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First direct identification of the barlens vertical structure in galaxy models

Applying spectral dynamics methods to one typical N-body model with a barlens, we dissect the modelled bar into separate components supported by completely different types of orbits. We identify at least four components: a narrow elongated bar, a boxy bar, and two components contributing to the barlens. We analyse the vertical structure of all components that make up the thick part of the bar (B/P bulge). We show that the `peanut' shape is mainly due to the orbits that assemble the boxy part of the face-on bar. We associate the X-shape with the narrow and elongated bar. The wider part of the barlens with square-like isophotes contributes to the boxy shape of the B/P bulge when we observe the galaxy edge-on. However, 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. Thus, for the first time, we demonstrate that the rounded face-on barlens can not be entirely associated with the B/P bulge.


Introduction
About half of the edge-on galaxies possess boxy/peanut-shaped (B/P) bulges (Lütticke et al. 2000, but see Yoshino & Yamauchi 2015). In some cases these bulges can exhibit an X-shape, therefore, the abbreviation B/P/X is sometimes used for such structures. The fractions of B/P bulges and bars in galaxies of Hubble types earlier than Scd are almost the same (e.g., Erwin & Debattista 2017;Li et al. 2017. It is widely believed that B/P bulges are the inner parts of bars that have thickened in the vertical direction (e.g., Salo 2016 andAthanassoula 2016, andreferences therein). Such a connection follows, first of all, from the results of N-body simulations which show that stellar discs are often subject to the bar-like instability. Subsequently, the evolution of the bar is accompanied by the formation of a vertically thin outer structure and a vertically thick inner structure (Combes & Sanders 1981;Raha et al. 1991). Viewed sideon (along the minor axis of a bar), the thick inner part of a bar appears as a peanut-shaped stellar structure. At intermediate position angles of a bar this structure shows a boxy-like shape.
Although B/P bulges are usually detected in the edge-on view, in some works it is accepted that they are counterparts of the so-called barlenses (Laurikainen et al. 2011), which are lens-like structures embedded in narrow and elongated bars and seen rather face-on (see the review by Laurikainen & Salo 2016). An understanding of the common nature of B/P bulges and barlens components came from an analysis of N-body simulations. Athanassoula et al. (2015) viewed a set of N-body snapshots with a barlens from different viewing angles and concluded that a barlens is the vertically thick part of the bar but viewed faceon, that is, a barlens and a B/P bulge are the same part of the bar, but simply viewed from a different viewing angle.
The main proof that the barlens is another way of thinking about the projected B/P bulge is the coincidence of their relative sizes. From photometric analysis of nearly face-on galaxies Erwin & Debattista (2013 found that B/P features extend from 1/3 to 2/3 of the bar radius while Laurikainen et al. (2011);Athanassoula et al. (2015); Herrera-Endoqui et al. (2015) obtained similar results using barlenses in face-on galaxies. Using stellar-kinematic diagnostic Méndez-Abreu et al. (2008) measured the location of vertical protrusions in the barlens galaxy NGC 98 at about 1/3 of the bar's radius. All these evidences led to the view that a barlens and a B/P bulge can be considered the same structure but viewed from different angles.
Erwin & Debattista (2017) expressed a slight doubt about such a coherent picture. First, it is not clear if barlens-like morphologies are always due to B/P structure. Besides that, the definition of a "lens" is somewhat ambiguous. It is unclear whether lenses and barlenses can be considered as similar structures. Some authors define the lens as a shallow feature on the brightness profile with further steeper declination (Kormendy 1979), while others identify the barlens by the steep quasi-exponential profile (Laurikainen et al. 2014). Additional argumentation can be found in Erwin & Debattista (2017).
Another major problem is that the thick part of the bar is a complex structure. Based on the spectral analysis of individual orbits of stars from different N-body models, Smirnov et al. (2021) dissected the modelled bars into separate components that are supported by completely different orbital groups. They identified at least four orbital components: the narrow elongated bar, the face-on boxy bar, and two components constituting the barlens. All these components may have their unique vertical structure and a prominent and large face-on component does not have to be thick in the edge-on view. Moreover, Smirnov et al. (2021) obtained that these components differ rather little in their Article number, page 1 of 5 arXiv:2103.02988v1 [astro-ph.GA] 4 Mar 2021 linear sizes but have quite different morphology. Thus, the coincidence of relative sizes also does not serve as an argument in favour of the structure coincidence. Therefore, observing bar at different viewing angles does not help to differentiate all its components. Dynamical arguments are required to reliably separate the components from each other. Smirnov et al. (2021) focused on the face-on morphology of bars. In this Letter we consider an N-body model with a barlens (Sect. 2) and analyse the edge-on structure of all the components that make up the bar (Sect. 3). In Sect. 4 we come to conclusions about the vertical structure of all identified components of the bar.

