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We have translated the results of $N$-body simulations of one barred model into the language of action variables and frequencies. Using this language, we analysed the behaviour of all orbits in the model on a large time scale at the stage of a mature bar. We show that the orbits join the bar while preserving their adiabatic invariant, which takes into account the 3D structure of the orbits. This allows us to apply the concept of the Lynden-Bell derivative for each of these orbits and trace how the sign of the derivative changes, i.e. how asynchronous changes in angular momentum $L_z$ and orbital precession rate $\Omega_\mathrm{pr}$ (normal orbital mode) change to synchronous (abnormal mode). The transition to the abnormal mode occurs when $\Omega_\mathrm{pr}$ reaches the angular velocity of the pattern $\Omega_\mathrm{p}$, after which the orbit becomes stuck in the bar trap. All this happens against the background of secular changes in actions ($L_z$ decreases, $J_\mathrm{R}$ and $J_z$ increase). At the same time, corotation particles near two stable Lagrange points are also subject to secular changes in their actions. They increase $L_z$ and drift to the periphery, shifting corotation outwards. We also show that a change in the orbital mode from normal to abnormal and the trapping of orbits in a bar is possible only when the bar speed decreases with time, regardless of what is causing the bar to slow down. Our findings clarify and expand the picture of bar formation and evolution in numerical models., Comment: Accepted in MNRAS

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., Comment: 18 pages

We draw attention to the bright galaxies that do not show a bar in their structure but have a flocculent spiral structure. Using the THINGS' and HERACLES' kinematic data for four barless galaxies (NGC~2841, NGC~3512, NGC~5055, NGC~7331) we built their mass models including dark halos. We concluded that the fraction of the dark matter does not exceed 50\% within the optical radii of the galaxies. This is too little to explain the lack of a bar in these galaxies. In an attempt to understand the featureless structure of these galaxies we constructed several $N$-body models with an initially reduced content of dark matter. We concluded that, in addition to the low mass of the dark halo, the decisive factor that leads to a barless disc is the start from an initially unstable state. An isolated dynamically cold disc (with the Toomre parameter $Q < 0.5$) settled into rotational equilibrium passes trough the short stage of violent instability with fragmentation and formation of stellar clumps. After that, it evolves passively and ends up with a featureless structure. We assume that the barless flocculent galaxies studied in the present work may be descendants of galaxies at high redshifts with rotation curves which are consistent with the high mass fraction of baryons relative to the total dark matter halo., Comment: 18 pages

Internal dynamics and kinematics of galaxies have imprints on the line-of-sight velocity distribution~(LOSVD). Gauss-Hermite parametrisation allows one to identify the kinematics features of the system in terms of skewness~($h_3$) and broadness~($h_4$) deviations of a LOSVD. Such a method provides information about the type of orbits since a $h_3-\overline V$ correlation is a sign of elongated orbits, and the anti-correlation is a sign of circular or near-circular orbits. In previous works, analysis of the $h_3-\overline V$ relation provided a tool to identify a hidden bar or B/PS bulge~(edge-on, $\mathrm{PA}=90^\circ$) and to probe their strength. We prepared two $N$-body galaxy models with clear B/PS bulges: one has an ordinary bar~(the X model), and the second one has a barlens embedded into a bar~(the BL model) to investigate the mechanism of formation of $h_3$ features at any position of an observer. We show that the $h_3-\overline V$ correlation appears in the regions where bar and disc particles are mixing. We also reveal that the model with a barlens has an $h_3-\overline V$ anti-correlation in the centre, and we show that barlens-specific orbits are responsible for this signal. Moreover, this feature can be observed only for galaxies with compact bulges and barlenses. The results of this work are applicable for the interpretation of future Integral-field unit (IFU) data for real galaxies with B/PS bulges, especially for objects with barlenses., Comment: 16 pages, 16 figures, accepted to MNRAS

