4 Radial distributions of detected bright lumps

4.1 Number distribution

In Fig. 2 we show for all galaxies of the sample the binned radial number distributions of the detected bright lumps. Histogram bins correspond to concentric elliptical annuli with semi-major axes successively growing by half a scale length $R_{\rm d}$ and shown out to 8 $R_{\rm d}$; the last bin included in the panels comprises all the detected lumps at radii larger than 7.5 $R_{\rm d}$. The number of counts per bin, $N_{\rm norm}$, is normalized by the largest bin value within 7.5 $R_{\rm d}$; the largest bin value as well as the total number of counts are printed within each panel. This kind of normalization was imposed to minimize the effects of different resolutions and seeing conditions for the following inquiry.

The accumulated radial number distribution $N_{\rm norm}^{\rm total}$for the lumps of all the galaxies, i.e. summing up all the profiles of Fig. 2, is shown as histogram in Fig. 4. The bright-lump distribution of many galaxies thrown together is represented by a radius-weighted exponential distribution that is indicated by the solid line obeying $N(R)\propto R\exp(-R/R_{\rm l})$ with $R_{\rm l}=0.86 R_{\rm d}$ (cf. Sect. 4.2). This basically reflects the exponential light profiles of dwarf irregular galaxies in general with, however, a slightly shorter scale length than is seen for the B-band continuum light (dashed line). Note that bright lumps or star-forming complexes not only are found way out to large radii, but that they constitute a nice tail in the radial number distribution out to at least six optical scale lengths, a point we come back to in the discussions of Sect. 7.

The similar exponential structure for two components of the disk is comparable to H$\;$II region distributions in other types of exponential-disk galaxies. In intermediate-type spirals (Athanassoula et al. 1993) and in irregular galaxies (Hunter et al. 1998) the azimuthally averaged radial distribution of H$\;$II regions follows the stellar light distribution as well. In our dwarf irregular galaxies bright lumps are exhibiting this same behaviour, indicating them to be representative for the distribution of H$\;$II regions, as expected. We can put this statement on a still firmer basis tracking down also the radial number density distribution of the bright lumps.

Figure 2: Radial number distribution of bright lumps in 72 dwarf irregular galaxies, based on B-band images. Bin width is 0.5 $R_{\rm d}$; the rightmost bin sums all counts beyond 7.5 $R_{\rm d}$. For each galaxy the counts are normalized by the highest bin value. In each panel the galaxy name, the highest bin value, and the total number of counts are given.

Figure 3: Radial number density distribution of bright lumps in 72 dwarf irregular galaxies, deduced from the number distribution shown in Fig. 2. In each panel the galaxy name and the distribution's approximate exponential-fit scale length are given (in units of B-band scale lengths; evaluated only if there is a total of at least 10 detected lumps, and represented by straight solid lines).

4.2 Number density distribution

Radial number density distributions for the bright lumps are obtained by dividing the number of counts in a given bin by the surface of the corresponding elliptical annulus (cf. Hodge 1969; Athanassoula et al. 1993). The bright-lump number density profiles for our galaxies are shown in Fig. 3. Solid lines represent exponential fits for galaxies with a total of at least 10 detected lumps; their scale lengths $R_{\rm l}$ are given in each panel in units of B-band continuum light scale lengths $R_{\rm d}$ and are listed in Table 2. For the whole sample we find a mean of

$${<R_{\rm l}/R_{\rm d}>} = 0.86\pm 0.06, \;\;\sigma = 0.41.$$ (1)

Thus, while the number density distributions of bright lumps are - at least partially - also exponential, their slopes are on average steeper than those for the density distributions of the underlying stellar light. We note that this very same quantitative behaviour was also observed for the number density distribution of H$\;$II regions in intermediate-type spiral galaxies: Athanassoula et al. (1993) found ${<R_{\rm H~II}/R_{\rm d}>} = 0.8$, $\sigma = 0.4$.

Concluding this subsection we state that dwarf irregular galaxies show azimuthally summed-up bright-lump profiles that are quantitatively comparable to those of H$\;$II regions in exponential-disk systems. Thus, as expected, bright star-forming complexes largely represent H$\;$II regions. In particular, the scale lengths for H$\;$II regions, for bright lumps on B-band images, and for B- and R-band continuum light images (Parodi et al. 2002) obey on average the ratio equation $R_{\rm HII}$:$R_{\rm l}$: $R_{\rm d}(B)$: $R_{\rm d}(R)\approx0.8$:0.9:1.0:1.1, i.e. the older the underlying population the larger the scale lengths. This general trend in star-forming dwarfs has been observed before, and we discuss some implications in Sect. 7.

