of the disc, blackwhile
around 45% belong to the bar component. The contributions of orbits supporting
the bar are approximately equal between x1-like orbits (responsible for a
narrow and elongated bar), boxy orbits (creating a face-on peanut), and
$\text{bl}_{\text{o}}$ and $\text{bl}_{\text{u}}$ orbits (main components of a
barlens, Smirnov et al., 2021) — each group contributes about 10% of orbits
(Table. 2). Although the $\text{bl}_{\text{o}}$ and $\text{bl}_{\text{u}}$
orbits are the largest groups of orbits belonging to the bar and supporting
the central barlens in the BL model, the exclusion of these types of orbits
does not change the main features of the kinematic maps of $h_{4}$. We only
note changes on the minor axis, where the number of pixels with negative
values decreased after the elimination of the $\text{bl}_{\text{o}}$ orbits.
On the other side, when the x1 orbits are removed, almost all pixels with
negative values of $h_{4}$ on the major axis disappear (upper row, the 2nd
column of Fig. 10). Thus, the characteristic minima of $h_{4}$ should be
associated with the narrow and elongated bar and not with the barlens for the
BL model. This result is consistent with the conclusions by Tikhonenko et al.
(2021) obtained by analysing snapshots of various orbital subsystems, but here
we use the kinematic analysis.

In its turn, the model with a pure face-on peanut bar (without a classical
bulge and without a barlens, labelled X) mainly consists of disc (44.71%) and
boxy orbits (45.19%). Accordingly, the elimination of boxy orbits for the X
model removes almost all kinematic features in the centre of the model galaxy,
along with four dark green rays with a small opening angle. All other groups
of orbits in the bar are not represented in large numbers (only about 10% in
total), and their exclusion has no effect on the $h_{4}$ maps.

For the BL model, a signal from pixels with negative values of $d_{4}$ is
associated almost exclusively with x1 orbits. For the X model, on the
contrary, a characteristic pattern in the area of the bar with negative values
of the parameter $d_{4}$ is created by a subsystem assembled from boxy orbits.
Thus, for models with different bar morphology, the features of the LOSVD
expressed in term of $h_{4}$ parameter are formed by different orbits.

Figure 10: The maps of $h_{4}$ for the BL (_top_ row) and X (_bottom_ row)
models at face-on position with orbital groups excluded in turn. The numbers
in the bottom left corner of each plot correspond to the relative contribution
of the currently excluded orbital group (indicated in the title of each plot)
to the total number of particles (see Table. 2 for details). The colors are
the same as in the Fig. 2.

###  5.2 Exclusion of orbital groups from inclined models

In this section, we focus on a more detailed discussion of the formation of
the ring of $h_{4}$ minima for barlens galaxies. As was discussed in previous
sections, the ring of $h_{4}$ minima is not characteristic feature of galaxies
with barlenses, and this ring is formed for all our models due to the
contribution of azimuthal and radial velocity components to the LOSVD. In
order to demonstrate that the pixels with negative values of $h_{4}$ on a
face-on maps for inclined models have different nature, we now focus on the
$h_{4}$ maps when the model has inclination $i=40^{\circ}$ and the position
angle $\mathrm{PA}=90^{\circ}$ (the most prominent ring). We will exclude the
same orbital groups as in Section 5.1 from the model.

Fig. 11 (top row) shows the results for the BL model. The figure shows that
exclusion of x1 orbits does not change the $h_{4}$ map888The same area as for
$i=0^{\circ}$ and $\mathrm{PA}=0^{\circ}$ case in Section 5.1., although at
$i=0^{\circ}$ and $\mathrm{PA}=0^{\circ}$ the x1 orbits are responsible for
the $h_{4}$ minima. The exclusion of the $\text{bl}_{\text{o}}$ orbits changes
the map of $h_{4}$, in contrast to the face-on case, producing some effect
along the minor axis, but still not eliminating the ring.

The bottom row of the Fig. 11 shows the $h_{4}$ maps for the inclined X model.
After eliminating the main orbital groups for the X model (‘boxy’ orbits), we
also do not observe the elimination of the ring of $h_{4}$ minima999Although
we note that there are practically no particles in this region.. Moreover, the
exclusion of any type of orbits does not remove the ring.

