THE FINE-SCALE STRUCTURE OF THE NEUTRAL INTERSTELLAR MEDIUM IN NEARBY GALAXIES

Nearly all of the H i hole studies reported above are based on identifications made "by eye." This method carries selection biases which are difficult to quantify and is therefore subjective to a certain degree. To address this, several attempts have been made to create automated detection algorithms. The first such attempt was made by Thilker et al. (1998) who used models of expanding structures to identify shells in observed data cubes; they applied their code to NGC 2403. Mashchenko et al. (1999) refined the code to include more realistic three-dimensional hydrodynamical simulations of shells and revisited NGC 2403 where they detected 601 expanding shells. The same technique was also applied to the Canadian Galactic Plane Survey (CGPS) by Mashchenko & St-Louis (2002). These codes, although they are quite robust in discovering regular, expanding shells around young stars fail to detect non-spherical or incomplete shells. These are the type of features we are predominantly finding in our data. A different approach to an automated procedure was proposed by Daigle et al. (2003) who used a neural networks algorithm to detect expanding shells based on velocity spectra. This method was successfully tested on data from the CGPS. More recently, Ehlerová & Palouš (2005) developed a model independent algorithm which identifies H i holes as depressions in the channel maps without any constraints on the shape or expansion velocity of the hole, but requiring the hole is "complete." They applied this code on the Leiden–Dwingeloo Survey data and identified about 1000 structures in the Milky Way.

A cursory inspection of our data on galaxies in the Local Volume, with varying orientation parameters, sensitivity, and linear resolution, quickly convinced us that none of the automated methods can be applied successfully to our entire sample. The human brain is superior when it comes to recognizing irregular and open structures. We therefore decided to identify the H i holes "by eye," despite the fact that it is less objective than an algorithm and quite a time-consuming process.

The search for H i holes in the 20 THINGS galaxies was conducted with the KARMA¹⁰ visualization software package which contains a large number of modules (tasks) several of which were used for this project. In particular, the task kvis was used to display and investigate the integrated H i maps as well as digital movies of the data cube. The task kpvslice was also extensively used as it offers many ways to look at a data cube by producing interactive position–velocity (pV) cuts in any orientation through a data cube.

The detection process involved several steps which were followed in a consistent way for all 20 galaxies. First, the task kvis was used to inspect the natural weighted integrated map for the most obvious, more extended structures. The second step was to search the higher resolution robust weighted integrated H i map for smaller holes. From these two steps, we generated a list of candidate holes. The task kvis was then used to inspect the natural weighted and robust weighted data cubes by displaying the channel maps sequentially, much like a movie. This was done for two reasons: although some of the holes are clearly visible in the integrated maps, many holes do not show up at all due, for example, to the inclination of the galaxy which means that holes along the line of sight tend to blend, or are "filled in" with emission at other velocities. Also, holes which are still fully contained within the H i disk can only be reliably detected in channel maps or pV diagrams. The second reason to display the channel maps as a movie was to discard holes that are chance superpositions of structures, such as parts of spiral arms, rather than genuine holes in the ISM. In order to set standard criteria to classify a feature as a genuine hole, we decided that a hole should be visible over at least three or five channels (depending on whether the velocity resolution is 5.2 or 2.6 km s⁻¹, respectively), which is of the order of the FWHM corresponding to the typical H i velocity dispersion encountered. This further eliminates spurious features.

In order to eliminate as much as possible any subjectivity, each hole in the revised list was examined individually using the task kpvslice. The signature of a hole in a pV diagram depends on the type of the hole as is discussed later in this paper, but overall this method is better suited for the detection of the smallest features. In addition, the inspection of pV diagrams is used to determine some of the basic properties of the holes, such as position, type, heliocentric, and expansion velocities. For every feature detected, two sets of properties were determined. The first set involves the observed properties and the second set the derived properties. In addition, a quality ranking Q is assigned, which represents a (subjective) judgement regarding the probability of the feature being actually a shell in the H i distribution; its values range from 1 (low quality) to 9 (high quality). Even though this property is not directly quantifiable, it depends on many aspects of the appearance of a hole and it follows certain criteria:

• 1.  
  the hole must be present in at least three or five consecutive channel maps depending on whether the velocity resolution is 5.2 or 2.6 km s⁻¹, respectively,
• 2.  
  the center of the hole must remain stationary across the channel maps in which it is observed,
• 3.  
  there must be sufficient contrast (at least a 50% drop in surface brightness) between the hole and its immediate surroundings, and
• 4.  
  the shape of the hole in pV space must be clearly defined and adequately described by an ellipse.