4.3 Concentration index

The concentration index CI of a galaxy is a convenient parameter to quantify galaxy morphology of low- and high-redshift galaxies. It compares the light content for different radial intervals. Various definitions have been used, none of which takes into account the exponential-disk constraint. It is, however, a trivial observation that in scale length-versus-luminosity diagrams the scatter in scale length around the mean relation correlates with the concentration index. To have a scale length-independent concentration index we will explicitely factor in this underlying disk feature. Following Heller et al. (2000) for the sake of comparision, the concentration index CI(R) is taken as the ratio of the flux or, in our case, the number of complexes within an elliptical aperture of semi-major radius R/2, i.e. from the inner part of the galaxy, to the flux or number of complexes from its outer annulus with inner and outer semi-major radii of R/2 and R, respectively. Opposite to Heller et al. (2000) - who suspect linear radial distributions of the (flux from) star forming regions - we do not bring the two numbers to an equal-area basis by dividing the outer number of lumps by a factor of three. Instead, we want to relate the measured lump CI to the corresponding one for an assumed underlying radial number distribution $N(R)\propto R\exp(-R/R_{\rm d})$ (or, as it is more adequate for lumps, $\propto$ $R\exp(-R/R_{\rm l})$; see below). We thus normalize our galaxy concentration index with $CI_0(R) = \int_0^{R/2} N(r)$d $r / \int_{R/2}^R N(r)$dr. Expressing the total radius R in terms of the disk scale length $R_{\rm d}$, $R=x~R_{\rm d}$, one finds

$$CI_0(x) = \frac{\left[ 1-(1+x/2){\rm e}^{-x/2} \right]} {\left[ (1+x/2){\rm e}^{-x/2}-(1+x){\rm e}^{-x} \right]}\cdot$$ (2)

Thus our normalization constant CI₀(x) depends on the chosen outer radius R, or x, respectively, used to determine the concentration index. On a statistical footing the 72 dwarf irregular galaxies of our sample share the mean ratios $R_{\rm d}$: $R_{\rm eff}$:R₂₅: $R_{995} \approx 1$:1.5:3:5.1 in the continuum B-band light, where $R_{\rm eff}$ is the effective radius, R₂₅ is the 25th-mag/arcsec²isophotal radius, and R₉₉₅ is the radius equivalent to an aperture with 99.5% of the total flux. Theoretically, pure exponential-light decays yield $R_{\rm d}$: $R_{\rm eff}$:R₂₅: R₉₉₅ = 1:1.7:4.1:7.4, where the value for R₂₅ was interpolated using the observed mean ratios given above. The fact that our galaxies exhibit on average smaller radii is a consequence of the exponential-disk description being only an approximation; in particular, there is a large fraction of bright galaxies with central light cusps. The appropriate normalization constants corresponding to the above radii are then CI₀(1)=0.52, CI₀(1.5)=0.64, CI₀(3)=1.23, and CI₀(5.1)=3.01.

Figure 4: Total radial number distribution of bright lumps in 72 dwarf irregular galaxies. Binning in units of a tenth of a scale length; the contributions from each galaxy are normalized by the value of its highest-value bin (cf. Fig. 2). The dashed line follows a distribution $R~{\exp}(-R/R_{\rm d})$ as it is expected from the continuum light exponential disks. A better match is provided for a distribution with a mean lump scale length of $R_{\rm l}=0.86 R_{\rm d}$, plotted as solid line.

Figure 5: Normalized bright-lump concentration indices CIvs. galaxy rotation velocity. Pure exponential-disk distributions correspond to CI=1. Filled circles represent galaxies of our sample, triangles at panel upper boundaries stand for infinite CI values resulting from galaxies with no lumps detected in the outer annulus, and open symbols is data from Roye & Hunter (2000).

With this normalization we expect the concentration indices to be $CI\approx1$. Deviations from this canonical value provide information on the actual shapes of the profiles. For example, the median values of the concentration indices obtained by Heller et al. (2000) for their dwarf irregular (BCD + LSB) galaxy sample are - corrected to our normalization by adopting their limit R=R₂₅ and thus applying CI₀=1.23 - (4.23/3)/ 1.23 = 1.15 and (3.43/3)/1.23=0.93 for the H$\alpha$-flux and the continuum images, respectively. Thus BCD and LSB galaxies are rather well represented by exponential light profiles on large parts. However, the values for the BCDs typically lie above and those for the LSBs below these median values, reflecting the fact that the former galaxies are more actively star forming in the center regions than the latter. Returning to our sample, we arrive at values CI=0.94, 1.26, 1.32, 1.73 for outer radii R=2, 3, 4, 5 $R_{\rm d}$, respectively. These values typically being larger than one and even increasing with larger outer radii is due to the normalization used so far that was based on the continuum light scale length $R_{\rm d}$.