Thus, we associate the ring of $h_{4}$ minima for $i>20^{\circ}$ and
$\mathrm{PA}>45$ not with the morphological features of the bar of one model
or another, but with the contamination of the LOSVD by the velocity components
$v_{\varphi}$ or $v_{\mathrm{R}}$. As was shown in Section 4, the main
contribution to the LOSVD contamination for cuts along the major axis of the
bar comes from the radial velocity component, if the galaxy is inclined around
the minor axis of the bar. As to the minor axis, the appearance of the
negative pixels comes from the the azimuthal velocity component. Thus, we
attribute the ring of $h_{4}$ minima for an inclined galaxy to a superposition
of these two effects. For pixels in an intermediate position between the major
and minor axes, the ring arises due to the contribution of all velocity
components to the LOSVD, but not due to the features of the vertical density
distribution.

Figure 11: The same maps of $h_{4}$ for the for BL (_top_ row) and X (_bottom_
row) models as in Fig. 10 but for $i=40^{\circ}$ and $\mathrm{PA}=90^{\circ}$.
Starting from the second column, a particular orbital group is excluded in
each panel. The colors are the same as in Fig. 2.

##  6 Discussion

The kinematic maps presented in this work are well suited for interpreting IFU
spectroscopy data. To date, one of the best spectral data on the kinematics of
galaxies is the TIMER data. The galaxies of this sample (21 objects) have
inclinations from $i=20^{\circ}$ to $i=60^{\circ}$ and a PA spread from
$0^{\circ}$ to $90^{\circ}$. Thus, to obtain an interpretation of these data,
one needs a grid of $N$-body models at intermediate inclinations. However,
Debattista et al. (2005) did not provide any maps with nonzero inclinations.
They only briefly discussed the inclination effects for $i<30^{\circ}$. Bureau
& Athanassoula (2005) demonstrated the kinematic cuts, in particular, of the
parameter $h_{4}$ (their Fig. 3), with different inclinations and PA (mainly
$\mathrm{PA}=90^{\circ}$) but only for $i\geq 75^{\circ}$. Iannuzzi &
Athanassoula (2015) also mainly discussed maps for models at high
inclinations. Thus, our choice of inclinations
($0^{\circ},20^{\circ},40^{\circ},60^{\circ}$) and position angles
($0^{\circ},45^{\circ},90^{\circ}$) fill in the gap that exists in previous
works. According to Laurikainen & Salo (2017), 11 from 17 clearly barred
galaxies are also barlens galaxies in Gadotti et al. (2020) TIMER sample. One
more galaxy (NGC 7755), which is not included in the sample by Laurikainen &
Salo (2017), has clear photometric signs of a barlens on the S4G image. To
interpret such kinematic data, it is very important to have numerical models
with a barlens. The models by Debattista et al. (2005) and Bureau &
Athanassoula (2005) do not include a pure barlens. Iannuzzi & Athanassoula
(2015) have barlens models (models GTR101 and GTR116), but there is no
detailed map grid for such galaxies at intermediate inclinations. We provide
these maps for our BL model with a barlens and demonstrate the kinematic
features associated with this subsystem. Moreover, our wide grid of
inclinations and position angles for all our models provide the possibilities
to interpret not only MUSE data but also apply it for future spectroscopic
surveys.

We have models that are close to the models analysed by Debattista et al.
(2005), and using their example we can confirm the absence of a relationship
between the extent of the B/PS bulge and the size of the area where the double
negative minima of $h_{4}$ are localised. Our model X resembles the R1 and R4
models by Debattista et al. (2005) in the Toomre parameter $Q$ and the initial
thickness, although Debattista et al. (2005) considered models with rigid
halos. The Xb model is analogous to the model B3
($M_{\mathrm{b}}/M_{\mathrm{d}}=0.2$, Debattista et al., 2005). As a result,
the $h_{4}$ maps for the models X, Xb are similar to the maps of the parameter
$s_{4}$ of Debattista et al. (2005) for their models R1, R4 and B3. The X
model has the extended B/PS bulge, and the negative minima of $h_{4}$ are
located close to the center and do not coincide with the thickest parts of the
B/PS bulge, as it is seen on the side-on snapshots (Fig. 1). To some extent,
this also applies to the Xb model. As was noted in the Section 3, the negative
values of the $h_{4}$ parameter are also partially localised in the bar itself
away the minima on the major axis along four rays and outline the dimensions
of the B/PS bulge. The models by Debattista et al. (2005), similar to the X
and Xb models, exhibit the same behavior (e.g., the rightmost panel of their
Fig. 13). This means that although the B/PS bulge is usually associated with
the thickest parts of the bar, the parameter $h_{4}$ does not always delineate
the boundaries of these thickest areas if we take cuts along the major axis.
In Section 6.1 we will discuss in more details the relationship between the
bar thickness, vertical density distribution, and kinematic signatures of a
peanut.

