3 Disk parameter analysis of the complete sample of dwarf irregular galaxies

Our long-term project of establishing a volume-limited data base on the photometry of nearby dwarf galaxies is approaching its final stage. We may now proceed with the exploitation of the accumulated information. Earlier analyses have already revealed a new effect which needs to be confirmed. With our larger, nearly complete sample at hand we may hope to find other interesting relations among the parameters available. Concerning photometry the present analysis focuses on exponential parameters of dwarf irregular galaxies. Where not stated separately the term "irregular'' denotes all late-type dwarfs, i.e. Im galaxies, BCD or BCD-like galaxies as well as spirals of type Scd or later. More photometric parameter relations, including dwarf elliptical galaxies, are discussed in Bremnes (2000).

3.1 The complete sample

Figure 7: Luminosity function, apparent magnitude distribution, and distance distribution for the Im galaxies (dark shaded), BCD or BCD-like galaxies (light shaded), and spiral galaxies later than Sc (white). Binnings are 0.5 mag for the magnitude distributions and 0.5 Mpc for the distance distribution.

In a series of five papers (Papers I, III, IV, VI, and this one) we have presented photometric results for more than 100 field and group dwarf galaxies in the nearby 10 Mpc volume. Among the galaxies with reliable distance estimates there are 71 dwarf irregulars with B- and R-band data and 4 with B-band data only. Three galaxies (UGC 5658, UGC 8914, and DDO 97) with distances clearly above 10 Mpc are excluded in the following. The three galaxies with the largest distances within the sample are then BK1N at 10.5 Mpc, UGC 4998 at 11.2 Mpc and Kar54 at 12.2 Mpc.

BR photometry and kinematic data for 72 irregular dwarf galaxies of our sample are listed in Table 3 (available only in electronic form at the CDS, see footnote on first page). Selected columns are taken over from our series of papers, complemented where necessary, converted to physical units where adequate, and two columns are newly added. The meaning of the individual columns is as follows:

Column 1 gives the reference to the original publication. Roman numbers refer to the papers of our series with VII being the present one. Columns 2 and 3 name the galaxy and its type. Columns 4 and 5 list the B apparent magnitude (corrected for galactic extinction) and B-R colour, while Col. 6 gives the B absolute magnitude calculated by means of the distance in Col. 7. With a few exceptions all the distances are from the catalog of nearby galaxies by Karachentsev et al. (1999) and given in Mpc; they are based on photometric, group-membership and Hubble law distance determinations. For five galaxies (KK45, ESO 555-G028, ESO 489-G056, ESO 308-G022, and ESO 558-PN011) H$\alpha$distance estimates are taken from Huchtmeier et al. (2000), and for seven galaxies (Kar54, UGC 4998, BK1N, NGC 4248, ESO 473-G024, IC 2038, and ESO 059-G001) distances are determined in accordance with Karachentsev et al. (1999) by means of a Local Group centroid correction and of the Hubble law using a local Hubble constant of $H_0=70\;$km$\;$s$^{-1}\;$Mpc^-1.

Column 8 informs about field (F) or group (G) membership. While most galaxies in Papers I, III, and IV were considered to be group galaxies and those in Paper VI to be field galaxies, we apply a common selection criterion to the galaxies of the present study: we define a group dwarf galaxy as one with a neighbour brighter than -17.5 in absolute magnitude and with a relative distance of less than 1 Mpc. As a pool for possible neighbouring galaxies the catalog of Karachentsev et al. (1999) was used which provides distance information for more than 300 galaxies within 10 Mpc.

Column 9 lists the axial ratio of the fitted ellipse at the 25th-mag/arcsec² isophote. Columns 10 and 11 give the B-band model parameters for an exponential light distribution, i.e. extrapolated central surface brightness in mag/arcsec²(uncorrected for inclination) and the corresponding scale length in pc (measured in terms of equivalent radii), respectively. Column 12 is a measure for the B-R colour gradient (see Sect. 3.5 for an explanation), and Col. 13 lists the magnitude difference between that of a virtual comparison galaxy with a purely exponential radial surface brightness profile and that of the actual galaxy (as defined in Sect. 2.3).

Unlike in Paper VI where external data from other authors was added to enlarge the sample we will rely only on our own photometric data set. This guarantees a consistent treatment (as exemplified in Sect. 2) which is particularly required for the inquiry of subtle effects that otherwise could be overseen in an increased extrinsic scatter. However, for a comparison with dwarf irregular galaxies residing in clusters we will refer to the compilation of Bremnes (2000) including Virgo cluster data from Binggeli & Cameron (1993) and Centaurus cluster data from Jerjen & Dressler (1997).

