Retrograde infall of the intergalactic gas onto S-galaxy and activity of galactic nuclei

1 Introduction

The galaxy formation process occurs within the gravitational potential of massive dark matter halos, a result of major and minor mergers involving systems of various properties. The growth of galaxies is driven by the substantial accumulation of gas, serving as the source for subsequent star formation. Hence, the characteristics of galactic stellar populations are intricately linked to the parameters of gas accretion.

There is a great variety of galaxies with misaligned kinematics of different stellar populations. Galaxies exhibiting counterrotation, where the rotational directions of gas and stars are opposing, hold a special significance in the field (Li et al. 2021, Katkov et al. 2022, Proshina et al. 2016). It is worth emphasizing that counterrotating discs attract attention even within large-scale cosmological models. For instance, data stemming from IllustrisTNG100 simulations reveal kinematically misaligned discs spanning a broad spectrum of galactic parameters (Starkenburg et al. 2019, Duckworth et al. 2020a, Khoperskov et al. 2021, Lu et al. 2021).

In recent years, the count of observed galaxies displaying counterrotation or some form of kinematic misalignment between their stellar and gas components has been on the rise. These phenomena are no longer as exotic as the initial discoveries suggested (Franx and Illingworth 1988). A specific class of such systems comprises galaxies with a clear misalignment in the kinematics of various stellar components, for instance, PGC 66551 and NGC 254 (Katkov et al. 2022, 2023). This phenomenon arises because the kinematic characteristics of the original gas are preserved during the star formation process.

Stellar discs with such kinematics in galaxies of different morphological types are associated with close interactions and mergers, which are most effective in dense environments. The kinematic reversal is found in PGC 046832, which is the central object of the cluster Abell 3556 (Brok et al. 2021). Generally speaking, mechanisms for the formation of counterrotating galaxies include a retrograde infall of gas onto the disc (Osman and Bekki 2017, Khoperskov et al. 2021) and a minor merger of a gas-rich dwarf galaxy (Di Matteo et al. 2008, Saburova et al. 2018, Bassett et al. 2017). As a result, the differences in the kinematics of stellar populations are often imprinted in the diverse age and chemical composition of the two stellar components (Pizzella et al. 2014, Nedelchev et al. 2019). Therefore, determining the specific mechanism responsible for the formation of misaligned stellar components in different galaxies can be accomplished by analysing its chemical composition, as outlined by Zinchenko (2023).

It is crucial to highlight that the accreting gas becomes essential for both star formation and central activity in early-type galaxies once their internal gas reserves are depleted. The fraction of lenticular galaxies displaying counterrotation between their gas and stellar components stands at approximately 30%. Recent observational data suggest a relative increase in the prevalence of such objects as ongoing research unfolds (Katkov et al. 2014, 2023, Raimundo 2021). The emergence of gas in early-type galaxies is commonly interpreted as a consequence of mergers and/or cold accretion (Davis et al. 2011). In contrast, the occurrence of counterrotation between gas and stars in late-type galaxies is around 10% due to the complicating presence of the original gaseous disc, which hampers the detection of misaligned kinematics (Pizzella et al. 2004).

Instances of misaligned kinematics or kinematic reversals can be observed in the context of active galactic nuclei (AGN), as exemplified by IC 1459 (Franx and Illingworth 1988) and NGC 5077 (Raimundo 2021). Analysing various samples enables the identification of kinematically distinct galactic cores (Krajnovic et al. 2011, Raimundo 2021, Ebrova et al. 2021), with IC 1459 serving as one of the prototypes in this regard (Franx and Illingworth 1988).

Counterrotation can enhance the conditions conducive to the activity of AGN by augmenting the inflow of gas towards the supermassive black hole. This phenomenon is substantiated by numerical simulations (Negri et al. 2014, Starkenburg et al. 2019). Recent analyses underscore the effectiveness of fuelling such nuclei, both on an individual galaxy basis (Gnilka et al. 2020, Raimundo 2021) and across larger datasets (Raimundo et al. 2023). One intriguing attribute of gas distribution in some Seyfert galaxies is the presence of nearly non-rotating ionized gas within the central region, as exemplified by NGC 5899 (Bianchin et al. 2022).

Traces of counterrotation in disc are a fairly common phenomenon both in early type galaxies (about 10–15%) and in blue galaxies ( ∼ 2 % ) (Barrera-Ballesteros et al. 2015, Chen et al. 2016). The central regions of late-type counterrotating galaxies exhibit increased rates of star formation due to the additional flow of gas into the centre. Major mergers of two spiral galaxies of comparable mass result in elliptical or lenticular galaxies with a kinematically distinct core (Bois et al. 2011).