One more feature on the $h_{4}$ maps, which is typical for all our models is
the ring of negative values of $h_{4}$ when $\mathrm{PA}>45^{\circ}$.
Signatures of a ring from negative values of $h_{4}$ can be noticed in figure
29 by Iannuzzi & Athanassoula (2015) (the case $\mathrm{PA}=90^{\circ}$,
$i=60^{\circ}$, models GTR101 and GTR116, which apparently have barlenses).
This is the main plot which is used to interpret the TIMER data. As the
appearance of a ring on the $h_{4}$ maps for our models is solely the effect
of the galaxy inclination and a special value of the position angle of the bar
major axis, in Section 6.2 we discuss the pitfalls regarding the
interpretation of available observational kinematic maps. We also warn of
difficulties regarding the interpretation of available observational kinematic
maps.

###  6.1 How kinematics captures the density distribution

In Section 3.1, we have shown that for the models with B/PS bulges but with
different bar morphology in a face-on view, the characteristic vertical
density distribution of B/PS bulges (the values of $d_{4}$) is followed by the
broadness of LOSVD ($h_{4}$ parameter). This result is in agreement with
Debattista et al. (2005). Moreover, we have shown that negative values of
$h_{4}$ follow only flat-topped distribution, but not the thickest parts (Fig.
2). For a more detailed analysis, we now discuss the behaviour of $h_{4}$ and
$d_{4}$ values in the thickest areas of the bar (top 10% of the pixels sorted
by $\operatorname{med}|z|$). Fig. 12 shows that the thickest part of the
galaxy is not characterised by a flat-topped density distribution and,
respectively, by negative values of $h_{4}$. Also, Fig. 12 demonstrates that
in the thickest regions of the bar, the most negative values of $h_{4}$ and
$d_{4}$ are localised in the same places for all models. At least, this is
true for pixels lying in a strip along the major axis of the bar. At the same
time, there is no connection between the kinematics ($h_{4}$, the top row of
Fig. 12) and the thickness of the models (bottom row of Fig. 12). All
considered models show both positive and negative $h_{4}$ values in these
selected pixels. The $h_{4}$ minima do not fall into the regions of the
largest thickness for the BLx and Xb models, especially for the X model,
although for the BL model, the $h_{4}$ minima are located in the pixels that
characterise the largest thickness. Fig. 13 shows the relationships between
three parameters, $h_{4}$, $d_{4}$ and $\operatorname{med}|z|$, for the pixels
in the thickest areas of the bar (10% of the pixels with the largest thickness
values). In the thickest zones, a clear anticorrelation between $h_{4}$ and
$\operatorname{med}|z|$ exists only for the BL model.

For the thickest parts of the bar in models without a barlens, the minima of
$h_{4}$ and $d_{4}$ shift from the centre to the periphery (compare Fig. 2 and
Fig. 12). In the X model, the prevailing orbits in the bar are boxy orbits,
which, unlike x1 orbits, have many self-intersections near the centre. They
give deep two-sided minima of $h_{4}$ near the centre on the maps in Fig. 2,
although these areas are not thickest. For the thickest areas of the bar, the
positions of the double minima of $h_{4}$ and $d_{4}$ are very similar for all
models. We refer it to the fact that the minima for our X model are very wide
and the negative values of $h_{4}$ are still present within boundary of the
most vertically extended B/PS bulge part (see Fig. 1).

Thus, although the flat-topped shape of the LOSVD (negative values of $h_{4}$)
in the face-on case for models with different types of B/PS bulges does not
unambiguously track the thickest areas of the B/PS bulges, in the thickest
regions, the $h_{4}$ parameter certainly is an indicator of the features of
the vertical density distribution ($d_{4}$). It follows that interpretation of
the real data has to be done carefully since the minima of $h_{4}$ do not
track the position of the thickest zones and, accordingly, do not track the
size of the B/PS bulge.

Figure 12: The maps of $h_{4}$ (_top_ row), $d_{4}$ (_middle_ row) and
$\operatorname{med}|z|$ (_bottom_ row) for the BL, BLx, Xb, and X models (from
the left to the right) in the thickest area of the bar (top 10% of the pixels
sorted by $\operatorname{med}|z|$). The colors are the same as in Fig. 2.