Finally, in Col. 14 we compile the measured rotation velocities where available. The majority stems from Karachentsev et al. (1999), and for the other about 20 percent galaxies left the 21-cm line width measurements were taken from Bottinelli et al. (1990), corrected for turbulent motion according to the prescriptions of Tully & Fouqué (1985), and inclination corrected assuming a mean intrinsic axial ratio of q₀=0.2 on behalf of Hubble's formula for the conversion of b/ainto an inclination angle (Hubble 1926) and allowing for a minimum inclination of $25^\circ$.

3.2 Luminosity function

The distance and apparent-magnitude distributions as well as the corresponding luminosity function for the 72 dwarf irregular galaxies are shown as histograms in Fig. 7. The binnings are 0.5 mag for the magnitude distributions and 0.5 Mpc for the distance distribution. The luminosity function exhibits a skewed shape which is due the different contributions of the Im galaxies, BCD or BCD-like galaxies, and the spiral galaxies. While the Im galaxies are distributed symmetrically around a pronounced peak at M_B=-14 mag, the brighter spiral and BCD galaxies add to the more luminous part of the distribution. The apparent-magnitude distribution shows a prominence of galaxies in the range between 13 and 16 mag with the Im galaxies providing a steadily increasing partition up to a cut-off at 16 mag. This 16 mag cut-off in apparent magnitude and the -14 mag peak in absolute magnitude combine to a distance modulus of 30 mag corresponding to our 10 Mpc volume limit. As a limit of completeness we therefore adopt M_B=-14 mag; beside for explicitely stated exceptions all the following fitting procedures will rely only on data related to an absolute magnitude brighter than this.

A peculiarity of the distance distribution is the three-peaked appearance which is basically due to the influence of the group and cloud galaxies belonging either to the M 81 group, the CVnI cloud, or the M101 group at distances around 3.5, 5, or 6.5 Mpc, respectively. The two galaxies within the sample with the largest distances are not included in the histogram.

3.3 No morphology-scale length relation

The well known relation of an increasing scale length for a brighter absolute magnitude does nicely show up in our sample, too. Free fits of the form log $(\alpha_B^{-1})=a M_B + {\rm const}$ to the subsamples of the Im, the BCD or BCD-like, and the spiral galaxies lead to similar slopes with values $a\approx-0.10\pm0.03$. As can be inferred from Table 4 the three morphologically distinct constituents of our sample (Im, BCD or BCD-like, and spirals galaxies) form a homogenous data base as concerned to scale length (and consequently to central surface brightness) and to colour.

  

Table 4: Photometric parameter relations of dwarf irregular galaxies specified to morphology, environment, and kinematic data.

$\!$The gently decreasing sequence of mean scale lengths for the subsamples at a given absolute magnitude is below statistical significance. Similarly, the mean colours <B-R> are the same within the errors to the mean of the whole sample, i.e. $<B-R>~=0.99~\pm~0.03$ mag (M_B<-14 mag). For our purposes we therefore see no need to differentiate between morphological classes in the following. Opposite to this, the galaxy environment definitely is of discriminating influence, as we will demonstrate now.

3.4 Structure-environment relation

Figure 8: a) Scale length vs absolute magnitude for the field (squares) and group (filled circles) galaxies. The solid line is a forced fit with slope -0.08 to all galaxies brighter than our limit of completeness (MB<-14, indicated by the vertical dotted line). This slope was previously found for the dwarf irregular galaxies in clusters (Bremnes 2000); their corresponding relation is plotted as the dashed line for comparison. Field and group galaxies each obey the same relation (solid line). b) Residuum in log( $\alpha _B^{-1}$) vs. B-R colour. The residuum is defined as the difference between the measured value of log( $\alpha _B^{-1}$) and the fitted value according to Fig. 8a. Symbols as in Fig. 8a, with those for galaxies brighter than MB=-14 mag additionally circled. The solid line is a least-squares fit to these brighter galaxies only. Also indicated by vertical lines are the mean colours for the brighter field (dot-dashed) and group (short-dashed) galaxies.

It was recently found (Bremnes 2000; Barazza et al. 2001 (Paper VI)) that dwarf irregulars in a field or group environment have brighter central surface brightnesses in the mean than cluster dwarf irregulars. Equivalently, the scale lengths of field and group dwarf irregulars are on average lower than those of dwarf irregulars dwelling in clusters. However, comparing field against group dwarf irregulars no such differences are seen. Here we report further evidence for this trend of a structural dependence on environment among dwarf irregular galaxies.

