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Astron. Astrophys. 325, 933-942 (1997)

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3. Numerical simulations

There are only a few investigations that illustrate several proposed scenarios of polar ring formation. This lack of theoretical study devoted to the question of ring origin is connected with the necessity of the usage of complicated 3D gas dynamics models, since polar rings contain a high amount of neutral gas (up to a few [FORMULA]), as seen from Table  1. One of the most fit tools to construct such models is a smoothed particle hydrodynamics algorithm - SPH, - originally proposed by Lucy (1977) and Gingold & Monaghan (1977) and considerably developed in the late eighties (e.g., Hernquist & Katz 1989). It is fully three-dimensional, flexible regarding the geometry of the gas distribution and consequently well-suited to modelling both the donor and recipient galaxies and the empty extended space between them. Applied to a range of polar-ring investigation problems, this method has already lead to many important results.

The main result of the ring stability investigations is that the time for gas in an axisymmetric or triaxial potential to settle into a steady state is small compared to the age of the Universe (see Christodoulou et al. 1992 and references therein). So there is sufficient time for accreted gas to form polar rings. However, these studies did not take into account time-dependent effects and realistic initial conditions. Some efforts to include these effects have been made by Rix & Katz (1991) and Weil & Hernquist (1993).

Rix & Katz (1991) have treated the polar-ring formation process as a gradual smearing of a small gaseous blob captured by a galaxy of spherically-symmetric structure on a distant circular orbit. Naturally, the diameter of the forming ring (up to 80 kpc) was quite completly dictated by the initial position of a gaseous satellite (the orbit radius). The approach of Weil & Hernquist (1993) was more self-consistent. They have considered a parabolic encounter of a low-mass gas-rich companion with a more massive galaxy and consequent merging of the former. The gaseous component of the satellite was settling into the ring after the total disruption of the satellite in the vicinity of the target galaxy. The forming ring turned out rather small (about 6-7 kpc in diameter) and its size was determined in the end by a hand-introduced description of details of the companion disruption.

The ability of the accreting material to dissipate energy is the crucial factor for ring formation. Test (noninteracting) particles (which imitate molecular clouds from intergalactic space or clouds belonging to a companion galaxy) lose energy due to ram-pressure in an extended halo of a spiral galaxy and form a gaseous ring rotating around the disk of the spiral (Sofue & Wakamatsu 1993, Sofue 1994).

3.1. Modelling of polar-ring characteristics during the ring-forming process

The main feature of the present investigation is a description of the full history of the gas stripping of the spiral galaxy outskirts and its consequent capture by a satellite during a parabolic encounter. Keeping in mind the recent observations of several interacting pairs of galaxies of comparable luminosities which demonstrate evidently ring-like structures in the making (see Introduction), we have considered a distant encounter of equal-mass systems. In the case of a distant encounter the effect of self-gravity may be ignored. The treatment of the gas hydrodynamics is described below.

We did not explore in a comprehensive manner the orbital parameter space and choose only one set of impact parameters for which the modelling encounter of an S0 galaxy with a gas-rich system of comparable mass unambiguously results in the polar-ring formation around the former. Motivated by the above observational data analysis, we have undertaken a numerical investigation of the ring-forming process for galaxies with different structures.

3.1.1. Method

Our simulations are based on a rather standard variant of the SPH code, which is the same as that in Sotnikova (1996). For simplicity, we adopt the smoothing length h (analogue of the particle size) to be independent of the local gas density. The smoothed values of hydrodynamical parameters are estimated by using a grid with cell length equal to [FORMULA]. We assume an isothermal equation of state, leaving thermal effects completely unexplored. The gas is always at a temperature [FORMULA], and the corresponding equation of state is

[EQUATION]

where P and [FORMULA] are the gas pressure and density, and [FORMULA] is the isothermal speed of sound. For [FORMULA] , [FORMULA].