Figure 13: The correlation matrix between $h_{4}$, $d_{4}$, and
$\operatorname{med}|z|$ for the thickest areas of the bar (top 10% of the
pixels sorted by $\operatorname{med}|z|$), according to Fig. 12

The $h_{4}$ parameter reliably tracks the vertical density distribution only
in the face-on case. As the inclination increases, other velocity components
($v_{\varphi}$ and $v_{R}$) contaminate the LOSVD, and $h_{4}$ no longer
follows $d_{4}$. Thus, for inclined galaxies, the analysis of the shape of the
LOSVD as an indicator of a peanut should be carried out with caution because
at $i\geq 40^{\circ}$ the connection between $h_{4}$ and $d_{4}$ is broken
(Fig. 5). There is another effect associated with the broadness of LOSVD.
Differently rotating components (for example, disc and bar) also lead to a
negative value of the parameter $h_{4}$, but this has nothing to do with the
features of the vertical density distribution. Such effects also may
contaminate the $h_{4}$ maps of inclined galaxies. We have shown that for all
our models with B/PS bulges, the effects of inclination and rotation of the
bar major axis affect the $h_{4}$ maps similarly. Accordingly, we conclude
that there are no significant differences in kinematics between different
types of B/PS bulges in our models.

###  6.2 Possible pitfalls in interpreting observational data

The $h_{4}$ parameter reliably tracks the vertical density distribution only
in the face-on case. However, even for galaxies with low inclinations and low
PA (as in the two upper plots on the left of Fig. 4 and Fig. 5), interpreting
the two-sided minima of $h_{4}$ along the major axis of a bar is
sophisticated. The reasonable question is, which component of the bar these
minima can be attributed to as the signatures of a B/PS bulge, i. e. to the
bar itself or to the barlens if each of them is present, as in galaxy NGC 98
(Méndez-Abreu et al., 2008; Méndez-Abreu et al., 2014). The analysis of the
orbital composition of the bar carried out in Section 5.1 showed that for the
BL model, only the component assembled from the x1 orbits shows the peanut
signatures. Inclusion in or exclusion from the model of orbits assembled into
a barlens ($\text{bl}_{\text{o}}$ and $\text{bl}_{\text{u}}$) has no effect on
the $h_{4}$ maps (Fig. 10). Signatures of a peanut, as a two-sided minima of
$h_{4}$ on the major axis of the bar, are always present, but they are not
related to the barlens.

The most demanded in kinematic applications may be our BL model, although all
our models show similar behaviour when the galaxy is inclined and rotated. We
refer to the fact that all our models are galaxies with B/PS bulges. The
question arises whether, it is possible to say that the edge-on B/PS bulge in
the BL model is identical to the face-on barlens based on the kinematics. For
galaxies with $i>20^{\circ}$ and $\mathrm{PA}>45^{\circ}$, our models predict
a ‘ring’ of negative values of $h_{4}$. The appearance of the ring in $h_{4}$
maps is consistent with the $h_{4}$ maps of the inclined galaxies NGC 4984 and
NGC 7755 ($\mathrm{PA}\approx 90^{\circ}$) from Gadotti et al. (2020). As an
illustration, we show the $h_{4}$ maps from Gadotti et al. (2020) and model
maps created from the BL model in Fig. 14. The model was positioned in the
same way relative to an observer as the galaxies in question.

However, the presence of the ring of negative values of the parameter $h_{4}$
is not an exclusive feature of the BL model. With this position of the bar,
the ring appears for all our models. The existence of such a ring is not a
signature of a peanut. It is associated with the complicated contribution of
other velocity components, $v_{\mathrm{R}}$ and $v_{\varphi}$, to the LOSVD,
in addition to $v_{z}$. As we have shown above, a signature of the B/PS bulge
through the $h_{4}$ parameter appears only along the major axis for low
inclinations of the galaxy.

