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3. Discussion


The main properties of the whole sample galaxies are presented in Table 2. The sample includes systems in a wide range of apparent inclinations, from nearly face-on objects (NGC 2217, tex2html_wrap_inline1800) to galaxies seen almost edge-on (NGC 4442, tex2html_wrap_inline1802). These galaxies have been selected from RC3 among the SB0 with tex2html_wrap_inline180413 and tex2html_wrap_inline1806.

As said in the introduction, our main purpose is to collect and make available a wide set of data on stellar kinematics of barred S0 galaxies. This data archive could be very useful to produce in the future more realistic models of stellar bars. We can, however, discuss here the main properties and problems encountered in analyzing our data.

3.1. Peculiarities of ionized gas

In the sample of 14 galaxies considered in this paper only 4 show the presence of [OIII] or Htex2html_wrap_inline1808 emission lines: NGC 2217, NGC 4546, NGC 4684 and NGC 7079. This low percentage is in agreement with the fact that early-type galaxies does not show the presence of ionized gas with high frequency (Bregman et al. 1992). It is interesting to note yet that in all the cases where emission lines were detectable, the gas kinematics appears decoupled from that of the stars. In two cases (NGC 4546 and NGC 7079) the gas is rotating in a direction opposite to that of the stars (gas counter-rotation). This feature, revealed in dozen of galaxies (see Galletta 1996 for a review), is generally attributed to the accretion of gas from outside. An anomalous gas structure has been also detected in NGC 2217; there, the ionized gas seems to rotate in a warped disk whose inner part is perpendicular to the bar, while the outer part lies on the main galaxy disk. Finally, in NGC 4684 strong non-circular motions are observed, with ionized gas in a thick central disk from where two opposite filaments arose, with radial motions.

This peculiarity of gas motions is quite singular; due to the small number of objects involved, one cannot say if there is a connection between these peculiarities and the presence of a bar. The highly warped gas disk of NGC 2217 may have an internal origin. Friedly & Benz (1993) suggested that this gas disk may be modeled by vertical resonances driven by the bar.

Table 3: Deduced properties of sample galaxies

3.2. Waving rotation curve of stars in bars


In five of the galaxies the rotation curve along the bar, when folded about the center of symmetry, show a "waving pattern" (see Bettoni 1989 for a discussion of the feature). The galaxies are: NGC 2983 (tex2html_wrap_inline1834), NGC 6684 (tex2html_wrap_inline1836), NGC 7079 (tex2html_wrap_inline1838), NGC 4596 (tex2html_wrap_inline1840), and NGC 4643 (tex2html_wrap_inline1842). For NGC 4596, this deviation appears also in the data of Kent (1990). Other SB0s not included in our sample but where waving shape is present are IC 456 (Bettoni 1989), NGC 936 (Kormendy 1983), NGC 1543 and NGC 4477 (Jarvis et al. 1988). This deviation appears superimposed to the general galaxy rotation, and it is confined to the region dominated by the bar.

The waving pattern in the rotation curve has been always detected in galaxies seen at intermediate inclination (between tex2html_wrap_inline1844 and tex2html_wrap_inline1846 ). The spiral NGC 6744 is close to this empirical upper limit but has no waving pattern, while the SB0 NGC 7079, with apparently the same inclination, have it. The lack of the waving pattern effect in galaxies with low inclination suggests that the motions responsible of this peculiarity predominate on the galaxy plane and have no vertical components. On the other hand, inside edge-on galaxies this effect may be submerged by the predominance of the disk dynamics.

The observed amplitude of the waving pattern is quite low, tex2html_wrap_inline1848 30 km stex2html_wrap_inline1850 and projected on the galaxy plane corresponds to non-circular deviation of 30-50 km stex2html_wrap_inline1854 in radial direction. A theoretical interpretation of this feature has been recently presented by Wozniak & Pfenniger (1996) using self-consistent models. They conclude that the waving pattern is due to the presence of retrograde orbits whose amount in the galaxy can vary from 14 to 30%.

Table 4: Velocity dispersion trends with the radius measured along the galaxy major axis and along the bars

3.3. Velocity dispersion in bars

The central value of the velocity dispersion is indicated in Table 2 (click here), mediated in the innermost tex2html_wrap_inline1856. One goes from "warm'' systems, with tex2html_wrap_inline1858 250 km stex2html_wrap_inline1860, to quite "cold'' bulges, with values lower than 100 km stex2html_wrap_inline1862. Inside the galaxy, the shape of the velocity dispersion profiles vary from almost flat (e.g. NGC 4596) to sharply falling with increasing radius (e.g. NGC 3271). Table 4 (click here) indicates the observed trend of tex2html_wrap_inline1864 measured along the bar and outside the bar (typically along the galaxy major axis). From these value we note that there is an agreement between the trend measured along the bar and the global trend in the galaxy. When a plateau of velocity dispersion is present, it is detected also outside the bar. The only exception seems to be NGC 4371, with a quite constant bar velocity dispersion, compared with a tex2html_wrap_inline1866 decreasing with radius in the rest of the galaxy. There is, however, a tendency of tex2html_wrap_inline1868 profiles to be more slowly decreasing inside the bar than outside.

3.4. Deviation from circular motions

We can suspect that a non-asymmetric structure such as the bar should modify the global galaxy rotation, introducing non-circular motions. These deviations are observed in gas kinematics, both from HI- and optical emission- data, generating an S-shape of the zero-velocity line in the velocity fields (see Peterson et al. 1978; Peterson & Huntley 1980; Huntley 1978). In our galaxies, we are interested to test the amplitude of the deviations from circularity present in the stellar velocity fields. To this purpose, we adopted a simple model, as described in the following.