A detailed analysis of the dynamics of counterrotating galaxies based on the Illustris TNG100 cosmological simulation data formed the basis for the scenario of their formation due to retrograde gas accretion processes onto a gas-rich disc galaxy (Khoperskov et al. 2021). The sample of 25 galaxies included objects with two counterrotating stellar discs and a gaseous component that rotates against one of the stellar discs. It was demonstrated that the counterrotation of stars in galaxies results from star formation in a gaseous disc accreted from the outer reservoir. This counterrotating gaseous component completely replaces the pre-existing gaseous disc, so the initial stage of the formation of counterrotating galaxies involves a significant loss of gas by the galaxy (Khoperskov et al. 2021). In addition, some of these galaxies look like polar ring galaxies (PRGs) as a result of gas accretion, which indicates a single mechanism for transporting gas to the centre and the formation of misaligned kinematics in the disc along with the outer polar ring. It must be emphasized that the occurrence of PRGs among nearby galaxies is approximately 0.1% (Moiseev et al. 2011, Reshetnikov et al. 2011, Reshetnikov and Mosenkov 2019) or even ∼ 1 % (Mosenkov et al. 2023), which, however, depends on the sample.

The connection between AGN and polar rings is clearly visible in observational data (Smirnov and Reshetnikov 2020, Hauschild et al. 2022). For example, SDSS data show an increased occurrence of central activity among PRGs. The edge-on galaxy NGC 4111 is an interesting example where the central activity is fed by gas directly from the inner polar ring with a diameter of only 450 pc (Hauschild et al. 2022).

Thus, counterrotations in the disc, central activity, and polar rings may be manifestations of a single process of minor merging of a gas-rich galaxy or accretion of a dense gas filament. There are numerical studies of such mergings aimed at identifying the mechanisms of the formation of multi-spin systems, including rings of different orientations and strong bends of discs (Bekki 1997, Bournaud and Combes 2003, Mapelli et al. 2015, Reshetnikov and Sotnikova 1997). The forming polar rings consist of gas and stars of a dwarf galaxy, the process of destruction of which takes a long time. The term “polar rings” traditionally includes systems with some deviation from the angle Θ = 9 0 ° between the ring and the host disc (tilted ring/disc or warped ring/disc) (Moiseev 2008, Finkelman et al. 2011, Khoperskov et al. 2014, Merkulova et al. 2012, Mosenkov et al. 2022). The Illustris TNG50 simulations allow us to study the mechanisms of formation of PRGs and the evolution of the angle of polar rings (Smirnov et al. 2023).

Our research is dedicated to conducting an in-depth examination of the process involving the replacement of a galaxy’s gaseous disc. This replacement is contingent upon the angle of incidence of retrograde intergalactic gas infall. The primary objective of this study is to establish the correlation between gas accretion onto the galactic disc and the efficiency of delivering gas to the central region, thereby sustaining an active galactic nucleus. The comprehensive problem formulation, the underlying mathematical model, and the numerical algorithm are expounded upon in Section 2. The outcomes of our numerical simulations are presented in Section 3, followed by a discussion of conclusions in Section 4. An accreting gas intergalactic filament can also generate an outer rotating ring of gas around the host disc galaxy, which is located in different planes. We briefly discuss this phenomenon of polar/tilted rings formed from the portion of gas that ends up outside the galactic disc after the interaction (Section 3).

2 Model and initial setup

The parameters of the main galaxy model are typical for large galaxies such as the Milky Way, with a maximum rotation velocity of about 220 km/s and a substantial amount of gas in the disc, which extends for about 20 kpc. The optical radius of the galaxy coincides with the size of the stellar disc and is equal to 10 kpc. We use the basic model of the host galaxy described by Khoperskov et al. (2021, 2024), Khrapov et al. (2021). The differences are related to the radial profiles of the stellar velocity dispersion and the thickness of the stellar disc in order to consider a gravitationally stable stellar-gaseous disc and exclude the development of spiral arms and a bar before the gas accretion. We also consider a spherically symmetric dark halo density distribution to avoid the formation of a spiral/bar structure at the initial stage (Butenko et al. 2015, 2022).