We stress that this work is primarily dedicated to the detection of the
vertical density distribution features from the stellar kinematics. To make
our point clear, we focus on the values of the $h_{4}$ parameter alone for
now. However, the LOSVDs of the models contain a lot of information, for
example, we do not use the ‘skewness’ of the LOSVD (the $h_{3}$ parameter). In
Sec. 3.1 and 3.3, we do find a minor difference in the $h_{4}$ maps depending
on the model, but eventually conclude that these differences are insufficient
to clearly distinguish them. This situation changes when one starts
considering the $h_{3}$ parameter too. In Fig. 15, we provide preliminary
$h_{3}$ maps for all our models, where one can see that the BL model has a
distinctive feature in its central part black(inside 0.5 units radius). This
feature is associated with the rotation of the inner parts of the barlens but
does not characterize its vertical structure. We tentatively connect this
feature with $\text{bl}_{\text{u}}$ component, which is also responsible for
the slightly elevated $h_{4}$ values in the same area (cf. the rightmost and
the leftmost panels of the Fig. 10), ultimately leaving the problem of the
detection of kinematic characteristics of barlens galaxies using both $h_{3}$
and $h_{4}$ parameters to a future paper focused on a more detailed comparison
of BL model kinematics with available observational data.

Figure 14:  The BL model rotated to mimic two galaxies from the TIMER sample:
NGC 4984 (top plot, $i=54^{\circ}$, $\text{PA}=-103^{\circ}$) and NGC 7755
(bottom plot, $i=53^{\circ}$, $\text{PA}=94^{\circ}$). For each plot, the left
panel shows the intensity map of the model. On right panels we reproduce the
maps of $h_{4}$ parameter from Gadotti et al. (2020) with synthetic $h_{4}$
maps of our BL model shown on the inset plots. blackThe $h_{4}$ colourbar is
the same as in Fig. 2.  Figure 15: The preliminary maps of $h_{3}$ parameter
for all considered models at $\text{PA}=0^{\circ}$, $i=40^{\circ}$

##  7 Conclusions

We have explored four $N$-body models that have a gradual progression in a
face-on bar morphology and imitate real galaxies from galaxies with a barlens
to featureless barred galaxies. Like our predecessors (Debattista et al.,
2005; Iannuzzi & Athanassoula, 2015), we focused on the analysis of 2D maps of
the $h_{4}$ parameter to probe the presence of a “peanut” in cases where this
morphological feature is not visually evident. These maps, produced on the
basis of model data cubes with a spectral resolution close to that available,
for example, for the data from the TIMER project (Gadotti et al., 2019), can
be compared with real observational data. We have studied in detail the 2D
kinematic maps for various orientations of the bar to the line of sight
($i=0^{\circ}-60^{\circ}$, $\mathrm{PA}=0^{\circ}-90^{\circ}$). Our uniform
grid of disc inclinations and position angles of the bar major axis covers
almost all possible variations in the position of the galaxy, except for the
edge-on view.

Based on the spectral analysis of the main frequencies of the orbits in the
bar, we have previously identified separate subsystems, or orbital groups,
assembled into a bar (Smirnov et al., 2021). The groups differ in structure,
primarily vertical, which means that they can also differ kinematically. We
have analysed the contributions of the LOSVDs of individual orbital components
to the cumulative LOSVD in order to correctly interpret the LOSVD moment maps
and judge which orbital groups make the main contribution to the negative
values of the parameter $h_{4}$.

The main conclusions of the work are as follows.

  * 1.

The characteristic vertical density distribution attributed to the B/PS bulges
subsists in models with and without barlens and can be detected using the
parameter $h_{4}$.

  * 2.

The data analysed to reproduce the well-established face-on kinematic
signatures of the presence of a “peanut” (B/PS bulge): at low inclinations
($i<20^{\circ}$) along the major axis of the bar, two symmetrical negative
minima of the $h_{4}$ parameter are observed. Unlike other negative pixels on
the 2D maps of the $h_{4}$ parameter, the negative pixels of this parameter
along the major axis of the bar really delineate the boundaries of the B/PS
bulge.

  * 3.

There are no significant morphological differences in the maps of the $h_{4}$
parameter for the model with a barlens (BL model) and without it (X model) at
all inclinations and $\mathrm{PA}>45^{\circ}$, when a ring of negative $h_{4}$
values begins to form. An interpretation of observational data at this viewing
position of the galaxy only on the basis of these maps does not allow one to
probe the vertical structure of a barlens if it is present and co-exists with
a B/PS bulge within the same galaxy.

  * 4.

With an increased inclination, the diagnostics based on the $h_{4}$ parameter
is ruined due to a break in the connection of the $h_{4}$ parameter with the
features of the vertical density distribution.

  * 5.