Overall we detected more than 4000 candidate H i holes across the 20 galaxies. We decided to stay on the conservative side and only keep holes detected at high significance, i.e., above a certain threshold. This threshold was determined by comparing the holes we found in IC 2574 with those found in a parallel study by Rich et al. (2008). Both authors looked for holes independently assigning a threshold parameter Q value to each hole. We then compared our catalogs based on the position and size of each hole. A hole was considered a match when its position was within half a beam size and its diameter agreed to better than 75%. From this comparison, we found that over 75% of our Q>5 holes had been identified by both authors and we therefore decided to use this value as our quality threshold. Once all the low Q holes were removed from our lists we were left with ∼1000 H i holes.

In what follows we describe the observed characteristics of each hole.

• 1.  
  The position of the center of the hole in right ascension and declination (J2000 coordinates) determined by the (average) pixel where the flux density is at a minimum. The uncertainty in the position is taken as half the robust beam size for each galaxy and is of the order of 3''.
• 2.  
  The heliocentric velocity (V_hel) of the hole. This was defined by the velocity channel in which the contrast between the H i hole and its rim was highest. The uncertainty here is equal to the velocity resolution (2.6–5.2 km s⁻¹).
• 3.  
  The semimajor and semiminor axes (b_maj, b_min) of the hole. To determine the major and minor axes, the approximate orientation of the hole was estimated and the task kpvslice was then used to draw cuts along these directions through the center of the hole, either as seen directly on the integrated H i map or inferred from the pV diagram. These cuts revealed the decrease in H i at the position of the hole. The axes were taken as the FWHM of this depression. The uncertainty is of the order of half the robust beam size or 3''.
• 4.  
  The axial ratio of the hole was defined by the ratio of minor axis to major axis. The uncertainty of the axial ratio is of the order of 25%.
• 5.  
  The position angle (P.A.) of the hole on the sky measured from north through east. We estimate the uncertainty to be of the order of 20°.
• 6.  
  The type of the hole (1–3). Following Brinks & Bajaja (1986), the holes were assigned to one of three different types, based on their appearance in the pV diagrams (Figure 2). A type 1 hole is a hole that has completely blown out of the disk of the galaxy, and therefore neither the approaching nor the receding sides of the hole can be observed as can be seen in Figure 2, top. In a type 2 hole, we only observe one side of the hole, which is usually seen in pV diagrams as a deviation toward either higher or lower velocities with respect to the velocity of the bulk of the H i within that area of the galaxy (Figure 2, middle). Finally, in a type 3 hole we see both sides of the hole. Its signature in pV diagrams is that of an elliptical structure (Figure 2, bottom). Typically, type 1 holes are larger than type 2 and 3 as they represent a later stage of bubble evolution, i.e., type 3 holes are expected to evolve into type 1 (and possibly type 2) after growing.
• 7.  
  The expansion velocity (V_exp) of the hole. This was measured from the signature of the hole in the pV diagrams. For type 1 holes, it is impossible to measure an expansion velocity. However, it is very likely that these have actually stalled and their final expansion velocity will be equal to the velocity dispersion of the gas. For type 2 holes, the expansion velocity was taken as the difference between the velocity of the H i from that side of the hole which is seen to deviate from its measured velocity and the mean velocity of ambient gas surrounding the hole. In the case of type 3 holes, the expansion velocity was taken as half the difference between the velocity of the approaching and receding sides of the hole. We adopt an uncertainty that is equal to the velocity resolution (2.6 or 5.2 km s⁻¹). We tested the validity of the derivation of the expansion velocity for type 2 holes by applying the method explained for type 2 to the type 3 holes. The derived velocities were the same to within the uncertainties.
• 8.  
  The average flux density (I) around the hole. This was determined by averaging the pixel values on the integrated H i map over an area around the hole, twice the area of the hole itself. This value is needed in order to derive the approximate density before the creation of the hole. Errors here are estimated to be of the order of 10%.