The motion of [FORMULA] particles, which represent elements of the continous gaseous medium, is traced by means of equations of momentum conservation:

[EQUATION]

where [FORMULA] and [FORMULA] are the spatial coordinate and velocity of the i -th particle, and [FORMULA] is the gravitational potential. The term [FORMULA] describes the acceleration due to viscosity [FORMULA]:

[EQUATION]

where [FORMULA]. We adopt one of the standard forms for the artificial viscosity [FORMULA] which serves to represent the sudden deceleration of the gas motion when strong shocks arise in the gas (e.g., Hernquist & Katz 1989):

[EQUATION]

In the expression (3), the viscosity depends on the divergence of the velocity field. The first term is analogous to a bulk viscosity, the second, indroduced to prevent particle interpenetration at high Mach number, is of the Neumann-Richtmyer type. Parameters [FORMULA] and [FORMULA] are free parameters. According to Hernquist & Katz (1989), one can satisfactorily reproduce the change of the density and pressure across the shock front if the values of [FORMULA] and [FORMULA] are equal to 0.5 and 1.0, respectively. We used just these numbers in our simulations.

The procedure of a smoothed-value estimation of the gas density is quite usual, as well as the transition from hydrodynamical equations (1) and (2) to SPH-equations (for a detailed description of the adopted numerical scheme, see Sotnikova 1996).

3.1.2. Model

Donor galaxy

To minimize the number of free parameters, we chose a very simple model for the donor galaxy - its potential was taken as that of a softened point mass

[EQUATION]

where G is the gravitational constant, [FORMULA] the mass, and [FORMULA] the softening scale length of the potential.

[FORMULA] particles are used to represent the gaseous medium. They are initially gathered in a thin disk with a [FORMULA] distribution from the center of the donor galaxy up to an outward radius [FORMULA] and are placed in dynamically-cold circular orbits. The total gas mass is 0.2 [FORMULA]. The gaseous particle size h, which determines the numerical spatial resolution, is 300 pc.

Throughout, we employ the following system of units: the gravitational constant [FORMULA], the mass of the donor galaxy [FORMULA], the outer radius of the initial gas distribution in the donor galaxy [FORMULA]. In terms of physical units, a unit mass can be defined as [FORMULA], and a unit distance corresponds to 15 kpc. Combined together, they give a unit time to be 86.6 Myrs, and a unit velocity to 169.3 km [FORMULA].

Accreting galaxy

We have considered the two-component model of a galaxy, consisting of a bulge and a disk. The light distribution of the bulge is well-provided by the model used, for example, in Weil & Hernquist (1993). For this model, the potential is

[EQUATION]

where [FORMULA] is the bulge mass, and [FORMULA] the scale length of the potential.

The gravitational potential of the disk component of the galaxy is approximated by a Miyamoto & Nagai (1975) potential

[EQUATION]

where [FORMULA] is the disk mass, and [FORMULA] and [FORMULA] are scale radii. This model was favored by the simplicity of its practical usage. In our simulations, [FORMULA].

The ratio [FORMULA] determines the range of models from bulge-dominated systems (large values of the ratio) to disk-dominated galaxies (small values of the ratio). Four models for an accreting galaxy have been considered, with values of [FORMULA] equal to: 2.0, 1.0, 0.5, 0.2. The total mass of the galaxy is taken the same for all models and equal to that of the donor galaxy: [FORMULA].

As shown in Sect. 2, main galaxies of PRGs with extended rings are similar to spiral galaxies in global photometric structure (and probably mass distribution). According to de Jong (1996), the bulge effective surface brightness shows the best correlation with morphological type of spiral galaxies. Therefore, the total bulge luminosity (mass) also correlates well. Scaling radii show a large scatter for all types of spiral galaxies. Thus, we fixed the scale lengths of galaxy components, leaving only the mass of the bulge to be variable. Scale lengths of the bulge [FORMULA] and disk [FORMULA] potentials are 1 kpc and 5 kpc, respectively. For these values, the half-mass radius of the bulge is [FORMULA] kpc, and the maximum of the rotation curve of the disk component falls on the radius [FORMULA] kpc.