We clearly distinguish between the influence of two factors on the morphology
of the LOSVD moment maps. The first is the effect of the inclination
(contamination of the LOSVD with velocity projections other than $v_{z}$). The
second is the contamination of the cumulative LOSVD by the LOSVDs of different
orbital subsystems or with different kinematics (fast and slow rotating
components).

  * 6.

The wide ring of negative values of $h_{4}$ is observed on 2D model maps at
intermediate inclinations ($i=20^{\circ}-60^{\circ}$) and
$\mathrm{PA}>45^{\circ}$. Such rings can be noticed on the $h_{4}$ maps for
the barlens galaxies (e.g., NGC 3351, NGC 4984, NGC 7755, Gadotti et al.,
2020). However, rings are present in all our models, and their appearance can
not be a distinguishing feature of the barlens model.

  * 7.

We analysed the impact of different orbital groups in the bar on the kinematic
signatures of a peanut for all our models. For our models, different orbital
groups make different contributions to the minima of $h_{4}$ on both sides
along the major axis of the bar. For the barlens model, the minima of $h_{4}$
are associated with the x1 family of orbits, the main building material of a
long narrow bar. If this orbital family is well represented in the bar, then
it is this family that will give two-sided negative minima of $h_{4}$ along
the major axis of the bar and signals about the features of the vertical
density distribution (B/PS bulge). Thus, the orbits supporting the barlens are
not responsible for the kinematic signatures of the peanut.

## Acknowledgements

The authors express gratitude for the financial support from the Russian
Science Foundation (grant no. 22-22-00376). blackWe are grateful to the
anonymous reviewer for a careful and thorough reading of the manuscript. We
are also grateful to the TIMER team (Gadotti et al., 2019) for making their
data publicly available. This work made use of
Astropy:101010http://www.astropy.org a community-developed core Python package
and an ecosystem of tools and resources for astronomy (Astropy Collaboration
et al., 2013, 2018, 2022), as well as NEMO stellar dynamics toolbox (Teuben,
1995). The authors also thank Roberto Saglia for reading a preliminary version
of this manuscript and a fruitful discussion regarding the optimal way to
present the results of this work.

##  8 Data availability

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

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##  Appendix A Effects of inclination: $h_{4}$ maps

In Section 3.2, we provide the $h_{4}$ maps for different inclinations and
position angles for the BL model. Figs. 16, 17, 18 show the maps similar to
those presented in Fig. 4, for the BLx, Xb, and X models. Despite the
differences between $h_{4}$ maps for a face-on view, which depend on the exact
bar morphology, the inclination effects are similar and lead to the same
kinematic features for all presented B/PS bulges. As was shown in the Sections
3.2, 3.3, 5, 5.2 all B/PS bulges form the rings of $h_{4}$ minima for
$i>20^{\circ}$ and $\mathrm{PA}>45^{\circ}$. We also note that the size of the
ring depends on the exact size of the B/PS bulge.

Figure 16: Effects of the disc inclination ($i$) and PA in the $h_{4}$ maps
for the BLx model. See Fig. 4 for all details. Figure 17: Effects of the disc
inclination ($i$) and PA in the $h_{4}$ maps for the Xb model. See Fig. 4 for
all details.  Figure 18: Effects of the disc inclination ($i$) and PA in the
$h_{4}$ maps for the X model. See Fig. 4 for all details.

##  Appendix B Dissection of the bar into orbital groups for BLx and Xb model

Section 5 contains the $h_{4}$ maps for the BL and X models. Here we provide
the same maps for the BLx and Xb models. Both of them mainly consist of a
quarter of boxy orbits, which contribute to the $h_{4}$ maps in the face-on
case (Fig. 19). However this contribution is hardly to notice. We refer it to
the smearing of the peanut signatures by the classical bulge in the centre
(for the BL model the classical bulge is compact, the X model has no a
classical bulge). Fig. 20 shows that the exclusion of any of orbital group of
the bar does not change the $h_{4}$ maps for the inclined galaxies.

Figure 19: The maps of $h_{4}$ for the BLx (_top_ row) and Xb (_bottom_ row)
models at face-on position with orbital groups excluded in turn. See Fig. 10
for all details. Figure 20: The maps of $h_{4}$ for the BLx (_top_ row) and Xb
(_bottom_ row) models at $i=40^{\circ}$, $\mathrm{PA}=90^{\circ}$ with orbital
groups excluded in turn. See Fig. 11 for all details.