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The observed characteristics have been used to determine the derived properties as described in the following.

• 1.  
  The diameter (d) of the hole. Since most of the holes are not circular but elliptical, the diameter was defined using the geometric mean:

  $$\begin{equation} d=2\sqrt{b_\mathrm{maj}\, b_\mathrm{min}}, \end{equation} \tag{ 4 }$$

  where b_maj and b_min are the semimajor and semiminor axes of the hole, respectively. The uncertainty in d is of the order of 10%.
• 2.  
  The kinetic age t of the hole. Assuming that over its lifetime the hole was expanding with V_exp, this is

  $$\begin{equation} t_\mathrm{kin}\,\mathrm{(Myr)} =0.978\frac{d/2}{V_\mathrm{exp}}, \end{equation} \tag{ 5 }$$

  where d is the diameter of the hole in pc and V_exp is in km s⁻¹. Since the expansion velocity was higher at earlier stages of the expansion, the kinetic age is an upper limit. Its statistical uncertainty is around 20%.
• 3.  
  The H i column density (N_H i) around the hole. The column density is given by

  $$\begin{equation} N_{\rm H\,\mathsc{i}}\,\mathrm{(cm^{-2})}=1.823\times 10^{18}\sum T_\mathrm{B}\, dV, \end{equation} \tag{ 6 }$$

  where T_B is the brightness temperature in units of Kelvin, V is the velocity in km s⁻¹ and the summation is over the velocity channels which contain signal. The relation between brightness temperature and flux density at 21 cm wavelength is given by

  $$\begin{equation} T_\mathrm{B}\,\mathrm{(K)}=\frac{S}{1.66\times 10^{-3}(B_\mathrm{maj} B_\mathrm{min})},\, \end{equation} \tag{ 7 }$$

  where B_maj and B_min are the FWHM of the beam, respectively, in arcsec and S is the flux in mJy beam⁻¹. We calculated T_B from the average flux density (I) around the hole which we converted to Kelvin using the above equation.
• 4.  
  The midplane H i volume density (n_H i). This is given by

  $$\begin{equation} n_{\rm H\,\mathsc{i}}\,\mathrm{(cm^{-3})}=\frac{N_{\rm H\,\mathsc{i}}}{3.08\times 10^{18}\,l(r)}, \end{equation} \tag{ 8 }$$

  where N_H i is the column density in cm⁻² and l is the effective thickness of the H i layer in pc which is assumed to have a Gaussian profile with a 1σ scale height h (see Section 3). Thus,

  $$\begin{equation} l(r)\,\mathrm{(pc)}=\frac{h\,\sqrt{8\ln 2}}{\cos i}. \end{equation} \tag{ 9 }$$
• 5.  
  The volume (V) of the hole. For holes that are contained within the disk (type 3) this is simply

  $$\begin{equation} V\,\mathrm{(pc^3)} = (4/3) \pi (d/2)^{3}, \end{equation} \tag{ 10 }$$

  where d is the diameter of the hole in pc as given in Equation (3). Type 3 holes are intrinsically smaller (and younger) and for those it is justified to assume that their extent perpendicular to the disk is the geometric mean of the other two axes. As is shown a posteriori (see Section 5.2), this is a very reasonable approximation given that their shape cannot be greatly distorted during their short lifetime. For holes that are larger than the thickness of the H i disk (almost all type 1 and type 2 holes), we assumed that these have taken a cylindrical shape and therefore the volume is

  $$\begin{equation} V\,\mathrm{(pc^3)} = 2 \pi (d/2)^{2} \sqrt{8 \ln {2}}\,h, \end{equation} \tag{ 11 }$$

