# Gasflows in Barred Galaxies with Big Orbital Loops—A Comparative Study of Two Hydrocodes

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## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Orbital Analysis

#### 3.1. Stellar Response Models

#### 3.2. Orbits

#### 3.2.1. Peridic Orbits (POs)

- The family x1. The main building block for the bar of the model is, as expected, the x1 family. However, we have to underline that, due to the presence of the sine terms in Equation (1), its orbits cross the $y=0$ axis with $\dot{x}\ne 0$. We find that essentially all x1 orbits are cuspy and they develop loops, which are conspicuous for ${E}_{J}>-1.4\times {10}^{5}$. Representatives of the x1 family are depicted with the black color in Figure 4a. All of them are stable, with the orbit with the largest loop and the longest projection on the y-axis being at ${E}_{J}=-$121,468.
- The family that we indicate as “f” in Figure 3 has a stable and an unstable part. As we observe in Figure 3, it changes its stability practically at ${E}_{J}\left({\mathrm{L}}_{1}\right)$. Orbits of this family are given in Figure 4b. The cyan- and black-colored orbits are stable at ${E}_{J}=-$124,157 and −120,000, respectively. The grey orbit, which has developed loops reaching the L${}_{1}$ and L${}_{2}$ regions, is at ${E}_{J}=-$111,723 and is unstable. As energy increases, the ${x}_{0}$ initial condition of the orbits of family f increases, leading to hexagonal, rhomboidal shapes. We have, in this case, along the charactristic of f in Figure 3, a transition from a 4:1 to 6:1 resonance morphology. Changes in the morphology of POs along a characteristic are observed as the curve passes through the region of a resonance (see, e.g., Figure 3.7 in [14]). For ${E}_{J}\u2a86-1.21\times {10}^{5}$, the POs of family f are unstable. Thus, there are no stable, rectangular-like orbits, which could help the bar reach corotation. The x1 orbits at these energies already have big loops at the increasing part of the characteristic, for ${E}_{J}>-$121,468. They also have shorter projections on the y-axis than the outermost x1 orbit, drawn in black in Figure 4a. The situation with the orbital loops at and beyond the 4:1 resonance region is summarized in Figure 4c, where we give x1 at ${E}_{J}=-$121,468 (black), x1 at ${E}_{J}=-$112,503 (cyan) and f at ${E}_{J}=-$111,723 (gray).

#### 3.2.2. Non-Periodic Orbits

## 4. Gas Response

#### 4.1. SPH

#### 4.2. RAMSES

## 5. Discussion

#### 5.1. Code Dependence of the Responses

#### 5.2. Stellar vs. Gas Response

## 6. Conclusions

- The basic conclusion of the present study is that the dust-lane shocks in the gas responses of barred galaxies models avoid the regions in which the stable POs of the x1 family have developed sufficiently big loops. They deviate from these regions and, as they bypass them, they form extensions at an angle with the straight-line shocks. Ahead of them, in the direction of rotation, we find low-density regions, which in many cases have a “triangular”-like shape, as, e.g., model 12 in [1]. This morphology is encountered in both codes we used (SPH and RAMSES).
- Responses during the growing-bar phase are smoother, in the sense that we observe fewer regions with strong density gradients. As the amplitude of the perturbation during this phase is always smaller than the final, maximum one, the models do not develop x1 POs with big loops (Figure 2a). Thus, the straight-line, dust -lane shocks extend to larger distances.
- A characteristic feature of the models are the “tails”, which are dense regions that appear bifurcating from the straight dust-lane shocks at specific points along them. The bifurcating points are identified with the points at which the x1 POs start having considerable loops (indicated with arrows in Figure 8). Gas is streaming along these lanes towards the points where the three density enhancement join.
- The SPH models can be followed for a long time only with replenishment of particles that are manually removed mainly from the overdense regions, but it gives, after a certain time, an invariant response. RAMSES, on the other hand, has a short, relatively turbulent phase for $t\u2a86{T}_{f}=3T$ and then reaches a repeating cycle, where the morphology of the straight-line dust-lane shocks and their extensions characterize the snapshots. Nevertheless, there is a dominating “mean” morphology, given in Figure 15.
- Both codes give information valuable for understanding the dynamics of the model and should be used when comparison with the morphology of specific galaxies is attempted. With the Lagrangian SPH method, we can obtain detailed velocity fields in the the dense regions of the model, while with the grid code RAMSES, we can obtain an overall picture of the kinematics of the models. Both modeling techniques are useful for understanding gas dynamics in barred-spiral galaxies.
- Finally, as regards the stellar response, in this model, like in the case of the models for NGC 4314 we studied in [6] and for NGC 1300 in [9], we find a second bar, which is a “chaotic” envelope around the x1 bar. It is “chaotic”, in the sense that it is supported by chaotic orbits. The consistency of its appearance in the models indicates that this is rather a common feature in barred galaxies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PO(s) | Periodic orbit(s) |