In Fig. 8a scale length is plotted against absolute magnitude in the B band. Squares and circles represent field (50) and group (22) galaxies, respectively. The solid line is a fit to all our field (32) and group (15) galaxies brighter than -14 mag and with a forced slope of -0.08. This slope corresponds to a least-squares fit to cluster late-type galaxies in the magnitude interval $M_B \in [-13, -17]$ (Bremnes 2000) shown as the dashed line. A free fit to our dwarfs yields with values of $-0.095\pm0.027$ essentially the same slope. But there obviously is a shift in the sense that - at a given absolute magnitude - a denser environment is related to a higher scale length (fainter central surface brightness). Table 4 summarizes our numerical findings. While in Bremnes (2000) and in Paper VI the mean differences $\Delta$log( $\alpha _B^{-1}$) were 0.15 and 0.14 ( $\Delta\mu_0^B=0.82$ and 0.76 mag), respectively, we find the shift amounting to 0.12 mag ( $\Delta\mu_0^B=0.52$ mag). Despite the lower value as compared to the previous determinations (which also included galaxies from other sources) the shift is clearly statistically significant given the uncertainty of 0.03 mag for the field and group irregulars or 0.02 mag for the cluster irregulars. Note that when comparing field and group dwarf irregulars one finds no such separation; field and group galaxies are thus not repeating the trend described above. However, the scatter $\sigma$ for the (logarithmic) scale length is twice as small for the group than for the field dwarf irregulars.

Figure 9: a) Exponential-fit accuracy parameter versus central surface brightness in the B band. b) Colour gradient versus scale length in the B band. The horizontal line marks the mean for the galaxies brighter than MB=-14 mag; the curved lines are proportional to the inverse scale length: the dotted one is for a constant factor of proportionality while for the short-dashed line the factor of proportionality is a linear function of scale length (see text for details). - Symbols as in Fig. 8b.

The scatter - particularly for the field irregulars - is considerable and must be partly intrinsic. Given the shift between field and group and cluster galaxies we may wonder if it is related to different stellar populations dominating galaxies in different environments. Due to a lack of B-R colour indices for the cluster dwarf irregulars we have to circumvent a direct B-R colour comparison between the two environments. Following Bremnes (2000) we define the residuum $\delta~$log( $\alpha _B^{-1}$) as the difference of the measured value to the fitted value, i.e. $\delta~$log( $\alpha _B^{-1}$) $\equiv$ log( $\alpha _B^{-1}$) -(-0.08 M_B +1.493), and plotting it against colour B-R some relation becomes manifest. This is revealed in Fig. 8b where a least-squares fit applied to the galaxies brighter than -14 mag (circled symbols) is shown as the solid line, obeying

$$\delta~{\log}(\alpha_B^{-1}) = (0.16\pm0.12)(B-R)-0.159~.$$ (5)

Within the errors both field and group galaxies independently exhibit this same behaviour, albeit the scatter of 0.17 is considerable. Because the B-R colours show no dependence on absolute magnitude M_B, there is thus an indirect indication for a gentle but systematic trend of galaxies with higher scale lengths being redder on average and for a given absolute magnitude. Additionally, as indicated by the vertical lines in Fig. 8 b and as can be inferred from Table 4, group galaxies with a mean colour of <B-R> =1.05 mag are on average slightly redder than field galaxies with <B-R> =0.97 mag. As with scale length the scatter $\sigma$for the mean <B-R> colour is smaller for the group than for the field dwarf irregulars.

3.5 Variations in the radial light distribution

The radial surface brightness profiles of most dwarf irregular galaxies (and dwarf elliptical galaxies) as well as those of spiral disks are exponentially decreasing. This is particularly true at intermediate galactocentric distances which may serve as a fitting domain. The overall accuracy of an exponential fit to a given surface brightness profile may then be quantified by the difference between the total magnitude of a purely exponential intensity law (Eq. (4)) and the actually measured total magnitude. In the previous papers of our series this quantity was named either $M_{\rm exp}-M$ or $\Delta m$ (cf. Sect. 2). In Fig. 9a) $\Delta m_B$ is plotted against central surface brightness $\mu_0^B$. The majority of galaxies shows the exponential law being nominally a good description for the radial intensity distribution ( $\Delta m_B\approx0$). There are, however, quite a few aberrant galaxies. As was discussed in Paper III deviations from the exponential-fit description are frequently seen in the innermost regions and/or at large radii, the reasons ranging from strongly asymmetric distributed star formation regions to uncertain sky subtraction. Therefore it is not surprising that these deviations are not correlated with either the central surface brightness, shown here, or with other global parameters such as absolute magnitude, scale length, or colour.