  where h (in pc) is the 1σ scale height and the term $\sqrt{8 \ln {2}}\,h$ expresses the thickness of the disk.
• 6.  
  Indicative H i mass (M_H i). This is an estimate of the mass that filled the hole, assuming the hole is spherical and completely empty and that the gas had a uniform distribution with volume density n_H i before the creation of the hole. In addition, assuming that only a small fraction of the surrounding H i is ionized, this value indicates the amount of H i mass which is expected to have been swept up to form a rim for those holes not blown out of the disk. M_H i was calculated using

  $$\begin{equation} M_{\rm H\,\mathsc{i}}\,{(M_{\odot })} = 2.9 \times 10^{5}\, n_{\rm H\,\mathsc{i}}\, V, \end{equation} \tag{ 12 }$$

  where n_H i is the volume density of the gas around the hole in cm⁻³ and V its volume in pc³. The uncertainty of M_H i is around 30%.
• 7.  
  The galactocentric distance (R) of the hole. This was calculated using

  $$\begin{equation} R\,\mathrm{(pc)}=D\times 10^{6}\sqrt{(x^{\prime \prime })^{2}+(y^{\prime \prime })^{2}} \,\mathrm{pc,} \end{equation} \tag{ 13 }$$

  where D is in Mpc and

  $$\begin{eqnarray} &&x^{\prime \prime }=x\sin \theta +y \cos \theta \\ &&y^{\prime \prime }=\frac{ y\sin \theta - x\cos \theta }{\cos i} \end{eqnarray} \tag{ 14 }$$

  and

  $$\begin{eqnarray} &&x = (\alpha -\alpha _\mathrm{0}) \cos \delta _\mathrm{0} \\ &&y = \delta - \delta _\mathrm{0}, \end{eqnarray} \tag{ 16 }$$

  where θ is the P.A. of the galaxy defined by the direction of the receding major axis, i its inclination, (α, δ) are the celestial coordinates of the center of the hole, and (α₀, δ₀) are the celestial coordinates of the center of the galaxy (see Table 2).
• 8.  
  The estimated energy requirements (E_E) of the hole. Chevalier (1974) derived an equation for the energy requirements of an expanding shell, assuming this energy is provided by a single explosion:

  $$\begin{equation} E_\mathrm{E}\,\mathrm{(erg)} = 5.3 \times 10^{43}\, n_\mathrm{o}^{1.12}\, (d/2)^{3.12}\, V_\mathrm{exp}^{1.4}, \end{equation} \tag{ 18 }$$

  where V_exp is the expansion velocity in km s⁻¹, n_o is the density of the ambient medium in particles cm⁻³, and d is the diameter of the hole in pc. As was done in the case of M 31 (Brinks 1981), Ho II (Puche et al. 1992), IC 2574 (Walter & Brinks 1999), and DDO 47 (Walter & Brinks 2001), n_o was replaced by n_H i, which means we ignored contributions by He and H₂. In order to correct E_E for an He contribution, a factor of 1.5 would need to be applied. Due to the large number of assumptions, in Chevaliers equation, the uncertainty in E_E is of the order of 50%. Despite the large uncertainty, for reasons of consistency with previous papers on this subject, and for want of a better alternative, we decided to derive energies using this equation.

The set of figures we will describe next shows the number distributions of the observed and derived properties of the holes. We start with showing the diameter distribution of the holes in order to investigate if there are any obvious trends regarding their size distribution, both within a galaxy as well as from target to target. The result is shown in Figure 3 which shows a mosaic of 20 panels, each panel showing a histogram representing the relative (normalized) frequency distribution as a function of diameter of each of the 19 targets, plus a final panel showing the distribution summing over all targets. The bins are 200 pc in size. There are obvious differences among the galaxies. However, in this and subsequent figures one has to guard against selection effects. The absence of small diameter holes in, e.g., NGC 2841, NGC 3521, and NGC 3184 is due to their larger relative distance which sets a lower limit to the linear resolution. We therefore add a vertical dotted line to indicate the resolution limit for each galaxy. Also there are other factors influencing the detection of holes such as the signal-to-noise ratio, and the inclination of the object under study (e.g., NGC 2841 and NGC 3521). The large diameter end of the distributions varies greatly among the galaxies. In three cases, NGC 2976, NGC 3031, and NGC 4736, we see a sharp cutoff at around 400 pc which, at first glance, does not seem to correlate with galaxy type or any other global property. On the other hand, it appears that Scd type galaxies tend to have small numbers of large (>1 kpc) holes in their H i distribution. We return to this below.