However, if the lump scale length $R_{\rm l}$ is used for the normalization instead of $R_{\rm d}$, one indeed recovers $CI\approx1$. This is shown in Fig. 5 where we have plotted the normalized concentration indices CI(x) of our galaxies for various aperture radii $R=x R_{\rm l}$ against their rotational velocities, now adopting the mean scale length $R_{\rm l}=0.86 R_{\rm d}$ found for the lumps in Sect. 4.2. Actually, one would prefer to adopt for each galaxy its particular lump scale length, but given the uncertainties in determining them, we are content with the mean value given in Eq. (1). Only galaxies with at least five detected lumps are included, leaving about 50  galaxies. Infinite values result for CIs in the case of no outer-annuli lump detections; in the plots they are included as triangle symbols with values fixed at CI=13. Data for the upper right panel is listed in Table 2. The panel's median CI values, plotted as dashed lines and ignoring the $CI=\infty$ cases, are 0.83, 1.24, 0.95, and 1.04. Lying all in the vicinity of one, this is consistent with the annulus-integrated exponential distribution for the summed lump number distribution seen in Fig. 4 out to large radii.

Thus, applying the analytic tool of the concentration index, we have again demonstrated that the radial distribution of star forming regions is non-linear but follows an annulus-integrated exponential distribution. This implies a non-uniform random spread of the star forming regions throughout the disk, which explains the discrepancy found by Heller et al. (2000) between the $CI_{\rm H\alpha}$ values for actual galaxies and the lower ones for simulated galaxies with random star formation region positions. Roye & Hunter (2000) pointed out an increased scatter of concentration indices for faster rotating galaxies of their sample. In the upper right panel, data from Roye & Hunter (2000), adopted to our normalization using CI₀(3.5)=1.53(assuming $R_{25}=3R_{\rm d}=3.5R_{\rm l}$) and with a median CI value of only 0.58, are plotted, too; however, we no longer see this effect in any panel with our larger sample.

4.4 Peak number distribution: Shear-enhanced star formation at work?

Figure 6: Left: peak number distribution of bright lumps in dwarf irregular galaxies, deduced from the number distributions shown in Fig. 2; included are only galaxies with peaks corresponding to at least three counts. The histogram for all the primary (i.e. highest) peaks is shown as bars bordered with thin lines; the dashed line is a Gaussian eye fit forced to peak at one scale length. The histogram for the primary peaks of those galaxies exhibiting secondary peaks as well is overplotted with thick lines; the shaded bars indicate the distribution of the corresponding secondary peaks. Note the pronounced signals at the center and in particular at two scale lengths. Right: peak number distribution for 20 simulated galaxies, once for galaxies with solid-body rotation (upper panel) and once for galaxies with a transition to strong differential rotation at two scale lengths (lower panel). The simulated galaxies are generated by means of a stochastic self-propagating star formation (SSPSF) disk model. As in the figure to the left, thick-lined bars represent primary peaks whereas shaded bars inform on secondary peaks. Note the reversed peak distribution.

For the sample as a whole the radial locations of bright lumps mirror the exponential intensity distribution of the underlying population. This is, however, only a rule-of-thumb. Individual galaxies may exhibit strong deviations from this mean statistical behaviour (cf. Fig. 2). For example, the primary peak in the radial number distribution (i.e. the bin with the highest value) appears not around one scale length, but is shifted to smaller or higher radii. The histogram for the radial distributions of primary peaks only is shown in Fig. 6, left (thin-lined bars). Only galaxies with peaks corresponding to at least three counts are included. The expected maximum of occurences of primary peaks around one scale length is clearly recovered; a Gaussian with a mean at one scale length and a standard deviation of 1.4 $\;R_{\rm d}$ is overplotted as the dashed line. However, another particular feature of individual lump number histograms is the frequent presence of a secondary peak that is lower than the main peak (instead of monotonicly smaller bin heights on both sides of the main peak); this is the case for about a third of our galaxies (cf. Fig. 2). The radii of primary peaks (with at least three counts) and secondary peaks are listed in Table 2; in a few cases of equal height peaks we refered to 0.1 $R_{\rm d}$-bin width number distributions to decide which of the peaks is the primary or the secondary one. While the primary peaks of those galaxies exhibiting a second, minor peak as well are crowded around one scale length (thick-lined bars), the corresponding distribution of secondary peaks reveals a pronounced maximum at about two scale lengths (shaded bars).