Orbit

Before modelling a ring-forming encounter, we first solved the two-body problem for the donor and ring-host galaxies. The numerical procedure employed is quite similar to that used in Weil & Hernquist (1993).

Two interacting galaxies are initially separated by a rather large distance ([FORMULA] 75 kpc), so that tidal effects be negligible. The initial velocities are taken as for parabolic encounter. The primary galaxy passes in a zero-inclination (in the plane of the gaseous disk of the donor galaxy - xy plane), prograde orbit (that is, the orbital angular momentum is parallel to the spin of the donor galaxy) with a pericenter distance [FORMULA] initially set as 1.6. The polar axis of the ring-host galaxy disk component (parallel to the x -axis) lies in the orbital plane, so the orbit is polar regarding the ring-host galaxy. The calculated orbit somewhat differs from parabolic (due to non-Keplerian potentials of galaxies), but its form is nearly the same for all models of a ring-host galaxy.

3.1.3. Results

During the encounter, the primary galaxy strips the outskirts of the donor object and a ring, rotating in the direction of the orbital motion, eventually forms around the galaxy in the encounter plane. As the equatorial plane of the disk component of S0 is taken to be perpendicular to that of the orbital motion, the ring is polar. The total amount of accreted gas is about 10% of all gas in the donor galaxy (or about [FORMULA]) and there is not any significant difference in the stripped mass for all considered models of the ring-host galaxy. Some amount of the gas fell into the central part of galaxy. The mass of this gas is estimated as [FORMULA] within 1  kpc from the center. The timescale of the ring formation is a few [FORMULA] years. This time is somewhat shorter for rings forming around bulge-dominated galaxies and reaches up to [FORMULA] years for a disk-dominated model. The interacting galaxies are separated by a sufficiently large distance (more than 120 kpc) by the time of the ring steady-state settling, so that there are no any direct evidences of interaction between the galaxies and the ring-host system as an isolated object.

The simulations have revealed an interesting feature which appears during the ring-forming process in different galaxian potentials. As a steady state sets in, ring sizes begin to diverge. The results for four runs are presented in Fig. 2.

[FIGURE] Fig. 2. Time 10 (after the pericenter passage moment) frame for 4 runs. The values of the bulge to disk mass ratio of the ring-host galaxy are shown in the upper right-hand corner of all frames. All frames display the orbital plane (xy -plane) spatial projection and measure 30 kpc per edge. The equatorial plane of the ring-host galaxy disk component (parallel to yz -plane) is perpendicular to that of the orbital motion.

Fig. 2 gives a final stage of polar-ring formation for all considered models. This is the orbital plane projection of the gaseous rings. The marked difference in the ring structure is the difference in ring size, which rises when the bulge mass decreases and ranges from about 7 kpc in diameter for bulge-dominated systems (see Table 1) to approximately 13 kpc for disk-dominated objects.

The ring radius is finally determined by the angular momentum of donor galaxy gaseous particles forming the ring with respect to the galaxy at the moment of the pericenter passage. As the impact parameters are nearly the same for all runs, the value of the total angular momentum of ring-forming particles is almost the same for all models. But one can see from Fig. 3 that the positions of particles with the same angular momentum are different for galaxies with different structures. Particles are closer to the center of galaxy if there is a more concentrated mass distribution - the bulge-dominated model. This implies, in particular, that under the same conditions rings forming around elliptical galaxies with a strong concentration of mass to the center (see, for instance, the curves marked as ([FORMULA]) and ([FORMULA]) in Fig. 3) should be less extended (on the average)  -  rather  internal  - than those around galaxies with a more gently sloping density profile (this fact was first mentioned in Sotnikova 1996).