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, which has a boxy/peanut shape (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 cannot be entirely associated with the B/P bulge., Comment: 5 pages, 5 figures, accepted to Astronomy and Astrophysics on March 26, 2021

Domestication of horses fundamentally transformed long-range mobility and warfare1. However, modern domesticated breeds do not descend from the earliest domestic horse lineage associated with archaeological evidence of bridling, milking and corralling2–4 at Botai, Central Asia around 3500 bc3. Other longstanding candidate regions for horse domestication, such as Iberia5 and Anatolia6, have also recently been challenged. Thus, the genetic, geographic and temporal origins of modern domestic horses have remained unknown. Here we pinpoint the Western Eurasian steppes, especially the lower Volga-Don region, as the homeland of modern domestic horses. Furthermore, we map the population changes accompanying domestication from 273 ancient horse genomes. This reveals that modern domestic horses ultimately replaced almost all other local populations as they expanded rapidly across Eurasia from about 2000 bc, synchronously with equestrian material culture, including Sintashta spoke-wheeled chariots. We find that equestrianism involved strong selection for critical locomotor and behavioural adaptations at the GSDMC and ZFPM1 genes. Our results reject the commonly held association7 between horseback riding and the massive expansion of Yamnaya steppe pastoralists into Europe around 3000 bc8,9 driving the spread of Indo-European languages10. This contrasts with the scenario in Asia where Indo-Iranian languages, chariots and horses spread together, following the early second millennium bc Sintashta culture11,12., Domestication of horses fundamentally transformed long-range mobility and warfare1. However, modern domesticated breeds do not descend from the earliest domestic horse lineage associated with archaeological evidence of bridling, milking and corralling2–4 at Botai, Central Asia around 3500 bc3. Other longstanding candidate regions for horse domestication, such as Iberia5 and Anatolia6, have also recently been challenged. Thus, the genetic, geographic and temporal origins of modern domestic horses have remained unknown. Here we pinpoint the Western Eurasian steppes, especially the lower Volga-Don region, as the homeland of modern domestic horses. Furthermore, we map the population changes accompanying domestication from 273 ancient horse genomes. This reveals that modern domestic horses ultimately replaced almost all other local populations as they expanded rapidly across Eurasia from about 2000 bc, synchronously with equestrian material culture, including Sintashta spoke-wheeled chariots. We find that equestrianism involved strong selection for critical locomotor and behavioural adaptations at the GSDMC and ZFPM1 genes. Our results reject the commonly held association7 between horseback riding and the massive expansion of Yamnaya steppe pastoralists into Europe around 3000 bc8,9 driving the spread of Indo-European languages10. This contrasts with the scenario in Asia where Indo-Iranian languages, chariots and horses spread together, following the early second millennium bc Sintashta culture11,12.