The accreting intergalactic gas in the numerical model is given by eight spheroidal gas clouds at the initial time (Figure 1). These gaseous clumps are located at a distance of more than 30 kpc, far beyond the optical radius of the galaxy, and move along a parabolic trajectory that crosses the disc of the main galaxy. The initial inclination of the trajectory of gas clouds relative to the galactic plane equals Θ g (the angle Θ g = 0 corresponds to motion in the disc plane). Their initial velocity of 0.8 of the circular velocity at a given radius ensures the infall of gas into the central zone of the galaxy.

Figure 1

Scheme of intergalactic gas infall onto S-galaxy in computational experiments.

We consider both the retrograde infall of the gas and the direct infall with respect to the initial rotation of the galaxy, which are determined by the initial parameters of the orbit of the intergalactic gas flow. The evolution of all components is calculated over 7 billion years. The structure and kinematics of the gaseous and stellar subsystems are modelled to understand the mechanism of gas delivery to the central kiloparsec of the galaxy, which is one of the key problems of the galactic nuclei activity.

3 Simulation results

The general picture of the process of intergalactic gas accretion onto the galactic disc is shown in Figures 2 and 3. Time is given in dimensionless units ( t = 1 → ℓ t = 76 × 1 0 6  years). The infall of gas clouds is accompanied by their rapid stretch along the trajectory, with the formation of a highly non-homogeneous stream (Figure 2). The motion of the clouds chain starts in the region z > 0 . The impact on the disc comes from the zone z < 0 , which is due to the choice of the initial orbit and results in the collision with a retrograde velocity and subsequent counterrotation of the gas. The first collision of the accreting gas with the gaseous disc of the galaxy occurs at time t ≃ 2 ( ∼ 152 Myr). The counter-impact of gas flows leads to effective mixing and strong heating of the gas. We see the destruction of the galactic gaseous disc around t ≳ 4 (300 Myr).

Figure 2

Surface density distributions of the gaseous (orange colour at the top) and stellar (blue colour at the bottom) components in various projections ( X Y and Y Z ) at the initial stages of collision of retrograde intergalactic gas with the disc galaxy ( t = 2 – 4 ).

Figure 3

Surface density distributions of the gaseous (orange colour at the top) and stellar (blue colour at the bottom) components in various projections ( X Y and Y Z ) at the stage of formation of a new gaseous disc with opposite rotation in the precessing disc system ( t = 6.6 – 52.6 ).

We emphasize that we study the evolution of a rather strong gas flow with twice the mass of the initial gaseous galactic disc. The velocity of the intergalactic gas at the moment of impact is ∣ v ∣ ≃ 2.5 with components v r ≃ − 1.6 , v φ ≃ − 1.8 , and v z ≃ 0.8 (Figure 1). Most of the galactic gas and the accreting matter are ejected outside the galaxy’s stellar disc. The remaining part of the gas forms a relatively dense central core with a size of r = 0.4 – 0.6 , which is caused by a significant loss of angular momentum during the impact. The consequences of the collision of gas flow at the initial stage lead only to weak bending oscillations of the stellar disc (Figure 2).

The subsequent stages of the evolution of the gaseous and stellar components are shown in Figure 3. The gas ejected due to the collision gradually returns to the galactic plane of the stellar disc ( t = 6.6 ) and forms a retrograde rotating gas disc around the central core ( t = 13.2 ). The radius of the new gaseous disc reaches R g ≃ 0.8 by the time t = 52.6 ( ∼ 4 billion years). In addition, some of the gas forms a ring-like structure with counterrotation ( 1 < r < 2 ), inclined at an angle ∼ ( 30 – 60 ) ° relative to the plane of the stellar disc. Such polar gaseous rings are similar to those observed in some galaxies (Khoperskov et al. 2014) and reproduced in cosmological simulations (Snaith et al. 2012).

The following evolution of the stellar disc is accompanied by bending oscillations and the appearance of a strong precession, the inclination of its rotation plane reaches ∼ 4 5 ° by the time t = 52.6 . Thus, three gaseous components are highlighted as a result of accretion. First, an increased concentration of gas occurs in the central kiloparsec ( r ≤ 0.1 ), which will be denoted by “core.” The rotating gas within r < 1 forms a disc in the galactic plane of the stellar disc (the core is an integral part of this disc). Finally, the gas outside the optical radius ( r > 1 ) has the shape of a rotating, slightly flattened ring, which is inclined to the plane of the galaxy.