The fitted exponential scale lengths provide a definition of the colour gradient in the surface brightness representation by means of differentiating Eq. (3), leading to $\nabla (B-R)\equiv 1.086(\alpha_B-\alpha_R)$. As mentioned in Sect. 2.3, and as can be seen in Fig. 9b, most dwarf irregulars exhibit no or small, mainly positive colour gradients. For the galaxies brighter than M_B<-14 mag the mean has a value of only 0.16 mag/kpc (horizontal line). Several galaxies, particularly those with small scale lengths, have strong positive colour gradients, up to more than 1 mag/kpc. This is, however, not too surprising: rewriting

$$\nabla (B-R)=1.086(1-\alpha_B^{-1}/\alpha_R^{-1})/\alpha_B^{-1}$$ (6)

one immediately expects the colour gradients to be inversely proportional to the scale lengths provided the ratios of the blue and red scale lengths would be about constant for all galaxies. As indicated by the curved dotted line this is approximately true for most of our galaxies (drawn for an assumed proportionality $\alpha_B^{-1}=0.9 \alpha_R^{-1}$). Applying a linear relation $\alpha_B \approx 1.4 \alpha_R - 0.5$ for the inverse scale lengths numerically gives a more realistic fit, including the region of negative colour gradients (short-dashed line). But this seems not to hold for a handfull of galaxies with negative colour gradients and with scale lengths shorter than about 0.6 kpc. Among them the most extremes are also faint and rather blue. However, as we will argue in Sect. 3.7.3, the colour gradients of most of these suspicious galaxies are simply inadequately determined by means of the above deduced gradient measure (as opposed to a gradient fitted directly to the colour profiles).

3.6 Linkage of photometric and kinematic parameters

Figure 10: Residual plots revealing the linkage of photometric with kinematic properties. The residuals for a parameter are defined as the deviations from a mean at given luminosity, while the mean is determined from an ordinary least-squares bisector fit of the parameter to absolute magnitude (see Table 4). Plotted are the residuals for scale length and surface brightness versus the residuls of three distinct circular velocities: the two leftmost panels use the residuals of the conventional rotational velocity $v_{\rm rot}$, measured at or beyond the optical extent of the galaxy; the middle panels use the residuals of the peak circular velocity of a hypothetical self-gravitating exponential disk, at an inner disk radius of about two scale lengths (Eq. (7)); and the rightmost panels use the residuls of the halo contribution to the rotational velocity at the same inner disk radius, as inferred from the Burkert (1995) halo model for dwarf irregular galaxies. See text for details.

Different models on the formation of galaxy disks all predict systematic dependencies on kinematic conditions for the resulting scale lengths and central surface brightnesses (e.g., Dalcanton et al. 1997; Weil et al. 1998; Zhang & Wyse 2000; Silk 2001). In this section we therefore seek to validate these expectations using observed quantities for the dwarf irregular galaxies of our sample. We particularly ask whether the scatter in the scale length or central surface brightness versus absolute luminosity diagram may be explained by deviating rotational velocities at fixed luminosity. In the following a variable's linear dependence on absolute magnitude defines its mean at a given luminosity. The difference of an individual data point from this mean we call the residual, in accordance to the definition given in Sect. 3.4. In Table 4 the relations for five variables are given including the particular circular velocities described below. The given relations have been determined applying ordinary least-squares bisector fits (Isobe et al. 1990) to all the 62 galaxies of our sample with known rotational velocity data. We chose working with bisector fits for reasons of statistical consistency: working with simple least-squares or with orthogonal fits results in spurious dependencies of the photometric variable residua on some of the photometric variables themselves. While for an exponential intensity profile the surface brightness residua have to correlate with scale length residua, other pairs of parameters should not correlate with each other. Only for ordinary least-squares bisector fits no significant correlations were found in plotting the scale length residua (surface brightness residua) against scale length (surface brightness) or absolute magnitude. This was even more evident if we relaxed the completeness condition, allowing for a longer fitting intervall for the strongly scattered data in determining the mean relations listed in Table 4. Therefore we work in the following with all the galaxies for which kinematic data is available, including also the galaxies fainter than the limiting magnitude imposed so far.

