In many galaxies the kinematical major axis changes its position angle with radius, while the velocity field remains predominantly symmetric. In Fig. 1 most of the galaxies in the upper part of the diagram exhibit this characteristic; their velocity fields are symmetric about the centre. As discussed briefly in section 2, we think that two different phenomena can both result in such a central symmetry : 1) oval-shaped structures in an otherwise planar disk, e.g. bars, and 2) warping of the plane of the disk. For simplicity we call the first phenomenon "oval distortion" and the second phenomenon "kinematical warp".
It is sometimes difficult to decide on the basis of the velocity field alone whether a galaxy has an oval distortion or a kinematical warp. In the case of an oval distortion we think that the gas moves in more or less elliptical orbits in the plane of the disk in the case of a kinematical warp the gas in the outer parts moves in circular orbits in different planes. The projection of a galaxy on the plane of the sky makes the distinction between the two situations ambiguous, but additional criteria can be found. One criterion comes from the structures seen on optical photographs, another one from the HI column density distributions. We favour an interpretation in terms of an oval distortion if on optical photographs structures can be found whose major axes do not align with each other or with the kinematical major axis. Usually these structures are found in the inner parts. We favour an interpretation in terms of a kinematical warp if the.kinematical major axis in the inner parts is well aligned with the one of the optical disk, but changes its position angle in the outer parts. Along with this distinction goes one which is less clear: for oval distortions the kinematical major axis is not perpendicular to the kinematical minor axis, for kinematical warps the axes are perpendicular. Further, it turns out that for galaxies with an oval distortion the HI distribution shows sometimes distinct features in the outer parts, which are coincident with optical structures; for galaxies with a kinematical warp the HI distribution continues smoothly outwards, usually beyond the optical emission visible on sky-limited IIIaJ-plates, and changes its major and minor axis in the same way as does the velocity field.
Obviously a warped barred spiral can fit most of the criteria for both phenomena. Usually in the inner parts the criteria for an oval distortion are met, while in the outer parts such a galaxy exhibits the features of a kinematical warp. We then simply have to accept that both phenomena occur. For most of the galaxies in Fig. 1, however, one or the other phenomenon dominates.
In the remainder of this section we concentrate on galaxies with an oval distortion. We present a brief summary of the observations, then we present a simple kinematical model, and finally we draw a number of general conclusions.
The following galaxies are thought to have an oval distortion:
NGC 3198. This galaxy has a small central bar, which is not resolved by the WSRT beam. The kinematical minor axis is not perpendicular to the kinematical major axis. In the inner parts the residual velocity field has a central symmetry: the sign of the residuals is the same in opposing quadrants.
NCC 4236. The kinematical major axis is not perpendicular to the kinematical minor axis. The bar is roughly aligned with the major axis of the optical disk.
NGC 3359. A hint of non-circular motions with central symmetry can be found in the observations of this barred spiral, but the resolution is rather low.
NGC 5383. Data from Peterson et. al. (1977) and Sancisi et al. (1978) indicate large deviations from axial symmetry associated with the bar. Again central symmetry is maintained. Van der Kruit and Bosma (1978) find two faint outer arms, which coincide with HI emission.
NGC 4258. This galaxy is known to have anomalous H-arms and a peculiar radio continuum structure (see Van der Kruit et al., 1972). The residual velocity field has a central symmetry in the inner parts. The HI distribution closely coincides with the optical arms, especially in the outer parts. If no peculiar arms were present we certainly would have regarded this galaxy as the type example of an oval distortion in a spiral galaxy.
NGC 4151. The HI emission in the outer parts coincides with the faint outer optical arms. The velocity field indicates- that the, central elliptically shaped structure is a fat bar. The nuclear disk (size 30") has an orientation similar to that of the outer regions.
NGC 4736. This galaxy is rather complex. In the inner parts (< l' radius) the kinematical major axis seems to be aligned with the optical structure. Beyond 1' the kinematical major axis changes position angle. The optical structure between 1' and 3.5' radius has a position angle of the major axis which differs from both the one in the inner parts and the one of the velocity field. Outside 3.5' the optically bright area is surrounded by an outer ring with yet another orientation, which connects at two sides with the structure inside 3.5'. HI emission has been found in this ring, as well as in the main optical parts, but not in the "gaps" in the light distribution between the ring and the main body.
