4. THE STRUCTURAL PROPERTIES OF BARS

Although bars are ubiquitous and might account for a significant fraction of a galaxy total luminosity, only recently studies dedicated to a more detailed modelling of the structural properties of bars started to come out more often (see e.g. [54, 78, 36]). This is partially because of the significant increase in complexity when one includes another component in the structural modelling of disc galaxies. These studies usually make use of the full data contained in 2D galaxy images, rather than only 1D surface brightness profiles, in order to obtain more accurate results. The rise in complexity thus usually means that automated procedures become much less reliable. In [36], BUDDA v2.1 [23] is used to individually fit galaxy images with model images that include up to three components: a Sérsic bulge, an exponential disc and a bar. Bars are modelled as a set of concentric generalised ellipses [8], with same position angle and ellipticity:

(1)

where x and y are the pixel coordinates of the ellipse points, a and b are the extent of its semi-major and semi-minor axes, respectively, and c is a shape parameter. Bars are better described by boxy ellipses (i.e. with c > 2). The surface brightness profile of the model bar is described as a Sérsic profile, as bulges. The Sérsic index of bars n_Bar is often in the range 0.5-1, with lower values representing flatter profiles. Another bar parameter fitted by the code is the length of the bar semi-major axis L_Bar, after which the bar light profile is simply truncated and drops to zero.

Similar fits were individually done to 1000 galaxies in a sample carefully drawn from the Sloan Digital Sky Survey (SDSS - see [42]). The sample spans from elliptical to bulgeless galaxies, with stellar masses above 10¹⁰ M (k-corrected z-band absolute magnitudes -20 AB mag), at a typical redshift of 0.05, and includes 300 barred galaxies. All galaxies are very close to face-on (axial ratio 0.9) and do not show morphological perturbations, thus assuring that dust extinction is minimised and that the sample is suitable for image decomposition. This also avoids the uncertainties in obtaining deprojected quantities. Fits were done in g, r and i-band images. The distributions of several bar structural parameters obtained in this work from the i-band images are shown in Figs. 1 and 2.

Figure 1. Distributions of bar structural parameters obtained from 2D bar/bulge/disk i-band image decomposition of 300 barred galaxies in the Sloan Digital Sky Survey (SDSS). The sample spans from lenticular to bulgeless galaxies at a typical redshift of 0.05, and excludes galaxies with stellar masses below 1010 M. From top to bottom and left to right: bar ellipticity, bar length (semi-major axis in kpc - H0 = 75 km s-1 Mpc-1), bar length normalised by disc scalelength h, and bar length normalised by r24 (the radius at which the galaxy surface brightness reaches 24 mag arcsec-2 in the r-band). Noted at each panel are the median and standard deviation values of the corresponding distribution, as well as the mean 1 error of a single measurement, when available. Bin sizes are 1-2.

Figure 2. Same as Fig. 1, but for bar-to-total luminosity fraction, bar Sérsic index, and bar boxyness [parameterised as c - see Eq. (1)].

Models of bar formation and evolution should be in agreement with these results. Conversely, models where bar properties are imposed can use these results as a guide in the adjustment of the bar properties. One must note, however, that, due to the relatively poor spatial resolution of the SDSS, these results are biased against bars shorter than L_Bar 2-3 kpc, typically seen in very late type spirals (later than Sc [25]). These results are thus representative of the prototypical, bonafide bars seen mostly in early type spirals (earlier than Sc) and lenticulars. Interestingly, the median bar ellipticity is 20% higher than the value found via ellipse fitting to galaxy images in [59]. This is exactly what was predicted in [36] as ellipse fits systematically underestimate the true bar ellipticity due to the dilution of the bar isophotes by the rounder, axisymmetric light distribution of bulge and disc. When fitting ellipses to galaxy images one does not separate the contributions to the total galaxy light distribution from the different components, but this is done in image fitting with different models for each component. This shows that results based on the ellipticity of bars measured via ellipse fitting should be considered with this caveat in mind. For instance, there is an indication from ellipse fits that, for faint galaxies, disk-dominated galaxies have more eccentric bars than bulge-dominated galaxies [9]. It it is not clear, however, if this result holds if the axisymmetric light contribution from bulge and disk is taken into account. Figure 1 also shows that most of these bars have a semi-major axis between 3 and 6 kpc (but with a long tail to longer bars), and that bars do not extend further than 3 times the disc scalelength h, or 1r₂₄ (the radius at which the galaxy surface brightness reaches 24 mag arcsec^-2 in the r-band - see also [29]). Figure 2 shows that a typical bar is responsible for 10% of the total galaxy light, and has a quite flat luminosity profile. Interestingly, one also sees that, indeed, bars have very boxy shapes.

These data also reveal interesting correlations. The left-hand panel in Fig. 3 shows a correlation between bar ellipticity and boxyness, i.e. more eccentric bars are also more boxy. Both quantities contribute to the strength of the bar. Thus, the product of both, × c, can be used as a measure of bar strength. The right-hand panel shows that the effective radius of the bar, normalised by r₂₄, is correlated with × c (see also [65]). Thus, longer bars are thinner and stronger, as expected from the theoretical models mentioned above. The left-hand panel in Fig. 4 shows that the length of the bar, normalised by r₂₄, is correlated with the bulge-to-total ratio of the galaxy. Considering these results and the theoretical expectations together, they indicate that bars grow longer, thinner and stronger with age, as a result of angular momentum exchange, and that bars have had more time to evolve in galaxies with more massive bulges. In fact, the right-hand panel in Fig. 4 shows that the normalised effective radii of bars and bulges are correlated (see also [6]). Hence, the growth of bars and bulges seems to be somehow connected. Through different paths, these conclusions are also reached by others [87, 27]. A more thorough analysis of these data will be published elsewhere.

Figure 3. Left: correlation between the bar ellipticity and boxyness. Right: correlation between the effective radius of the bar normalised by r24 and the product of the bar ellipticity and boxyness.

Figure 4. Left: correlation between the length of the bar normalised by r24 and the bulge-to-total ratio of the galaxy. Right: correlation between the effective radius of the bar and the effective radius of the bulge, both normalised by r24. The top panel shows all bars, the middle panel shows bars with ellipticity < 0.7, and the bottom panel shows bars with ellipticity 0.7 (which, interestingly enough, include several outliers). The solid lines are a fit to the data in the middle panel.