It has been known for more than three decades that axes of isophotal contours in many elliptical galaxies rotate (e.g. Liller 1960; Bertola & Galletta 1979; Nieto et al. 1992). Such twists can be explained either by intrinsic misalignement of isophotal surfaces (which are ellipsoidal in the first approximation) or by projection effects: in the latter case, the aligned ellipsoids must be triaxial and their eccentricity must vary with radius at the same time. Models (e.g. Madejsky & Möllenhoff 1990) show that even a moderate triaxiality can produce a considerable twist if one looks at an elliptical galaxy under oblique view.
Figure 1: Ellipticities (1-b/a), position angles (PA) and contour plots for a single 2D bar: a) Face-on view - full lines in plots of radial profiles (dotted lines indicate the same quantities measured after the bar is first analytically projected and then numerically deprojected back; see the text), b) Projection with and , c) Projection with and
It is natural to expect the isophote twist due to projection effects also in the case of galactic bars since they are obviously triaxial and their eccentricity shows a radial variation (as seen in galaxies viewed face-on). In looking for a correlation between central activity and isophote rotation, one should separate the intrinsic twists, related to dynamics, from mere projection twists.
The solution of such a task is outside the scope of this paper. Nevertheless we would like to initiate the discussion on that topic by several simple illustrative examples of projection effects on artificially constructed single and double bars.
Figure 1 (click here)a shows the ellipticity and PA profiles of a face-on viewed 2D bar, whose isodensity contours are perfect ellipses with axial ratio a/b varying radially from 1 to 3. After projecting (with only moderate inclination, ) about the line with (i.e. coinciding with the bar minor axis), the along the bar becomes two-fold, with two plateaus separated by a sharp -transition at which the ellipticity falls locally to zero (Fig. 1 (click here)b). With the same I but , one obtains a gradual twist of (Fig. 1 (click here)c).
A 2D double barred system with the inner component perpendicular to the outer one is presented in Fig. 2 (click here)a: the large-scale bar is the same as in the above case; the small one is 7 times shorter and its axial ratio a/b varies linearly from 1 to 2. The projection with and is shown in Fig. 2 (click here)b: the local ellipticity maximum corresponding to the secondary bar nearly disappears; the PA varies along both primary and secondary bars.
Finally, a system of two parallel bars (with the same parameters as above) is shown in Fig. 3 (click here)a. The projection by about the minor axis of both bars (i.e. ) results in an illusion of two perpendicular bars (Fig. 3 (click here)b).
The above examples clearly demonstrate that the projection is a crucial factor for classifying twists and double bars. To disentangle projection effects from intrinsic distortions, one can try to deproject the observed images, making use of two advantages spiral galaxies have with respect to ellipticals: a) they are fairly two-dimensional except the bulge region and b) the inclination I and position angle can be deduced from the shape of the outer disk under the assumption that it is intrinsically circular. A two-dimensional body with known I and can be deprojected without ambiguity: if conditions a) and b) were strictly met, the problem would be solved. Nevertheless many complications exist: the bulge is clearly three-dimensional; the primary bar may also be significantly thickened close to the center due to the scattering on vertical resonances (e.g. Combes et al. 1990); the secondary bar, when it exists, is confined to that bulge-bar 3D region; the outer disk has not necessarily the intrinsic circular shape which can result in substantial errors in determining I and .
Figure 2: Ellipticities (1-b/a), position angles (PA) and contour plots for a a 2D double bar - bars perpendicular: a) Face-on view, b) Projection with and
Figure 3: Ellipticities (1-b/a), position angles (PA) and contour plots for a a 2D double bar - bars parallel: a) Face-on view, b) Projection with and ,
In this paper, we have deprojected, under the assumption of two-dimensionality, galaxies with inclination lower than , and we present, in the appendix, the deprojected radial profiles together with the projected ones. Since outer disks are usually located outside our images, we have used disk inclination and position angles quoted in the Lyon-Meudon Extragalactic Database (LEDA, Paturel et al. 1989).
The deprojection can be done in two ways: either the image is first deprojected and then a new ellipse fitting is carried out or the ellipses fitted to the projected image are deprojected analytically. The two approaches are not completely equivalent because of discreteness of the detector array and because the isophotes are not perfect ellipses. Our experiments have shown that the first method is less reliable: after deprojecting the image, one has to interpolate to get intensity at pixel positions which causes numerical errors resulting in spurious variations of ellipticity and position angle in the subsequent ellipse fitting. We show this effect by dotted curves in Fig. 1 (click here)a: they correspond to the bar which is first projected with and (Fig. 1 (click here)c) and then deprojected back to the face-on position. Both ellipticity and position angle profiles significantly differ from the correct ones (full lines) inside , possibly giving illusion of a small secondary bar. This numerical error is expected to occur in regions with high density contrasts, e.g. close to the center or at the edges of bars. Therefore we have preferred the second approach.
Whether the deprojected profiles are meaningful or not, depends on how closely individual galaxies fulfil the conditions a) and b) given above. Clearly, in regions with non-negligible thickness, the error resulting from the deprojection will grow with the galaxy inclination. As can be seen from the profiles of observed galaxies, the deprojection does not look reasonable in the bulge region if I exceeds . Additional error is introduced by uncertainties in I and .
Being aware of big uncertainties in the deprojection procedure, we do not rely on it to draw firm conclusions about the nature of a twist but use it only as a secondary help: if a double bar (or gradual twist) seen on the projected image remains after deprojection, we consider the probability of its existence to be strengthened; if it disappears, while , we take it to be a projection effect; on the other hand if a double bar structure appears only after deprojection, we do not classify it to be a double bar.