A Tully-Fisher-like relation between the conventional rotational velocity $v_{\rm rot}$, measured at or beyond the optical extent of the galaxy, and absolute magnitude M_B is exhibited by the data (Table 4). Its scatter is rather large and, at a fixed luminosity, is not related to the scatter in scale length or central surface brightness. This is shown in the two leftmost residual plots of Fig. 10. And it is not surprising: dwarf irregular galaxies are known to be dark matter dominated at all radii (Carignan & Freeman 1988; Burkert 1995), thus it is not the baryonic matter that determines  $v_{\rm rot}$.

A more convenient kinematic measure to work with would be given by the disk and/or halo contributions to the velocity at some characteristic inner disk radius. The predicted rotation curve for an exponential mass distribution corresponding to the observed exponential surface brightness profile has been analytically given for a thin, self-gravitating exponential disk (Freeman 1970). Providing a characteristic inner disk radius this description may serve here as a simple model for the luminous matter distribution and the related Keplerian kinematics despite dwarf irregular and low-surface brightness galaxies being known to be better characterized by expanded disks (Sung et al. 1998). The velocity profile exhibits a peak at radius $R_{\rm peak}=2.14~ \alpha^{-1}$ where it has a value of $v_{\rm d}=0.62(2 \pi G {\mit\Upsilon} I_0 \alpha^{-1})^{0.5}$; here G is the gravitational constant, ${\mit\Upsilon}$ the disk mass-to-light ratio, I₀ the face-on central mass surface density, and $\alpha^{-1}$ the scale length (e.g., Chiba & Yoshi 1995). Converted to B band observables we have

$$v_{\rm d} = 2.62\times10^4~ \sqrt{10^{-0.4\mu_{0,i}^B}~\alpha^{-1}_B~{\mit\Upsilon}}~~~{\rm km~s^{-1}}$$ (7)

with $\mu_{0,i}^B[{\rm mag}/\ifmmode\hbox{\rlap{$\sqcap$ }$\sqcup$ }\else{\unskip\nob... ...$\sqcap$ }$\sqcup$ } \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi\arcsec]$ the inclination-corrected central surface brightness, $\alpha^{-1}_B[{\rm pc}]$ the disk scale length measured in parsecs along the major axis, and ${\mit\Upsilon}\equiv\frac{M}{L_B}[\frac{M_\odot}{L_{\odot}^B}]$ the disk mass-to-light ratio in solar units. In the middle panels of Fig. 10 the deviations of this peak velocity from its mean value at a given galaxy luminosity is compared to the corresponding residuals in scale length and surface brightness. Now the two residual plots clearly suggest a linkage between photometric and kinematic properties: at a given luminosity deviations from the mean are related to different rotation velocities, in the sense that higher scale lengths or fainter central surface brightnesses correlate with lower-than-mean peak rotational velocites divided by the square root of an unknown disk mass-to-light ratio. The plots shown correspond to a constant $\Upsilon =1~M_\odot/L_{\odot}^B$. There are several caveats with our finding: First, the issue of the unknown individual mass-to-light ratios is a non-trivial one and its possible relation with surface brightness, luminosity, or color is not tackled in the above approach. Second, one must be cautious in applying the model parameter $v_{\rm d}$ which is deduced from photometric quantities only and which does not relate to an independent, actually observed kinematic quantity. Finally, the thin disk assumption and, above all, ignoring the presence of dark matter leaves the model rather unrealistic.

However, we may approximately infer the contribution of the dark-matter halo to the total rotational velocity at $R_{\rm peak}=2.14~ \alpha^{-1}$ (in fact, at any radius) on purely phenomenological grounds. The total rotational velocity v(r) is conventionally decomposed into a dark matter ( $v_{\rm h}(r)$), a stellar ( $v_{\rm d}(r)$), and an HI component. Burkert (1995) and Salucci & Burkert (2000) successfully describe the rotation curves of dark matter-dominated dwarf galaxies by means of an empirically found one-parameter family of halos: Provided the rotational velocity v₀of the dark matter halo is known at a particular density scale radius (corresponding to the halo core size of typically four to seven optical scale lengths), then the rotational velocity due to the halo  $v_{\rm h}(r)$ is calculable at every radius r. One can show that the seven selected dwarf galaxies of Burkert (1995) obey $<v_0/v_{\rm rot}>\;=0.78\pm0.05$, i.e., the input parameter v₀ typically is lower by about 20 percent than the measurable circular velocity $v_{\rm rot}$ of the flat or nearly flat part of the rotation curve. Thus we may adopt $v_0 \approx 0.78\;v_{\rm rot}$ as a working approximation and determine the halo contribution to the rotational velocity at $R_{\rm peak}$, $v_{\rm h} \equiv v_{\rm h}(R_{\rm peak})$, by means of the Burkert-halo parametrization. The rightmost panels of Fig. 10 plot its residuals versus the residuals in scale length and surface brightness. It is revealed that the photometric exponential-disk parameters do indeed correlate with the corresponding halo-related kinematics: at a fixed luminosity higher-than-mean halo-induced rotational velocities are related to larger scale lengths and fainter central surface brightnesses. Note, however, that there is an island of nine galaxies with lower-than-mean scale lengths and with corresponding slower-than-mean circular velocities but with central surface brightnesses too faint instead of too bright; they all are brighter than about M_B=-16 mag but, as far as we can tell, share otherwise no distinctive feature.