[FIGURE] Fig. 3. The angular momentum of a test particle on a circular orbit versus the distance from the center of a galaxy in the equatorial plane of the disk for 4 bulge-disk models. The short, medium and long dash lines conform to [FORMULA], correspondingly; the solid line gives [FORMULA] relation for [FORMULA]. Two additional lines are shown as the bench-marks: ([FORMULA]) - for diskless model, and ([FORMULA]) - diskless model with more concenrated bulge ([FORMULA] kpc). The horizontal line marks the value of the mean angular momentum per ring particle for 4 runs.

In Fig. 4, the surface density profiles along the major axes of the rings are shown. One can see one more interesting feature of simulated rings: rings farther from the center of the galaxy are more extended.

[FIGURE] Fig. 4. Surface density profiles of rings. The short, medium and long dash lines conform to [FORMULA], correspondingly; the solid line - [FORMULA].

These results give the post factum justification of our impact parameter choice. Indeed, a closer passage results in less extended rings. The angular momentum of a gaseous particle at the moment of the closest approach of galaxies with respect the ring-host system is significantly affected by the term proportional to [FORMULA], where [FORMULA] is the ring-host galaxy velocity relative to the donor galaxy. For parabolic encounters, it leads to [FORMULA]. As the ring size monotonically decreases, the angular momentum decreases; a closer encounter gives a ring lying rather inside the galaxy. It is known that S0 galaxies possess some original gas, so the interaction between two gaseous systems would result in ring destruction. The further decreasing of pericenter distance leads to merging of the galaxies and total change of their structures. As to the more distant encounters, they do not allow capture of substantial amounts of gas, due to the sharp decrease of disturbing forces. We have repeated our runs with [FORMULA] kpc and have obtained that the mass of the resulting very diluted rings did not exceed [FORMULA]. Hyperbolic galaxy encounters are not efficient for ring formation, as the time of strong galaxy interaction is short. Let us note also that we discuss the properties of classic PRGs with optical star-forming rings. So we discuss the origin of relatively dense rings - with gas density large enough for star formation. This gives an additional restriction on the impact parameter, since the matter captured during the distant encounter will spread along a more extended orbit and will have lower mean density. Thus, galaxy encounters favourable to extended optical ring formation are extremely rare events. Such interactions must be between galaxies of specific types and within restricted geometry (polar encounters and a very narrow range of impact parameter).

We found in our simulations that, with the same impact parameter, the rings forming around bulge-dominated galaxies are less extended (by factor two) than the rings forming around disk-dominated galaxies. Therefore, having the same accretion history in the samples of elliptical and disk-type galaxies, we will have on average more extended rings around disk-dominated galaxies. This inference may explain in a quite natural manner the absence of extended polar rings around elliptical galaxies. It can also explain a difference in morphology of ionized gas in elliptical and S0 galaxies (Macchetto et al. 1996), assuming external origin of gas in these galaxies.

One can remark, however, that the obtained segregation of ring sizes (see  Fig. 2) is not as clear as the observed dichotomy (see  Table  1). The simple bulge-disk model giving internal rings for bulge-dominated objects fails to explain the existence of very extended (up to 30 kpc in diameter) polar rings around disk-dominated galaxies. Probably, there exists one more physical factor leading to the strong differences of the PRG structural properties.

3.2. Exploration of dark halo inclusion effect

As it was found from the observational data analysis, there exists for PRGs with extended rings a remarkable structural resemblence of host galaxies to late-type spiral galaxies. It is known that dynamical properties of late-type spiral galaxies are determined to a considerable extent by invisible massive halos (e.g., Freeman 1992). Hence, we can suppose that PRGs with extended rings possess a third global component (besides the bulge and the disk) - a dark massive halo. The existence of dark massive halos also follows from the analysis of ring kinematics (e.g., Schweizer et al. 1983, Reshetnikov & Combes 1994a). To account for the rotational velocities in the rings of UGC 7576 and UGC 9796 inside radii of 17 kpc and 21.4 kpc, respectively, one requires amounts of dark mass reaching 1.6 and 3 times the luminous masses (Reshetnikov & Combes 1994a).