N-body model and orbital frequencies
We analyse an N-body model of a barred galaxy evolved from an initially axisymmetric and equilibrium state with no gas involved. The model initially consists of an exponential disc isothermal in the vertical direction embedded in a live dark matter halo. The halo is modelled by a truncated sphere with the density profile close to the NFW profile (Navarro et al. 1996). During its evolution, the disc naturally forms a bar. Since the initial model contains a compact classical bulge, the emerging bar has a face-on feature, namely, the barlens that formed in its central area (Salo & Laurikainen 2017). The precise details of our model can be found in Smirnov et al. (2021) where the model described was labelled as a BL model.
The details of numerical modelling can be summarised as follows. We consider dynamically cool disc with Toomre parameter value Q = 1.2 at R = 2R d , where R d is the disc radial scale length. We scale the model to obtain a Milky Way-like galaxy and assume R d = 3.5 kpc and M d = 5 · 10 10 M , where M d is the disc mass. Then the time unit will be t u ≈ 14 Myr. The parameters of the dark halo were chosen to produce a reasonable dark halo profile with M h (r < 4R d )/M d ≈ 1.5. The model possesses a classical bulge with a Hernquist (1990) density profile. The total mass of the bulge is M b = 0.1M d , with the scale parameter r b = 0.05R d .
We use 4kk particles for the disc, 4.5kk for the halo and 0.4kk for the bulge. The evolution of the model was followed up to 8 Gyr.
To study the orbital composition of the bar we applied the methods of spectral dynamics (Binney & Spergel 1982) and obtained orbital frequencies in the evolving bar/disc. As the pattern speed varies, we have estimated the frequency shifts and found that for the time interval t = 400 − 500 (when the bar pattern speed is more or less established) the shift is small for most of the particles and its value is about the frequency measurement error (Parul et al. 2020). We have chosen this time interval to apply the frequency analysis to all disc particles and obtained f x , f y , f z and f R which are frequencies of oscillations along x, y, z coordinates and cylindrical radius R. In Fig. 1 xy, xz and yz snapshots for the whole BL model are presented at the time moment t = 450. A barlens is clearly seen in the xy snapshot while thick B/P structure is visible in the xz view. In Smirnov et al. (2021), the BL model was disassembled into several orbital groups to analyse the face-on morphology of an overall bar. We assume that each individual group is characterised by a unique set of frequency ratios f R / f x and f R / f y . In Smirnov et al. (2021) we have focused on the face-on morphology of the inner parts of bars in our models. For this purpose, at the initial stage of our orbital analysis we have discarded orbits from the rest of the disc (outside the ring at the co-rotation radius (CR) R c ≈ 3) as well as of the thin part of the bar itself. Orbits from the thin part of a bar are located inside the CR and can be distinguished by the frequency ratio f z / f x > 2.25 (Smirnov et al. 2021). Although the number of these orbits is about 14% of 4kk particles of the entire disc, all these orbits do not contribute to the thick B/P bulge but only to the most remote and vertically thin parts of the bar (see Fig. 2). For the thick part of the bar we have identified all separate orbital groups (see Fig. 3) based on frequency ratios f x / f R and f y / f R and have shown that structures which they contribute to are completely different in morphology. In this Letter, we analyse the edge-on morphology of the thick bar and its components in our BL model. Smirnov et al. (2021) identified the following components constituting a complex bar with a barlens: a 'classic' elongated bar ( f R / f x = 2.0 ± 0.1, 21.8% of particles from 4kk) and two components contributing to the barlens (bl u orbits, in our notation, with f R / f x ≤ 1.9, 9.2%, and bl o orbits, with f R / f x ≥ 2.1, 13.5%). In its turn, a 'classic' bar can be divided into two subsystems. One is assembled from the x1-like orbits ( f y / f x = 1.0±0.1, 11.4%), and another one is the so-called boxy bar with orbits that are characterised by the ratio f y / f x ≥ 1.1 (10.4%). In Fig. 3 all these orbital groups occupy separate regions (a spot and straight rays) and can be easy distinguished.