Figure 4 shows the mass evolution in three kinematically distinct subsystems. The new gaseous disc lies in the region r ≤ 1 with a mass approximately three times larger than the initial mass in this region. Moreover, most of the gas in the disc is located in the core region (approximately M core = 2 ∕ 3 M disc at later times). The formation of the outer polar ring requires more time and completed by t ≃ 45 . The core emerges very rapidly, and there is only very slow growth after t = 10 (760 Myr) due to viscous forces leading to a small accretion flow to the centre (see the slope of the red line in Figure 4). The formation of the ring and a new galactic gaseous disc ends after t ≃ 45 (3.5 billion years).

Figure 4

The evolution of the gas mass in the galactic centre M core ( r < 0.1 , red line 1), in the disc M disc ( r < 1 , blue line 2) and in the ring M ring ( 1 < r < 2 , green line 3) for the model with the retrograde gas and Θ g = 2 0 ° .

The evolution of the system is followed by mixing of the initial gas of the galaxy with the intergalactic gas. We highlight the fractions of the source gas in each of the three subsystems ( n core , n disc , n ring ) (Figure 5). The core and the disc masses include about 60–65% of the original gas of the galaxy after the establishment of a quasi-stationary picture. The intergalactic gas makes a contribution in the range of 35–40%. The ring is mainly made of intergalactic gas (more than 90% of the mass). There is less than 1% of the gas outside the two optical radii ( r > 2 ) that kicked out to the larger distances as a result of the stream’s impact into the disc.

Figure 5

Dynamics of the mass fraction of the initial galactic gas in the core n core ( r < 0.1 , red line 1), in the disc n disc ( r < 1 , blue line 2) and in the ring n ring ( 1 < r < 2 , green line 3) for the retrograde gas infall model at Θ g = 2 0 ° .

Understanding the complex 3D dynamics of multiple components is facilitated by calculating the angular momentum for each subsystem. Figure 6 shows the temporal variations of the two angular momentum components of the stellar disc, the gas disc, and the outer gas ring (Figure 3). The negative angular momentum corresponds to a retrograde rotation relative to the prograde rotation of the stellar disc ( L > 0 ). Since the gas kinematics in these regions are very different, the gas component is divided into the inner part r < 1 and the outer zone 1 < r < 2 . The external gas after the impact forms a kinematically isolated ring outside the plane of the stellar disc. The outer ring is tilted towards the stellar disc at an angle that changes slowly. Figure 7 shows the dynamics of the relative angular momentum of different components. The total angular momentum of the stellar subsystem varies within 1% over 7 Gyr (Figure 6), despite noticeable radial density redistributions in the stellar disc and its precession.

Figure 6

Evolution of the components L z and L y of the specific angular momentum for stellar disc ( L z , magenta line 1; L y , light-blue line 2) and gaseous disc ( L z , red line 3; L y , blue line 4) with Θ g = 2 0 ° .

Figure 7

Specific angular momentum L vs time for the gas core ( r < 0.1 , green line 1), the gas disc ( r < 1 , red line 2), the gas ring ( 1 < r < 2 , blue line 3) and the stellar disc ( r < 1 , magenta line 4) in the model with Θ g = 2 0 ° .

We observe a possible connection between the kinematic displacement of galactic components and the activity of the central black hole due to the inflow of a large amount of gas into the central kiloparsec. Figure 8 shows the time dependence of the gas mass in the centre of the computational domain, r < 1 kpc, for the retrograde and prograde gas infall simulations. Figures 4–7 contain simulation results in models without a bulge. Figure 8 also shows our calculations in bulge galaxy models for comparison.

Figure 8

Influx of gas mass M core into the inner region of the core ( r < 0.1 ) at different angles Θ g for retrograde infall of intergalactic gas into the disc of the bulgeless galaxy: 1 – Θ g = 2 0 ° ; 2 – Θ g = 1 0 ° ; 3 – Θ g = 5 ° ; 4 – Θ g = 3 0 ° ; 5 – Θ g = 4 0 ° ; 6 – Θ g = 4 5 ° ; 7 – Θ g = 6 0 ° . For comparison, the model with prograde infall (8 – Θ g = 2 0 ° ) is also shown. Curves 9 and 10 describe galaxy models with the bulge at Θ g = 2 0 ° for retrograde and prograde infall, respectively.