Domestication of horses fundamentally transformed long-range mobility and warfare1. However, modern domesticated breeds do not descend from the earliest domestic horse lineage associated with archaeological evidence of bridling, milking and corralling2–4 at Botai, Central Asia around 3500 bc3. Other longstanding candidate regions for horse domestication, such as Iberia5 and Anatolia6, have also recently been challenged. Thus, the genetic, geographic and temporal origins of modern domestic horses have remained unknown. Here we pinpoint the Western Eurasian steppes, especially the lower Volga-Don region, as the homeland of modern domestic horses. Furthermore, we map the population changes accompanying domestication from 273 ancient horse genomes. This reveals that modern domestic horses ultimately replaced almost all other local populations as they expanded rapidly across Eurasia from about 2000 bc, synchronously with equestrian material culture, including Sintashta spoke-wheeled chariots. We find that equestrianism involved strong selection for critical locomotor and behavioural adaptations at the GSDMC and ZFPM1 genes. Our results reject the commonly held association7 between horseback riding and the massive expansion of Yamnaya steppe pastoralists into Europe around 3000 bc8,9 driving the spread of Indo-European languages10. This contrasts with the scenario in Asia where Indo-Iranian languages, chariots and horses spread together, following the early second millennium bc Sintashta culture11,12., Domestication of horses fundamentally transformed long-range mobility and warfare1. However, modern domesticated breeds do not descend from the earliest domestic horse lineage associated with archaeological evidence of bridling, milking and corralling2–4 at Botai, Central Asia around 3500 bc3. Other longstanding candidate regions for horse domestication, such as Iberia5 and Anatolia6, have also recently been challenged. Thus, the genetic, geographic and temporal origins of modern domestic horses have remained unknown. Here we pinpoint the Western Eurasian steppes, especially the lower Volga-Don region, as the homeland of modern domestic horses. Furthermore, we map the population changes accompanying domestication from 273 ancient horse genomes. This reveals that modern domestic horses ultimately replaced almost all other local populations as they expanded rapidly across Eurasia from about 2000 bc, synchronously with equestrian material culture, including Sintashta spoke-wheeled chariots. We find that equestrianism involved strong selection for critical locomotor and behavioural adaptations at the GSDMC and ZFPM1 genes. Our results reject the commonly held association7 between horseback riding and the massive expansion of Yamnaya steppe pastoralists into Europe around 3000 bc8,9 driving the spread of Indo-European languages10. This contrasts with the scenario in Asia where Indo-Iranian languages, chariots and horses spread together, following the early second millennium bc Sintashta culture11,12.

Domestication of horses fundamentally transformed long-range mobility and warfare. However, modern domesticated breeds do not descend from the earliest domestic horse lineage associated with archaeological evidence of bridling, milking and corralling at Botai, Central Asia around 3500 bc3. Other longstanding candidate regions for horse domestication, such as Iberia and Anatolia, have also recently been challenged. Thus, the genetic, geographic and temporal origins of modern domestic horses have remained unknown. Here we pinpoint the Western Eurasian steppes, especially the lower Volga-Don region, as the homeland of modern domestic horses. Furthermore, we map the population changes accompanying domestication from 273 ancient horse genomes. This reveals that modern domestic horses ultimately replaced almost all other local populations as they expanded rapidly across Eurasia from about 2000 bc, synchronously with equestrian material culture, including Sintashta spoke-wheeled chariots. We find that equestrianism involved strong selection for critical locomotor and behavioural adaptations at the GSDMC and ZFPM1 genes. Our results reject the commonly held association between horseback riding and the massive expansion of Yamnaya steppe pastoralists into Europe around 3000 bc driving the spread of Indo-European languages. This contrasts with the scenario in Asia where Indo-Iranian languages, chariots and horses spread together, following the early second millennium bc Sintashta culture.

Based on the spectral analysis of individual orbits of stars from different $N$-body models, we show that the face-on morphology of the so-called `face-on peanut' bars (boxy bars) and barlenses is supported by different types of orbits. For `face-on peanut' bars, the so-called boxy orbits come to the fore, and they are responsible for the unusual morphology of the bar in the central regions. In the models with compact bulges, the bars show a barlens morphology in their central parts. We found that the barlens supporting orbits come in two types, one of which gives a square-like shape and the other have a rosette-like shape in the frame co-rotating with the bar. Such a shape is typical for orbits around stable loop orbits in nearly axisymmetric potentials only slightly distorted by the bar. They were already known from some of the previous studies but their role in barlens shaping was barely investigated. Although quite simple, the rosette-like orbits are found to be the main building block of a barlens in our models. The detailed analysis of all bar orbits also allowed us to distinguish the x2 orbital family and isolate the structure supported by orbits trapped around this family. The x2 family is well-known, but, apparently, for the first time in $N$-body models we have revealed the structure it supports by means of spectral dynamics and highlight its contribution to the barlens. We found that the x2 family population increases with an increase in central matter concentration., Comment: Published in MNRAS