What are the possible consequences of taking into account this structural component in the context of our investigation? As the gravitating mass of the primary galaxy increases, the orbital velocity of this galaxy relative to the donor object rises also. Then angular momentum arguments for a galaxy with such a structure lead to more distant orbits for captured particles.

We changed our bulge-disk model by adding a smoothed third component - a halo. The structure and the shape of dark halos in early-type galaxies appears to be, at present, an open question (Sackett et al. 1994, Combes & Arnaboldi 1996). For lack of unambiguous knowledge about the halo shape, we suppose the halo mass distribution to be spherical. As usual, halos are characterized by isothermal spheres over some radial interval. For simplicity, the folowing phenomenological cumulative mass profile is used to represent a halo in the present study:

[EQUATION]

where [FORMULA] is the total halo mass, [FORMULA] a cutoff radius, and [FORMULA] a core radius.

Our choice of halo parameters was determined by characteristic values obtained by Reshetnikov & Combes (1994a) from the ring kinematics investigation. The core radius [FORMULA] was taken as 9 kpc. The value of the cutoff radius [FORMULA] is somewhat arbitary and was taken to be equal to [FORMULA]. We choose this value to give a reasonable halo mass outside the radius comparable with [FORMULA]. We have let the dark mass to be equal to twice the luminous mass (bulge [FORMULA] disk) inside a radius of 15 kpc. The total mass of the bulge and disk have been reduced, that is, [FORMULA], and we let [FORMULA], [FORMULA] kpc, [FORMULA] kpc.

The geometry of the encounter was the same as in previous numerical experiments. Expecting the capture of gaseous matter on distant orbits and formation of a more diluted object, we have increased the total number of particles (up to 20 000) as well as their size h up to 375 pc to keep the gas treatment as a continuous medium.

The morphology of the new run is shown in Fig. 5.

[FIGURE] Fig. 5. Ring formation history for a galaxy with a massive halo. Dimensionless time (counted out from the moment of the minimum approach of galaxies) is given in the upper right-hand corner of all frames. The center of each frame coincides with the center of the ring-host galaxy. All frames show the orbital plane projection of the ring. The edge length of the frame corresponds to 80 kpc.

By the time of [FORMULA] after the pericenter passage, the captured gaseous matter is azimuthally smeared in a annular configuration around the galaxy. This structure gradually evolves into a closed ring. We have followed its evolution up to [FORMULA]. By this time, the ring was completly closed but not quite symmetric. According to Rix & Katz (1991) for a ring forming as a result of the gaseous satellite disruption, the smearing process takes up to seven orbital periods - up to 3-4 Gyrs or two times the final time of our run. The total mass gathered in the ring-like structure does not significally differ from that obtained in haloless runs. The ring has a total mass of about [FORMULA]. To explore the sensitivity of this value to the structural properties of the donor galaxy, we have constructed an additional model in which the spiral galaxy has its own halo. The parameters of this halo were taken the same as for the host galaxy. The mass of the captured gas turned out to be slightly smaller - [FORMULA]. (This value is somewhat small in comparison with that obtained from observations. We omit the discussion of this question, because this value is obtained for one set of orbital parameters and we did not investigate the role of initial conditions, impact parameter, and structure of the donor galaxy. It may be impossible to understand the formation of very massive rings in the frame of the pure accretion scenario and the merging event is needed.)

The most remarkable feature of this annular structure is its size. The ring material lies well outside the luminous material of the host galaxy. We estimated the diameter of the ring of about 30 kpc. This value is typical of the objects of the first group of PRGs (see Table  1). The annular configuration obtained is rather narrow. Its further long-lived evolution will lead not only to azimuthal smearing of the gas but also to flattening of the radial density profile, forming a disk-like structure. Two factors promote the formation of rings with extended density distributions - the viscosity (see Rix & Katz 1991) and the nonsphericity of the potential (see Fig.5 in this paper and Fig.1 in Katz & Rix 1992).

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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