SPH | Smoothed particle hydrodynamics |

## Note

1 | Available in https://bitbucket.org/rteyssie/ramses/src/master/, accessed on 16 May 2022. |

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**Figure 1.**The circular speed ${v}_{c}=\sqrt{r\frac{d{\Phi}_{0}}{dr}}$ (

**a**) and the k = 2 (

**b**), k = 4 (

**c**) and k = 6 (

**d**) components of the potential, normalized by ${\Phi}_{0}$, as a function of the radius r. From (

**b**–

**d**), magenta curves refer to the cosine and green to the sine terms.

**Figure 2.**(

**a**–

**h**) Successive snapshots of the stellar response model. Lighter regions correspond to denser regions, according to the logarithmic color bar at the bottom of the panels. The number of pattern rotations is given at the upper-left corner of the frames. The red dots indicate the location of L${}_{1}$ (

**top**), L${}_{2}$ (

**bottom**), L${}_{4}$ (

**right**) and L${}_{5}$ (

**left**) of the bar. The bar rotates counterclockwise.

**Figure 3.**Characteristics of the main families of POs, x1 and f, projected in the (${E}_{J}$, ${x}_{0}$) plane. Stable parts of these curves are plotted with black, while unstable with red, color. The zero-velocity curve is the thick green curve, having at its local maximum the L${}_{4}$ point. The area between the ${E}_{J}$’s of L${}_{1}$ and L${}_{4}$ is shaded with cyan color. In the embedded frame, we give an enlargement of the characteristics of all main families close to the center of the system (see text). The x2-x3 loop is plotted with magenta, the x1 characteristic with balck and that of x4 with blue color.

**Figure 4.**Periodic orbits with loops. (

**a**) Stable orbits of x1 are plotted with black and an unstable f with pairs of loops at its apocenters is plotted with grey. (

**b**) Two stable f orbits (black and cyan) and an unstable in grey. (

**c**) POs with loops beyond the 4:1 resonance. The grey belongs to f, while the two others (black and cyan) to x1.

**Figure 5.**(

**a**) Three orbits with loops, plotted with black, green and red color) that follow different paths and eventually cross the corotation region $(({x}_{{L}_{1,2}},{y}_{{L}_{1,2}})=(\pm 1.21,\pm 9.65))$. (

**b**) The central part of the $(x,\dot{x})$ surface of section at ${E}_{J}=-$111,723. The location of x1 and f are indicated with a green and magenta dot, respectively. The left empty part (roughly for $x<-0.5$) is occupied by invariant curves around x4 (not plotted), while the empty part for $x>-0.5$ is due to the fact that orbits with initial conditions in this region are practically escape orbits (they intersect the surface of section at large distances, outside the frame of panel).

**Figure 6.**The f PO for ${E}_{J}=-$118,092 (in black) together with two sticky periodic orbits (red and green) at the same energy, integrated for about three periods of f. The orbits do not reach the unstable Lagrangian points, located at $({x}_{{L}_{1,2}},{y}_{{L}_{1,2}})=(\pm 1.21,\pm 9.65)$.

**Figure 7.**The SPH response without redistribution of the particles, during the growing phase of the bar, for $t=0.52$ (

**a**) and for $t=0.65$ (

**b**). The artificial viscosity parameters we used are $(\alpha ,\beta )=(0.5,1)$. In this, and in all relevant subsequent figures, the bar rotates counterclockwise. Red dots indicate the Lagrangian points of the model.

**Figure 8.**Characteristic snapshots of a typical model with redistribution of particles in action, at $t=0.53$ (

**a**) and $t=9.83$ (

**b**). In (

**c**) we give a sketch that outlines the dense parts of the model. In the text, the green features, indicated with arrows, are described as “$\Gamma $/」”. We draw the straight-line dust lane shocks and its continuations with cyan and yellow color respectively.