X-structures are often observed in galaxies hosting the so-called B/PS (boxy/peanuts) bulges and are visible from the edge-on view. They are the most notable features of B/PS bulges and appear as four rays protruding from the disk of the host galaxy and distinguishable against the B/PS bulge background. In some works their origin is thought to be connected with the so-called banana-shaped orbits with a vertical resonance 2:1. A star in such an orbit performs two oscillations in the vertical direction per one revolution in the bar frame. Several recent studies that analyzed ensembles of orbits arising in different $N$-body models do not confirm the dominance of the resonant 2:1 orbits in X-structures. In our work we analyze two $N$-body models and show how the X-structure in our models is gradually assembled from the center to the periphery from orbits with less than 2:1 frequency ratio. The most number of such orbits is enclosed in a 'farfalle'-shape (Italian pasta) form and turns out to be non-periodic. We conclude that the X-structure is only the envelope of regions of high density caused by the crossing or folding of different types of orbits at their highest points, and does not have a "backbone" similar to that of the in-plane bar. Comparing the orbital structure of two different numerical models, we show that the dominance of one or another family of orbits with a certain ratio of the vertical oscillations frequency to the in-plane frequency depends on the parameters of the underlying galaxy and ultimately determines the morphology of the X-structure and the opening angle of its rays., Comment: Accepted for publication in ApJ

Based on the spectral analysis of individual orbits of stars from different $N$-body models, we show that the face-on morphology of the so-called `face-on peanut' bars (boxy bars) and barlenses is supported by different types of orbits. For `face-on peanut' bars, the so-called boxy orbits come to the fore, and they are responsible for the unusual morphology of the bar in the central regions. In the models with compact bulges, the bars show a barlens morphology in their central parts. We found that the barlens supporting orbits come in two types, one of which gives a square-like shape and the other have a rosette-like shape in the frame co-rotating with the bar. Such a shape is typical for orbits around stable loop orbits in nearly axisymmetric potentials only slightly distorted by the bar. They were already known from some of the previous studies but their role in barlens shaping was barely investigated. Although quite simple, the rosette-like orbits are found to be the main building block of a barlens in our models. The detailed analysis of all bar orbits also allowed us to distinguish the x2 orbital family and isolate the structure supported by orbits trapped around this family. The x2 family is well-known, but, apparently, for the first time in $N$-body models we have revealed the structure it supports by means of spectral dynamics and highlight its contribution to the barlens. We found that the x2 family population increases with an increase in central matter concentration., Comment: Published in MNRAS

X-structures are often observed in galaxies hosting the so-called B/PS (boxy/peanuts) bulges and are visible from the edge-on view. They are the most notable features of B/PS bulges and appear as four rays protruding from the disk of the host galaxy and distinguishable against the B/PS bulge background. In some works their origin is thought to be connected with the so-called banana-shaped orbits with a vertical resonance 2:1. A star in such an orbit performs two oscillations in the vertical direction per one revolution in the bar frame. Several recent studies that analyzed ensembles of orbits arising in different $N$-body models do not confirm the dominance of the resonant 2:1 orbits in X-structures. In our work we analyze two $N$-body models and show how the X-structure in our models is gradually assembled from the center to the periphery from orbits with less than 2:1 frequency ratio. The most number of such orbits is enclosed in a 'farfalle'-shape (Italian pasta) form and turns out to be non-periodic. We conclude that the X-structure is only the envelope of regions of high density caused by the crossing or folding of different types of orbits at their highest points, and does not have a "backbone" similar to that of the in-plane bar. Comparing the orbital structure of two different numerical models, we show that the dominance of one or another family of orbits with a certain ratio of the vertical oscillations frequency to the in-plane frequency depends on the parameters of the underlying galaxy and ultimately determines the morphology of the X-structure and the opening angle of its rays., Comment: Accepted for publication in ApJ