The retrograde accretion simulation, already 400 Myr after the collision, increases the gas mass in the central region by about 100 times compared to the initial mass in the gaseous disc of the unperturbed galaxy. This is due to the fact that the infalling intergalactic gas mixes with the gas of the galaxy, and this mixture rapidly loses its angular momentum during the first crossing of the disc. The dense gaseous core also forms very quickly (Figure 2–7). The prograde infall of the gas does not result in mixing, and a massive gaseous core does not appear. The growth rate of the mass of such a core is very small over the entire simulation interval so that the final mass of the core is about two orders of magnitude smaller compared to the retrograde infall simulation.

Models with the bulge do not qualitatively change the results compared to the fall of gas onto the bulgeless galaxy. Curves 1 and 9 in Figure 8 were calculated for the bulgeless and bulge models, respectively, at Θ g = 2 0 ° , which provides the highest rate of gas inflow into the centre of the galaxy. The model without the bulge gives the gas mass in the core only 10% less than with the bulge. This difference is possibly due to the central gradient of the disc rotation velocity. However, it must be taken into account that the addition of the massive bulge increases the mass of the entire stellar component and redistributes the gas in the disc along the radius during the evolution of the system. Cases of retrograde incidence (curves 8 and 10) also give similar results for the gas mass inside the core.

Therefore, the retrograde infall of gas on the galaxy provides a more efficient supply of gas in the galactic centre compared to the prograde infall, which connects the presence of kinematically displaced components in galaxies with AGN. A similar correlation was previously found for observed objects and numerical simulations in Starkenburg et al. (2019), Duckworth et al. (2020b), Gnilka et al. (2020), Raimundo (2021), Beom et al. (2022), Sil’chenko et al. (2022), and Raimundo et al. (2023). Our computational experiments confirm that the activity of galactic nuclei can be effectively enhanced due to the formation of counterrotating gas as a result of the accretion of the intergalactic medium. Mass growth within the central kiloparsec is quite sensitive to accretion conditions, which requires detailed studies (Figures 8 and 9).

Figure 9

Evolution of the gas density in the galaxy centre ρ c at different angles Θ g in the model of retrograde infall of intergalactic gas. The colours, symbols, and lines are as in Figure 8.

The description provided above aligns seamlessly with the behaviour exhibited by the central gas density ϱ c , as illustrated in Figure 9. The outcomes depicted in Figures 8 and 9 underscore a pronounced dependency of gas supply efficiency to the galactic core on the angle of incidence of intergalactic flow. Notably, the most effective gas delivery to the central region is realized at an angle of α = 2 0 ° . A smaller infall angle, such as α = 1 0 ° , results in a halving of both M core and ϱ c . In contrast, an infall trajectory strictly within the galactic plane exhibits even lesser efficacy. This phenomenon can be attributed to the compromised conditions for the disruption of the original gaseous disc. Furthermore, models employing an angle of incidence α > 2 0 ° yield unsatisfactory outcomes due to the reduction in the relative velocity upon collision of the flows. For instance, a perpendicular infall does not lead to gas counterrotation at all.

Figure 10 shows changes in the angle of inclination of the outer gas ring ( Θ R ) to the disc of the main galaxy at different gas incidence angles Θ g . We calculate the orientations of the gas ring and the host galaxy disc based on the construction of the vector of the total specific angular momentum, assuming that this vector is perpendicular to the plane of the rotating ring. This simple method for estimating the orientation angles of a rotating system is especially convenient for SPH/ N -body simulations. The condition Θ R > Θ g is satisfied at any angle of incidence Θ g after t > 10 = 760 million years (Figure 10). The outer rings are formed only after t > 10 in our models.

Figure 10

Dependences of the gas ring inclination angle ( Θ R ) on time. The dashed line describes the behaviour of the model for Θ g = 2 0 ° with the bulge. The vertical grey line at t = 10 approximately marks the end of the outer ring formation stage.

The ratio Θ R ∕ Θ g at the end of the simulation weakly depends on the incident angle of the gas flow and is within Θ R ∕ Θ g ≃ 1.4 − 1.6 . At incidence angles Θ g > 6 0 ° , we have an almost polar ring with an angle Θ R = ( 80 − 90 ) ° (Figure 11). Temporary changes in the functions Θ R ( t ) are insignificant, which is a reflection of the law of conservation of angular momentum in the absence of relative precession of the ring and disc. The solid curves in Figure 10 are based on simulations with bulgeless models. We also show the simulation result of galaxy with bulge for comparison at Θ g = 2 0 ° .

Figure 11

Relationship between the gas flow incidence angle ( Θ g ) and the polar ring orientation angle at the end of the simulation ( Θ R ).