Every point on the plane of the galaxy is seen at a projected distance from the center equal to:
being tex2html_wrap_inline1882 the position angle on the sky formed by a line crossing the point and starting from the center. R is the true distance on the galaxy plane and tex2html_wrap_inline1886 the PA of the line-of-nodes. The angular distance of the bar from the line-of-nodes on the galaxy plane is then
with tex2html_wrap_inline1888 the bar position angle.

We can describe the observed rotation curve along the line-of-nodes with the equation:
where i is the galaxy inclination with respect to the sky plane and A, B parameters of the curve. The above formula was adopted by Brandt (1960) to describe the rotation law of a galaxy represented by a sequence of flattened spheroid. If the mean motions are symmetric with respect to the galaxy rotation axis, every rotation curve observed at a position angle tex2html_wrap_inline1896 will be described by

Here, tex2html_wrap_inline1898 is the projection factor, equal to:

Actually, the observed stellar rotation curves should differ from the intrinsic rotation law, because of the asymmetric drift and the integration of the stellar light along the line of sight. However, if the velocity field and the light distributions are symmetric with respect to the galaxy's axis, these effects also must be axially symmetric. In our case, we expect that the bar will generate an asymmetry of the potential that should reflect itself on the orbits as much as the bar is strong. On the other side, differently from gas, stellar orbits tend to fill all the energy levels in the space phase, and this may minimize the deviations.

We assumed then for every galaxy a mean rotation curve along the line-of-sight, as described above and we projected it at all the PAs tex2html_wrap_inline1900 corresponding to our spectra. The disagreement between this circular rotation and the observed curves give us an indication of the non-circular motions present in the galaxy. For practical use, the Brandt formula was rewritten in function of two observable quantities: the maximum rotational velocity tex2html_wrap_inline1902 and the radius tex2html_wrap_inline1904 at which this velocity is reached. The parametric expression of the observed rotation curves is then:

The model curves were then fitted to the observed ones, deducing tex2html_wrap_inline1906, tex2html_wrap_inline1908, tex2html_wrap_inline1910 and i. They are represented in figures from 2 to 7 as full lines, while the observed data are represented by full squares. The galaxy inclination was deduced also in different way, from the observed axial ratio of the disk, measured at 25 mag/tex2html_wrap_inline1914, reported in RC3 (de Vaucouleurs et al. 1991) and assuming an intrinsic axial ratio 0.25 (see Table 2 (click here)). Also the line-of-nodes was, as a first guess, assumed to be coincident with the PA of the galaxy's major axis, according to the idea that the intrinsic structure of the disk is oblate. The derived properties under the assumption of circular motions are reported in Table 3 (click here). The angle tex2html_wrap_inline1916 between the bar axis and the line-of-nodes is reported in Table 3 (click here), while the intrinsic bar diameter tex2html_wrap_inline1918 is reported in Table 2 (click here).

Looking at the two tables, we can see that the differences between the inclination deduced in photometric way (Table 2 (click here)) and that obtained by fitting the mean rotation (Table 3 (click here)) do not differ more than tex2html_wrap_inline1920, a probable random deviation taking into account the uncertainties on the observed and intrinsic axial ratios of the galaxy. Also the angle tex2html_wrap_inline1922 do not differs more than tex2html_wrap_inline1924 from the PA of the apparent galaxy major axis, giving support to the hypothesis of oblateness (axial symmetry) of the sample galaxies. The only exception is NGC 6684, where a deviation of tex2html_wrap_inline1926 must be attributed to the presence of a triaxial bulge (Bettoni & Galletta 1988).

The existence of non-circular motions can be checked by means of the circular velocity represented by the model fit and the true velocities observed at different position angles. According to the tests made by Kormendy (1982b) in the case of NGC 936, we considered the following parameters: 1) the presence of deviations with respect to the systemic velocity along the disk minor axis; 2) the ratio tex2html_wrap_inline1928. The parameter tex2html_wrap_inline1930 represents the velocity difference between the major axis rotation curve, projected on the bar position angle, and the bar velocity curve. tex2html_wrap_inline1932 is the projected circular velocity expected along the bar. Positive values of both tests indicate the presence of elongated orbits.

In the first test, we found velocity deviations, in some case quite irregular, along the minor axes of seven galaxies (see figures). Among the remaining cases, NGC 2217, NGC 4267, NGC 4371, NGC 4643 and NGC 6684 exhibit bars close to the apparent major axis (as indicated by the angle tex2html_wrap_inline1934 in Table 3 (click here)) and the observed motions reflect the stellar motions inside the bars. Only NGC 4754 does not show appreciable deviations from systemic velocity along its minor axis.

The result of the second test is indicated in Table 3 (click here). Most part of deviations are due to the waving pattern presents along the bar, indicated in the last column of the table and described in the previous Sect. 3.2 (click here). The amount of the remaining deviation from circular motions is from 10% to 20%. Only NGC 4371, with a bar coincident with the galaxy minor axis, does not show deviations. Finally, in the almost edge-on galaxy NGC 4442 the predominant deviation is caused by a cylindric rotation, with equal velocities at different heights from the equatorial plane.

In conclusion, the stellar velocity field shows, in the bar region, deviations from circular velocities lower than 20%. In the outer regions and outside of the bar, the motions seems to be circular.


This work has been partially supported by the grant "Astrofisica e Fisica Cosmica" Fondi 40% of the Italian Ministry of University and Scientific and Technologic Research (MURST).

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