By means of self-consistent numerical simulations we investigated the dynamical impact of classical bulges on the growth of the secondary buckling of a bar. Overall we considered 14 models with different disc and bulge parameters. We obtained that a bulge with a quite modest mass $B/D=0.1$ leads to completely symmetrical evolution of the bar almost independently of the initial stellar disc parameters and even can damp the first bending. At the same time, the bars in all our bulgeless models suffer from the short primary and prolonged secondary buckling. Given the smallness of the mass suppressing secondary buckling, we conclude that a classical bulge along with the gas central concentration may be the main culprits for the rarity of bars with ongoing buckling in the local Universe., Comment: Accepted for publication in MNRAS

By means of self-consistent numerical simulations we investigated the dynamical impact of classical bulges on the growth of the secondary buckling of a bar. Overall we considered 14 models with different disc and bulge parameters. We obtained that a bulge with a quite modest mass $B/D=0.1$ leads to completely symmetrical evolution of the bar almost independently of the initial stellar disc parameters and even can damp the first bending. At the same time, the bars in all our bulgeless models suffer from the short primary and prolonged secondary buckling. Given the smallness of the mass suppressing secondary buckling, we conclude that a classical bulge along with the gas central concentration may be the main culprits for the rarity of bars with ongoing buckling in the local Universe., Comment: Accepted for publication in MNRAS

Recent observational studies of X-shaped structures revealed that values of their opening angles lie in a narrow range: from 20 deg to 43 deg with smaller X-shaped structures appearing to follow a characteristic opening angle ~ 27 deg - 31 deg. We use self-consistent numerical simulations to uncover what parameters of host galaxies govern the opening angle spread. We constructed a series of equilibrium models of galaxies with high spatial resolution, varying the dark halo and bulge contribution in the overall gravitational potential, the initial disc thickness of models and the Toomre parameter Q and followed their evolution for almost 8 Gyrs. Each model demonstrated the formation of clear X-structures with different flatness. We have obtained that opening angles lie in the range from 25 deg to 42 deg throughout the entire evolution. These values are roughly consistent with observational data. The greatest variation in the opening angles is obtained by varying the mass of the dark halo. The initial thickness of the disc and the Toomre parameter are responsible for smaller variations of the angle and shorter X-structures. An increase of both parameters changes the morphology of in-plane bars and X-structures. In some cases we observed even double X-structures. The main effect of the bulge is to prevent buckling at intermediate and late stages of the disc evolution. Comparison of models with different halo masses indicates that the smallest observable values of opening angles can be associated with the presence of a heavy dark halo (more than 3 masses of the disc within the optical radius)., Comment: 19 pages, 36 figures, accepted for publication in MNRAS

Recent observational studies of X-shaped structures revealed that values of their opening angles lie in a narrow range: from 20 deg to 43 deg with smaller X-shaped structures appearing to follow a characteristic opening angle ~ 27 deg - 31 deg. We use self-consistent numerical simulations to uncover what parameters of host galaxies govern the opening angle spread. We constructed a series of equilibrium models of galaxies with high spatial resolution, varying the dark halo and bulge contribution in the overall gravitational potential, the initial disc thickness of models and the Toomre parameter Q and followed their evolution for almost 8 Gyrs. Each model demonstrated the formation of clear X-structures with different flatness. We have obtained that opening angles lie in the range from 25 deg to 42 deg throughout the entire evolution. These values are roughly consistent with observational data. The greatest variation in the opening angles is obtained by varying the mass of the dark halo. The initial thickness of the disc and the Toomre parameter are responsible for smaller variations of the angle and shorter X-structures. An increase of both parameters changes the morphology of in-plane bars and X-structures. In some cases we observed even double X-structures. The main effect of the bulge is to prevent buckling at intermediate and late stages of the disc evolution. Comparison of models with different halo masses indicates that the smallest observable values of opening angles can be associated with the presence of a heavy dark halo (more than 3 masses of the disc within the optical radius)., Comment: 19 pages, 36 figures, accepted for publication in MNRAS