**Figure 9.**The velocity field of the snapshot of Figure 8 is given by zooming into its $(-5,11)\times (0,11)$ region. The green arrow indicates point A, where the tail of the lower $\Gamma $ feature joins the straight-line shocks. The green dot at $(x,y)=(1.21,9.65)$ indicates the location of L${}_{1}$.

**Figure 10.**The flow in the straight-line shocks region in two typical cases. (

**a**) During the growing bar phase. (

**b**) During the time the bar has a constant amplitude $(t>3T)$. We can observe in (

**b**) how the particles of the bifurcating tail help the shock extend to larger distances from the center, as they flow upwards in the direction opposite to the rotation of the system.

**Figure 11.**Stable x1 orbits (black, green and red) and the (grey) loops of an 6:1 f orbit (see Figure 4). The straight-line shocks are formed in the region occupied by the black x1 POs. The shocks avoid the loops, while the unstable f orbit does not affect the flow.

**Figure 12.**The evolution of a typical RAMSES model, with ${c}_{s}=5$ km s ${}^{-1}$. The snapshot in (

**a**) is in the growing bar phase, while in (

**b**–

**d**) the bar perturbation has already reached its maximum. Times are given in the lower-left corner of the frames. Densities increase from left to right according to the logarithmic scale of the color bar at the bottom of the figure. In red, we plot the stellar response particles at the corresponding times. The bar rotates counterclockwise. Red dots indicate the location of the Lagrangian points.

**Figure 13.**A typical snapshot of our RAMSES simulation, summarizing the main morphological features, together with the velocity vectors indicating the flow. The flow vectors are colored yellow, when they are in regions below the minimum density we considered. The two long white arrows indicate the point where the tails of the $\Gamma $ features join the dust-lane shocks in the bar region. The stellar response is given in the background using a red color for the test particles.

**Figure 14.**(

**a**–

**d**): Characteristic evolution of the flow at the end and beyond the x1-bar region within a $\Delta t=0.11$ period, during which the extension towards the $\Gamma $-like feature breaks and forms again. Times are given at the lower left corner of the panels. The flow in the upper branch of the $\Gamma $ feature splits, pointing to two opposite directions at the points indicated with heavy yellow arrows in (

**a**,

**b**). The long white arrow in (

**a**) indicates the point “A”, where the tail of the 」 feature join the dust-lane shock in the bar region. Red dots mark the location of L${}_{1}$.

**Figure 15.**A mean RAMSES response morphology, created by stacking the images of 100 snapshots within 4.35 rotational periods of the system. It has all the typical morphological features discussed in the text. Red dots give the locations of the Lagrangian points.

**Figure 16.**Typical responses in a RAMSES model with ${c}_{s}=10$ km s${}^{-1}$ (

**a**) and ${c}_{s}=20$ km s${}^{-1}$ (

**b**). The rest of the initial conditions are as in the model of Figure 13. In (

**a**), the response does not differ essentially from the model with ${c}_{s}=5$ km s${}^{-1}$, while in (

**b**) we observe instabilities in the shocks in regions with high density gradients.

**Figure 17.**A model in which the amplitude of the bar reaches only 50% of that of the main model. (

**a**) The stellar response. (

**b**) The gaseous RAMSES model. The disks of the models are populated initially homogenously. The model has features pointing to a more regular orbital background.

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**MDPI and ACS Style**

Pastras, S.; Patsis, P.A.; Athanassoula, E.
Gasflows in Barred Galaxies with Big Orbital Loops—A Comparative Study of Two Hydrocodes. *Universe* **2022**, *8*, 290.
https://doi.org/10.3390/universe8050290

**AMA Style**

Pastras S, Patsis PA, Athanassoula E.
Gasflows in Barred Galaxies with Big Orbital Loops—A Comparative Study of Two Hydrocodes. *Universe*. 2022; 8(5):290.
https://doi.org/10.3390/universe8050290

**Chicago/Turabian Style**

Pastras, Stavros, Panos A. Patsis, and E. Athanassoula.
2022. "Gasflows in Barred Galaxies with Big Orbital Loops—A Comparative Study of Two Hydrocodes" *Universe* 8, no. 5: 290.
https://doi.org/10.3390/universe8050290