4 Conclusions and discussion

The formation process of counterrotating stellar and gaseous discs has been explored through numerical simulations focused on the accretion of gaseous intergalactic clouds. Our study concentrated on models where the mass of the accreting flow is twice that of the galactic gas. This choice of parameters establishes favourable conditions for the occurrence of counterrotation. As gas retrogradely falls into the central region of the gas-rich S-galaxy, it triggers the swift disintegration of the original gaseous disc. Subsequently, the formation of a new but smaller disc with opposing rotation becomes evident.

The robust interaction between nearly opposing streams of galactic and intergalactic gas results in the efficient dissipation of angular momentum. Within a sphere with a radius of 1 kpc, a mass comparable to that of the entire initial gaseous disc can accumulate due to this phenomenon. This gaseous core exhibits low angular momentum and a high temperature, causing it to be predominantly supported by pressure forces. The presence of a supermassive black hole can induce radial flows that fuel central activity. The efficacy of gas inflow into the central zone, stemming from intergalactic cloud accretion, crucially hinges on the initial angle of incidence of the gas flow onto the disc ( Θ g ). Our calculations reveal a non-monotonic pattern in the relationship between the mass M core and Θ g . The most substantial mass aggregation within 1 kpc occurs at an angle of approximately Θ g ≃ 2 0 ° . Elevating the angle of incidence to 4 0 ° leads to a nearly tenfold reduction in M core .

The prograde infall of intergalactic gas lacks the capacity to significantly disrupt the disc and does not induce a substantial inflow of gas within a radius of 1 kpc. In terms of gas mass accumulation within this range, this accretion mode only yields around 1% of what retrograde infall achieves.

The numerical simulations conducted to explore the dynamics of counterrotating stellar-gas discs encompass various stages of interaction with the descending intergalactic flow over a span of approximately 7 billion years. Prior work by Starkenburg et al. (2019) demonstrated the enduring nature of counterrotating gas and stars, persisting for around 2 billion years, following their formation in Illustris numerical simulations. Our approach is rooted in the qualitative dynamics exhibited by a single galaxy subjected to an accreting flow.

It is important to note that our models do not incorporate star formation and feedback resulting from galactic core activity. The presence of ionized gas at different distances from the disc plane, arising from multiple supernova explosions and the influence of OB stars (Vasiliev et al. 2022), is also absent in our model. Our primary focus revolves around the dynamic interaction between the intergalactic flow and the gas-rich disc galaxy.

An additional intriguing aspect arising from our simulations pertains to the development of outer gas rings oriented at an angle to the galactic plane. These structures resembling polar rings naturally emerge as a consequence of retrograde intergalactic gas infall onto the gas-rich galactic disc. While our primary focus is on scrutinizing the mechanism of gas delivery to the central region to fuel galactic nucleus activity, the formation of polar rings necessitates a distinct and dedicated analysis.

Let us note, however, some preliminary results related to outer ring structures. The angle between the ring and the main disc Θ R is determined primarily by the angle of attack of the gas flow Θ g in the model under consideration. We always get a ring with an angle Θ R > Θ g , so this ring is slightly different from the polar orientation at Θ g ≳ 5 5 ° . The formation time of the outer ring occurs in approximately 1.5 billion years under the galaxy and gas filament parameters used. Further evolution is not accompanied by a change in the angle Θ R . Smirnov et al. (2023) analysed the evolution of two rings in the Illustris TNG50 simulation, formed around massive galaxies due to interactions with companions. An interesting result of this work is the secular growth of the ring’s inclination with a rate of 8 ° /Gyr. We see no change in the angle Θ R in our simulations, which may be due to the symmetry of the dark halo shape. The question of the influence of the triaxial shape of the dark halo on the properties of the polar rings was discussed, e.g., in the works of Snaith et al. (2012), Combes (2014), Moiseev et al. (2015), and Zasov et al. (2017).

There are deviations of the rings from polar orientation ( Θ R = 9 0 ° ) among the observed objects. The inclination of the ring in the galaxy PGC60020 is about 6 0 ° (Merkulova et al. 2012). The narrow inner ring in the galaxy PGC1806302 has a similar tilt angle (Finkelman et al. 2011). Various angles between the ring and the host disc are given for the sample of PRGs in the work of Mosenkov et al. (2022).

Given the intricacies of our simple model, which already encompasses numerous free parameters, such as initial masses, spatial distributions within each component, characteristics of the intergalactic flow orbit, and dark mass, the findings we have obtained warrant further expansion and deeper investigation in subsequent research.