Edge-on structure
The x1-like bar has a complex structure itself. Three main orbital blocks, which contribute to morphologically different structures, can be distinguished. There is a little number of orbits parented from the so-called x2 orbital family (1.7%). At the distribution over the ratio f z / f x (Fig. 4) these orbits have a wide peak at f z / f x = 1.4. Two other subsystems have a hump at f z / f x = 1.75 (orbits elongated along the bar) and a very narrow peak at f z / f x = 2.0. The last group consists of 'banana'-like orbits. and bl u components) in our BL model. We also provide unsharpmasked plots 1 of xz snapshots (third column) to highlight the vertical structural features of the bar and its components. It is seen that the peanut-like structure and the X-structure in the xz view ( Fig. 5; second column) are created by a boxy and x1-like parts of the "classic" bar. The peanut-like structure is more inherent in the boxy part of the bar, while the x1-like bar demonstrates four bright rays of the X-structure 2 , which outline the z-maxima of various quasi-periodic orbits (Parul et al. 2020).
Three orbital blocks of the x1-like bar demonstrate different structural features seen at xy and xz projections. First of all the xy projection clearly exhibits the existence of an inner perpendicular bar (the orbits around x2 family; they have a wide peak at f z / f x = 1.4). In the unsharp-masked xz plot for the 'x1like' orbital group this structure reveals itself as a flat subsystem. 'Banana'-like orbits ( f z / f x = 2.0 ± 0.05) delineate the most remote parts of a bar (e.g., Valluri et al. 2016;Patsis & Athanassoula 2019). They appear as two distant shell-like features in the xz unsharp-masked plot for the x1-like component. As to the orbits elongated along the bar and having a hump at f z / f x = 1.75, they are the main contributors to the X-structure and bright rectilinear rays protruding from the main plane of the disc (Parul et al. 2020).
The orbits from bl o group (a part of a face-on lens) are assembled into a smooth square-like structure in all three projections with slight traces of a 'peanut' in the central region (see an unsharp-masked plot in Fig. 5). The whole structure resembles a 3D cube with rounded corners and dents in the middle of the cube faces. If there are many such orbits, then, at intermediate P.A. of the bar, the structure assembled from such orbits can be responsible for the box-like isophotes of the B/P bulge because this structure retains its cube-like shape in all projections while other orbital subsystems take rounded shape. It is important to note that this subsystem is observed even in models without a classical bulge, but in such models the orbits inhabiting it are few (see, for example, Gajda et al. 2016).
The most striking thing is that the bl u orbits (the main part of a lens as was argured in Smirnov & Sotnikova 2018) constitute the rounded structure not only in the face-on view (Fig. 5) but also in the xz projection as well as in the yz projection. The whole structure resembles a ball-shaped fruit with a flat bone inside. This is especially evident in the unsharp-masked plot. The flat part and the halo around it stand out. Smirnov et al. (2021) have shown that the bl u orbits are the main contributors to the rounded face-on barlens in models with compact classical bulges. The orbits identified have rosette-like shape in the frame co-rotating with the bar. Such a shape is typical for orbits trapped around stable loop orbits from the nearly round families x2 or x4. The x2 orbits lie closer to the centre and rotate in the same direction as the bar in the inertial reference frame, and the x4 orbits rotate retrogradely, have more rounded shape and are located further from the centre. Examples of bl u orbits can be found in Smirnov et al. (2021) (their figure 14). They are not planar but have a rather small vertical extension and create a flat structure. If such orbits dominate in the central regions, they should manifest themselves as nuclear discs with kinematics which are characterised by near-circular orbits (Gadotti et al. 2020). In a subsequent study, we intend to go deeper into issues related to the kinematics of individual subsystems of B/P bulges. A smooth spherical halo from such orbits is also gradually formed but no signs of the off-centre vertical protrusions are observed.

Conclusions
We summarise our main results as follows.
Article number, page 3 of 5  4. Distribution of all orbits in the x1-like bar over the f z / f x ratio for the considered model at t = 450 (6 Gyr). The same distibution is indicated with the different color for the inner bar supporting family, which makes up most of the peak at 1.4 in the primary distribution.
1. For the first time, we have revealed the vertical structure of the bar component that appear as a barlens in the face on view. In our N-body model the rounded barlens is rather flat structure without signs of the vertical peanut shape. This part of the barlens has nothing to do with the B/P bulge. The coincidence of the barlens and B/P bulge sizes, which is often used to show the identity of these two structures, is rather accidental. As demonstrated by Smirnov et al. (2021), boxy orbits, which are assembled in a 3D 'peanut', are shortened with the transition to models with a compact classical bulge. Both the barlens and the B/P bulge are shorter than the elongated bar, but these components are not different projections of the same structure.
2. Apparently, the boxy shape of the B/P bulge is determined by bl o orbits which are assembled in a 3D cube-like structure with slight traces of a 'peanut'.
3. The 'peanut' component and the X-structure are mainly due to x1-like and boxy orbits. The 'peanut' component (boxy bar) is the main contributor to the B/P structure and this component is present even in the models without classical bulges and barlenses. Thus, the B/P bulge is a barlens-independent component of the bar vertical structure, and its presence in the edge-on view is not an indicator of the obligatory presence of a face-on barlens.