3.7 Discussion

3.7.1 Is the structure-environment relation significant?

No morphology-structure relation was surfacing from the data; the log $(\alpha_B^{-1})-M_B$ relations were similar for Im, BCD/-like, and late spiral galaxies, the individual free-fit slopes yielding $-0.1\pm0.03$ (Sect. 3.3). Guided by the work of Bremnes (2000) in comparing photometric mean values for field and group versus cluster dwarf irregulars, we adopted his slope of -0.08 found for Virgo and Centaurus cluster galaxies (Sect. 3.4). In an independent analysis of only the Virgo dwarf irregulars we find a slope about twice as steep. We checked that the structure-environment relation is present irrespective of the actually chosen slope value, as expected. For the inquiry on a linkage of photometric and kinematic data a steeper slope was used that emerged from double regression applied for reasons of statistical consistency (Sect. 3.6). Such a steeper slope also agrees better with a global $\log(\alpha_B^{-1})-M_B$ relation as determined for a combined sample of late-type irregulars and early type spirals (Makarova 1999, her Fig. 5). We stress, however, that the qualitative results reported in this paper are insensitive to the adopted fitting procedure.

3.7.2 Origin of the structure-environment relation

The structure-environment relation says that for cluster dwarf irregulars the exponential parameters differ on average from those of non-cluster galaxies. In particular, the scale lengths for field and group dwarf irregulars are on average shorter and thus the central surface brightnesses are brighter than for dwarfs in clusters. This was already reported in Bremnes (2000) and in Paper VI and is confirmed here with our complete data sample (Sect. 3.4). Several reasons may be responsible for the structure-environment relation. There are the manyfold influences of the cluster environment on a dwarf galaxy, including a relatively high peculiar velocity, galaxy harassment, ram pressure, and - quite important - tidal effects (stripping, stirring, induced angular momentum). In particular,

(i) an environmentally induced loss of a substantial amount of gas would lower the gravitational potential and thus expand the galaxy;

(ii) quenching of star-formation in cluster late-type galaxies may partly account for the observed effect. Late-type galaxies in clusters tend to be gas-deficient, implying a lowered present-day star formation rate (Gallagher & Hunter 1986; Cayatte et al. 1994). Thus Bremnes (2000) suggests a simple uniform fading scenario to explain the higher central surface brightness in field and group dwarf irregulars. According to this scenario star formation was much more efficient in cluster galaxies at earlier epochs while it is still ongoing at a moderately high rate in field and group galaxies. Such a fading in cluster irregulars would be accompagned by a reddening of the galaxies. Indeed, the cluster irregular sample of Gallagher & Hunter (1986) with $<B-V>~=0.51\pm0.02$ mag is by $0.11\pm0.03$ mag redder than their field and group comparison sample. This is also redder by $0.09\pm0.03$ mag than the equal means for the field and group irregular samples of Makarova (1999) and van Zee (2000), $<B-V>~=0.42\pm0.03$ mag. Consistently, Paper VI (excluding UGC 1281) provides us with a representative value of $<B-V>~=0.42~\pm~0.06$ mag for our sample of field and group dwarf irregular galaxies. The idea of quenched star-formation is additionally supported by our finding that for a given luminosity galaxies with higher scale lengths (or equivalently, with lower central surface brightnesses) are somewhat redder on average. This nicely agrees with the positive correlation between star formation or metallicity and central surface brightness in dwarf irregular galaxies as recently elucidated by van Zee (2001) and Grebel (2001), respectively. Thus both the colour trends that emerge from our photometric data (the sample mean colour values and the scale length dependence) provide some credits to a fading scenario;

(iii) due to tidal stirring, the initial gas dispersion of cluster dwarfs may be expected to be higher than in an isolated dwarf galaxy, naturally leading to a somewhat larger scale length for the stellar population (Andersen & Burkert 2000). Actually, tidal thickening of interacting galaxies has been observed by Reshetnikov & Combes (1997).