Is this excess of bright stellar complexes at radii larger than about two scale lengths a statistical fluctuation or is it a manifestation of some underlying mechanism? A candidate mechanism that deserves closer inspection is shearing due to differential rotation within the outer part of the disk. We thus discuss the physical plausibility for the influence of shearing on the generation of star-forming complexes within dwarf galaxies. Two questions will be adressed: first, is shearing in dwarf galaxies a viable mechanism? And second, can it account for the observed peak distribution?

A compilation of 20 high quality dwarf galaxy rotation curves by Swaters (2001) shows them to look much like those of spiral galaxies, with rotation curves rising steeply in the inner parts and flattening in the outer parts. In particular, most dwarf galaxy rotation curves start to flatten around two disk scale lengths, and no dwarf galaxy shows solid-body rotation beyond three disk scale lengths anymore. Concerning our observed occurence of minor peaks in the bright lump distribution, starting at and being most pronounced at about two scale lengths as well, we may wonder whether it is related to the transition from solid-body to differential rotation. Affirmative signals arrive both from theory and simulations. (i) Larson (1983) suggested that the SF rate increases with higher shear rate via the "swing amplifier'' mechanism: citing Toomre he points out that shear itself contributes strongly to the growth of gravitational instabilities, leading to gas density enhancements and subsequent star formation. For dIs, however, lacking spiral-density waves, swing amplification may seem an inappropriate mechanism to rely on. (ii) Alternatively, in their review Seiden & Schulman (1990, p. 40) remind that in models for stochastic self-propagating star formation (SSPSF) shearing increases the density of star-forming regions: gas-rich, potential star-forming regions are transported to and mixed with former star-forming regions, giving space for new star formation. (iii) Additionally, while it has been questioned whether shear may cause any visible effect at all given dwarf galaxies being rather slowly rotating systems (Hunter et al. 1998), it becomes more and more evident that some irregular galaxies like the Large Magellanic Cloud or NGC 4449 possess regular, large-scale magnetic fields (e.g., Otmianowska-Mazur et al. 2000), and thus it similarly becomes feasible that the magneto-rotational instability (e.g., Balbus & Hawley 1998) comes into play, effectively strengthening the effects of shear. Bearing in mind this possibility, we nevertheless restrain the discussion in the following on the second of these scenarios only.

Self-propagating star formation is observed with many galactic as well within many extragalactic objects. It is thought of as a locally important SF triggering mechanism in all types of galaxies. For example, modulated by density waves, long-lived spiral arms may be formed in bright disk galaxies (Smith et al. 1984); the surface filling factor of bubbles and the locations of molecular rings in observed disk galaxies can be quantitatively explained by SSPSF (Palous et al. 1994); age gradients in star-bursting galaxies can be accounted for by means of triggered star formation (Thuan et al. 1999; Harris & Zaritsky 1999); and last but not least, the general burst characteristics of compact and irregular dwarf galaxies is long known to partially be understood by means of SSPSF (Gerola et al. 1980; Comins 1984).

Relying on a two-dimensional SSPSF model, we numerically tested the hypothesis that the onset of shear-induced star formation around the turnover radius may leave its imprint in an overabundance of SF regions or of bright stellar complexes beyond two scale lengths. In the Appendix we describe the particular implementation. A general finding of our simulations is that the inclusion of shear (i.e. rotation) allows for about five to ten percent more star forming cells, the exact value depending on the particular parameters used. Being mainly interested in the azimuthally summed-up radial distribution of lumps under different rotational conditions, we compare simulation runs with and without a transition to a flat rotation curve. In the top panel of Fig. 6, right, a typical outcome for a simulation of 20 galaxies with rigidly rotating disks is plotted. The highest peaks are found to be located around one scale length, while the secondary peaks show occurences at many radii but with a preference for locations around two scale lengths. This coincides with our observed peak distributions. For comparision, in the bottom panel of Fig. 6, right, a representative peak distribution for a simulation run of 20 galaxies that exhibit a continuous transition from solid-body to (strong) differential rotation at two scale lengths is shown. Interestingly, the primary peaks now occur preferentially at around two scale lengths indicating a strong influence of shear on star formation around the turnover radius. While this is not the general picture observed with our sample, some of the brighter galaxies actually do match this pattern: IC 1959, ESO 154-G023, Ho I, NGC 5477.

We thus conclude that the observed pattern of primary peaks at one scale length manifests the underlying exponential-disk systems, and that the frequent occurrence of secondary peaks at about two scale lengths is not necessarily related to the onset of strong shear in rotating disks. As the simulations show, it is however consistent with the idea of triggered star formation based on a stochastic self-regulation scenario. Some of the larger galaxies are exhibiting pronounced primary peaks at two scale lengths but show minor peaks at one scale length; with these galaxies we may be directly witnessing shear-induced star formation. The possible role of bars will be reflected in Sect. 7.