IC 342. The HI extent along the kinematical minor axis in the map of Rogstad et al. (1972) is larger than the extent along the major axis. In the inner parts the kinematical major and minor axis are not perpendicular to each other. Harten's WSRT data (Harten, priv. comm.), suggest also that an oval distortion could be present in the inner parts, though the outer parts might be warped.
The above mentioned galaxies have some characteristics in common: they have been classified as SB, SAB, RSA, or PSA by De Vaucouleurs et al. (1976). This suggests that the central symmetry in the velocity field is a natural consequence of an oval or bar distortion in the potential field of the disk of the galaxy. As a crude attempt to describe the velocity field of a galaxy with an oval distortion we have made a simple kinematical model, based on a suggestion by Dr. A. Toomre (note that this model was made in 1974, before more elaborate, and physically better founded, models were constructed by others. It should not be-taken too seriously; and is only meant as a guideline to determine the geometry of the system).
Consider a circular orbit at radius r in the plane of the galaxy. We deform this circle into an ellipse by stretching it along the x-axis and squeezing it along the y-axis. The Cartesian co-ordinates (x, y) of a point P on the circle transform to (x', y') with:
a is the semi-major axis of the ellipse, b is the semi-minor axis. The circular velocity vector at P is (u, v) = (- V sin, Vc cos), with = arctan y / x and Vc the amplitude of the circular velocity at radius r. We can eliminate and transform this vector also with equations (1). Then we give the resulting ovals a constant pattern speed p, but we reduce the angular velocity at each mean radius to compensate for this. We then have:
where 2 = x2f2 + y2 / f2 and Vc is taken at . Finally we apply the correction for the orientation to the sky plane and calculate the radial velocity:
is the position angle of the major axis, and i is the inclination of the galaxy.
We have calculated models for a given rotation curve, Vc(r), and various values of , i, f and p. We can constrain a number of free parameters as follows: For.the projected ovals in the inner parts of a galaxy we assume that they outline the isophotes. This gives a relationship between the observed apparent axial ratio, g, and the parameters f, and i. The inclination i follows from the outermost observable isophotes, since at large radii the effect of an oval distortion in the inner parts must be small and hence the orbits nearly circular again. p can be chosen such that corotation occurs in the outer parts of a galaxy.
We have tried to fit the observations with this model. A reasonable fit is shown in Fig. 2a. The most significant results are that the position angle of the kinematical major axis changes with radius, and that there is a misalignment between the kinematical minor axis and that of the projected oval. In Fig. 2b we show a number of equipotentials for a homogeneous triaxial spheroid in its z = 0 plane (MacMillan, 1958), projected onto the sky with the same and i, in a frame rotating with p. The dimensions of the spheroid are a, af-2, and 0.la, in x, y, and z, respectively. The scale of this plot is arbitrarily adjusted,since we are only interested in the geometry of the projection. In Fig. 2c we reproduce the velocity field, superimposed on the optical picture, of NGC 4736. The similarities with Fig.'s 2a and 2b are encouraging.
Figure 2. a Velocity field from the model described in text. The contour interval is 20 km s-1, starting at -110 km s-1; the contour labeled 0 km s-1 has been added. The cross represents the major and minor axis of the projected inner ovals. b A few contours of the equipotential field described in text. c Velocity field of NGC 4736 superposed on a photograph taken by Arp.
We have made a similar calculation for a galaxy seen more edge-on. The results are shown in Fig. 3, where they are compared with the observed pictures of NGC 4258 (Van Albada and Shane, 1976). Again we can explain a number of characteristics of this galaxy in terms of our simple model. The pattern of residual velocities shows the central symmetry described earlier. The kinematical minor axis is not perpendicular to the kinematical major axis. Moreover, the "gaps" in the optical pictures and also in the HI distribution, especially visible in NGC 4258, NGC 4151 and NGC 4736, appear to be associated with Lagrangian points in the potential field in the rotating frame.
Figure 3. Left, top: the second contour (at a level of 1.2 × 1021 atoms cm-2) of the HI distribution in NGC 4258 (Van Albada and Shane, 1976); bottom: observed radial velocity field of NGC 4258. Right, top: a few contours of the equipotential field as described in text; bottom: model velocity field with the same contours as the observed one. The HI picture and the velocity field are not on the same scale.