For low surface brightness dwarf galaxies Taylor (1997) observes an increased rate of star formation depending on the small-scale environment, thus empirically supporting the hypothesis that galaxy interactions may trigger bursts of star formation which lead in turn to structural transformations. Recent simulations on the evolution of group dwarf irregular galaxies orbiting massive galaxies even suggest morphological transformations, caused by tidal stirring, to either dwarf spheroidals or dwarf ellipticals (Mayer et al. 2001). Within the group galaxies of our sample such influences are, however, not (yet) manifest: for the parameters under investigation the group and field galaxies are rather similar (cf. Table 4). Our empirical finding of late-type spiral galaxies having on average not significantly shorter scale lengths than Im galaxies at similar absolute magnitudes neither validates nor contradicts such an evolutionary scenario, too. The scatter may simply be too large for such subtleties to be uncovered. This coincides with the negative result of Karachentsev et al. (1999) who state that their "tidal'' or isolation index - which is a measure of local mass density surrounding a galaxy - is independent from the HI mass-to-luminosity ratio when applied to some hundred nearby galaxies of various types. Some hints for possible differences between our field and group data nevertheless arise from the slightly redder mean B-R colour for group galaxies than for field galaxies, and from the smaller scatter for the group galaxies both in scale length at a given luminosity and in colour;

(iv) for field galaxies we observe a linkage between the optical parameters for an exponential disk and the dark matter contribution to the rotational velocity at an inner disk radius of about two scale lengths (Sect. 3.6). Concerning the structure-environment relation this linkage hints to the possibility that cluster galaxies are embedded in slightly different (more massive) halos than field galaxies resulting on average in somewhat higher disk scale lengths. We discuss both our finding and its possible implication for the structure-environment relation further below (Sect. 3.7.4).

3.7.3 Change of colour gradients

Positive colour gradients are typical for most of our galaxies (see Fig. 9 b) and are naturally expected for galaxies with star formation concentrated to the denser inner regions. Despite of this, quite a few galaxies seem to have negative colour gradients. In particular, galaxies with scale lengths higher than about 1 kpc seem to enter the area of negative colour gradients. This is consistent with the observation that larger galaxies (LSB and normal spirals) have older central populations forcing negative colour gradients, with values correlated to bulge sizes (Bell & de Jong 2000; Matthews & Wood 2001; Andredakis et al. 1995). A handful of our dwarf irregular galaxies exhibit strong negative colour gradients, however. They all are faint, blue, and have relatively small scale lengths ( $\alpha^{-1}_B \le0.6$ kpc). While particularly the appearance of young and small sized isolated galaxies may be strongly influenced by outward-propagating supershells which induce peripheral star formation activity, this seems unlikely to be the case for most of the galaxies under investigation. Thus the negative colour gradients neither seem to be related to heavily off-centered star formation sites dominating galaxy colour and structure nor can they be explained by an overtly red central stellar population. After inspection of the surface brightness profiles and the corresponding colour profiles, and taking care of the estimated errors, one comes to the conclusion that the strong negative gradients are photometric artefacts resulting from an inconsistent scale length determination in the two photometric bands which is propagated to the indirect gradient determinition method adopted; thus the gradients are likely to be much smaller, i.e. closer to zero, or even positive. This statement seems to hold for some galaxies with weakly negative colour gradients as well, particularly those with small scale lengths. An example for an unintended mismatch between gradient definition according to Eq. (6) and the actual colour profile as shown in Fig. 3 is UGCA 148 with similar scale lengths in B and R (as determined beyond $\approx$25) but with a relatively strong positive colour gradient in the outer envelope.

Strong positive colour gradients appear for galaxies with very short scale lengths only (see Fig. 9b) and - according to van Zee (2001) - may be related to starbursts. In fact, any centrally concentrated young population imprints its influence on the colour gradient more pronounced in a small galaxy (i.e. in one with relatively short scale lengths) than in a bigger galaxy. Moreover, for galaxies with exponential intensity laws the colour gradients are positive and inversely proportional to the scale lengths in general as long as the blue-band scale lengths are shorter than those in the red band by a roughly constant value. We have illustrated that adopting a scale length difference of about 10% will indeed approximately result in the expected behaviour for the majority of our galaxies. We have extracted this same behaviour from the data on isolated dwarf irregulars in van Zee (2001) and independently in Heller & Brosch (2001) for Virgo cluster dwarf irregulars. If a slightly more complicated factor of proportionality is allowed for, i.e. an affine relation as described, then galaxies with larger scale lengths are predicted to exhibit also negative colour gradients, as indeed is observed with our sample. There are two immediate consequences from such a behaviour. First, it implies a constant B-R colour increase of about 0.1 mag for an interval of one scale length for dwarf irregular galaxies, irrespective of size or total colour. Second, the B-band scale lengths being systematically shorter than those in the R band (for the smaller galaxies only, i.e. those with short scale lengths) is in line with disk models that explain the formation and evolution of exponential gaseous and stellar systems by means of viscous radial infall (Firmani et al. 1996; Zhang & Wyse 2000; cf. also Hunter et al. 1998).

3.7.4 Kinematic couplings

The fair to tight correlations seen in Fig. 10 between optical disk parameters and the inferred accompagning disk dynamics are at first view rather astononishing, in particular for the rightmost panels given the crude estimate used for the input parameter v₀ of the Burkert model, i.e. $v_0 \approx 0.78\;v_{\rm rot}$. The quantitative results are, however, rather insensitive to the exact value of the adopted coefficient. Therefore it is not disturbing that the outermost observed rotational velocity of dwarf galaxies often resides on a still rising rotation curve and thus constitutes only a lower limit to the true maximum value where the rotation curve becomes flat. The tightness of the $\delta \log~\alpha^{-1}_B$- $\delta \log~v_{\rm h}$-relation is a consequence of applying the Burkert-model dark matter scaling relations at a small radius proportional to scale length: at small radii the imposed velocity profiles $v_{\rm h}(r)$ still form a rather similar family, thus reading off the velocities at scale-length dependent radii immediately translates into tightly correlated variations of circular velocity. Thus we are confident with our adopted working approximation and with the resulting correlations as presented.

Navarro (1997) already established an observational $\delta \mu_{0,i}^B$- $\delta \log~v_{\rm h}$-like relation for high- and low-surface brightness galaxies working not with the Burkert-halo density distribution but with a Navarro-Frenk-White profile and having available measured rotational velocities at inner disk radii, particular at the peak rotational velocity. Recently, van den Bosch et al. (2001) studied the angular momentum content of dwarf galaxies with a sample of 14 late-type spirals. Using a Navarro-Frenk-White profile they determined the spin parameters for the baryonic disks. One can show that a - admittedly weak - correlation emerges from their data between either the central surface brightness or the scale length and the disk spin parameter. Moreover, Chiba & Yoshi (1995) have already applied a tight correlation found between disk scale length and a combination of central surface brightness and measured rotational velocity at the characteristic radius of 2.14 scale lengths in order to determine extragalactic distances of early- and late-type spirals; their approach is similar to a Tully-Fisher relation study but working with inner disk circular velocities and assuming an exponential light distribution. This previous work and our own approach lead us to the conclusion that the scatter in the scale length or surface brightness versus absolute magnitude diagrams, particularly for dwarf irregular galaxies, seems to be correlated with the kinematic disk properties induced by the surrounding dark matter halo. The smaller kinematic contribution of luminous matter probably is related to the photometric structure as well, but due to the unknown disk mass-to-light ratios and ignoring effects of viscous infall we cannot make an argument as strong as above. Our conclusion is also in line with theoretical expectations: models on the formation of disk galaxies predict that, for a given mass or luminosity, increasing the halo angular momentum increases the exponential scale length or, correspondingly, decreases the extrapolated central surface brightness (Dalcanton et al. 1997; Zhang & Wyse 2000; Silk 2001). This applies to our case of measured rotional velocities, because halo angular momenta or halo circular speeds are indicative for the circular speed of the collapsed, centrifugally supported disks (Weil et al. 1998).

Accepting that higher-than-mean rotational velocities of field and group galaxies exhibit higher-than-mean scale lengths one wonders whether this trend should not show up in a direct comparison of the kinematics of cluster and field irregulars, too. This would point to a deeper cause for the structure-environment relation than a mere evolutionary difference, e.g. different halo populations. We are not aware of a study comparing observed rotational velocities in cluster and field environments in the sense that it supports or contradicts our interpretation; however, for galaxies in clusters Mould (1997) does not see a correlation of the intercept of the Tully-Fisher relation and cluster richness. Unfortunately, only 20 out of the 35 Virgo irregulars/late-type spirals mentioned in Sect. 3.7.1 do have kinematic data in the PGC, turning out to be too small a sample to exhibit the structure-environment shift. Thus we cannot rely on this cluster subsample in order to perform a kinematic analysis as above. Such a more complete and direct analysis is therefore postponed to future work with more galaxies at hand.