Sanders and Huntley (1976) have calculated periodic orbits in the presence of an oval distortion of the potential field of a disk. They find that the orientation changes with 90° when a resonance region is crossed. It is striking that in NGC 4151, NGC 4258, NGC 4736, and NGC 5383 several annular zones can be distinguished, each with a different position angle of the major axis. The position angle of the major axis in the inner parts (nuclear disk) is similar to the one in the very outer parts. In between is a region with a major axis which, after correction for spatial orientation, is roughly perpendicular to that in the outer parts. Sanders and Huntley (1976) have also calculated the gas response to an oval distortion, and find that shocks can quite easily be formed in the outer parts. We suspect that subsequent star formation might lead to narrow arms such as those in NGC 4151 and NGC 4258.
The discussion above suggests that oval distortions in the potential field are indeed present in several of the galaxies in Fig. 1. In our opinion galaxies like NGC 1068 (Hodge, 1968), NGC 1566 (De Vaucouleurs, 1973) and NGC 2903 (Simkin, 1975) can be described with a similar picture. Note that the prominence of the distortion changes with morphological type. In cases like NGC 4736 and NGC 1068 the faint outer arms are faint and touch each other after a turn of 180° and have the appearance of a "ring". In cases like NGC 4151 and NGC 1566 they do, not form a closed ring but a "pseudoring". In galaxies like NGC 4258, NGC 5383 and NGC 2903 the outer arms remain open. Along this sequence the oval (bar) becomes progressively smaller and spiral structures within the ovals less prominent. If we continue these trends even further we find galaxies like NGC 3198 and NGC 3359 with even smaller bars. The appearance of these galaxies is dominated by the arms.
A few other topics might be of some relevance in connection with oval distortions. The first is that in several systems the region of the oval (bar) coincides roughly with the extent of the lens in the radial luminosity profile. This seems to be the case for NGC 5383 (Van der Kruit and Bosma 1978), NGC 1068 (profile based on Hodge, 1968), NGC 1291 (De Vaucouleurs, 1975), NGC 4258 (Capaccioli, 1973) and M83 (Freeman, 1970). We do not know whether there is a unique correlation between ovals and lenses. For several systems illustrated in the Hubble Atlas (Sandage, 1961) two sets of spiral arms are found, each in a different part of the galaxy. Almost invariably Sandage comments that there is a sudden drop in luminosity between the two regions. (see for example NGC 5248, where the position angle of the inner parts differs from that of the outer parts, Burbidge et al., 1962). These galaxies might have ovals too.
We also emphasize that,on top of an oval distortion other non-axisymmetric distortions like spiral arms can occur. In NGC 4736, for example, the dust pattern in the inner parts (Lynds, 1974) outlines a spiral pattern, which can even be fitted with a model like one of those from Roberts, Roberts and Shu (1975), with a pattern speed of about 100 km s-1 kpc-1. Inside the inner Lindblad resonance a barlike structure (size ~ 20") is seen on short exposure photographs, see Chincarini and Walker (1967). This barlike structure might be associated with resonance effects like described by Contopoulos (1973). The dominant HII-regions in NGC 4736 occur in a "ring"-like structure which can only be partly fitted with the spiral pattern discussed above, and for which Van der Kruit (1976) claims that it expands. This illustrates the enormous complexity of this galaxy.
Another question is the relation between barred spirals and Seyfert galaxies: it has sometimes been stated that Seyfert galaxies have outer ring-like structures and therefore might be barred spirals (see De Vaucouleurs (1976) for a discussion). Our findings about the outer ring in NGC 4736 and other galaxies seem to strengthen this argument. Note, however, that the Seyfert phenomenon is associated with the nucleus. The frequency of occurrence of galaxies with outer rings as function of Hubble type indicates a preference towards the earlier types, hence those with presumably a high central mass concentration, but it is not clear how this is related to the presence of an oval distortion.
In conclusion, we suggest that oval distortions in the potential field of the disk of a galaxy manifest themselves in the velocity field by causing a centrally symmetric deviation from circular motion and a misalignment of the kinematical major axis of some of the structures seen on optical photographs. Oval distortions are rather common; they are probably related with some of the more complex phenomena in spiral galaxies like bars, two sets of arms, lenses etc. A detailed study of a sample of galaxies classified as SB or SAB galaxies, and also RSA and PSA galaxies, might prove fruitful in shedding more light on these phenomena.