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The relative number distribution of the expansion velocities is plotted in Figure 4. The bins correspond to 2 km s^-1 and only holes for which an expansion velocity could be measured are plotted. The distributions appear to vary significantly among the galaxies mainly because of small number statistics.

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Figure 5 shows the relative number distribution of the kinetic ages of the holes in the 19 galaxies in a format similar to Figure 3. The bins measure 10⁷ yr. The shaded histograms correspond to those holes for which an expansion velocity could be measured, i.e., types 2 and 3. For type 1 holes, which are likely to have stalled, we adopted a lower limit to the expansion velocity which was set to the one-dimensional velocity dispersion of the gas. The kinetic age distributions are similar to those of the diameters of the holes which is to be expected since the kinetic age is derived from the diameter and the expansion velocity which, across the targets, has a small range. M 81 (NGC 3031) shows a quite striking distribution. As mentioned in the previous paragraph, this galaxy predominantly has small holes. Their ages are all below 40 Myr. One possible explanation is that M 81 is actively interacting with M 82 and NGC 3077: numerical models by Yun (1999) provide an estimate of 2.2 × 10⁸ yr for the most recent perigalacticon and it is conceivable that this interaction eventually sets off a sequence of events which led to a phase of enhanced SF 0.5 × 10⁸ yr ago, leading to the current abundance of small, young H i holes. Similarly, NGC 2976 appears to have had an SF episode 1–2 × 10⁷ yr ago triggered by gas infall from an intergalactic cloud generated during encounters between members of the M 81 group (Bronkalla et al. 1992).

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The relative number distribution of the characteristic energies required to create the holes is shown in Figure 6. The energies are binned logarithmically. It is evident that nearly all distributions fall in a range that spans 3 orders of magnitude. The shift toward higher energies for the galaxies NGC 2841, NGC 3184, and NGC 3521 is a selection effect due to their relative large distance which reduces the linear resolution and sensitivity to pick up small holes. In the case of NGC 3627, the shift toward higher energies is due to the fact that this galaxy has the highest velocity dispersion, which is likely driven by the central bar, and hence the energies of all type 1 holes have probably been overestimated.

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Figure 7 shows the relative number distribution of the indicative H i masses, binned logarithmically. The shape of these distributions is dominated by the diameter distribution of the holes, as the volume densities are showing a much smaller range of values (2σ from ∼0.1 to ∼0.7 cm^-3). The indicative masses span a range of (2 × 10³–3 × 10⁷ M_☉). Apart from NGC 2403, NGC 2976, and NGC 3031 which have their distributions skewed toward smaller mass values, all other galaxies cover the same range.

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The radial location of the holes in the disk of the host galaxy is illustrated in Figure 8. The horizontal axis represents the galactocentric distance normalized to the maximum extent of the H i disk, R_max (given in Table 3), in bins of size 0.1R_max. From this figure, it is evident that wherever there is H i, holes are found, even in the outskirts of galaxies. The vertical line in the figure indicates the location of R₂₅, confirming the finding that holes are found well beyond the main optical body of a galaxy. The number of holes per area has its peak near the center, though, as can be seen in Figure 9 which shows the log of the number of holes per kpc² as a function of R/R_max. The surface number density of H i holes is fairly constant or slightly drops out to R₂₅, beyond which radius it drops significantly in most of the galaxies.

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Finally, Figure 10 illustrates the relative number distribution of the axial ratio of the holes in each galaxy. It appears that dwarf galaxies have rounder holes, on average, than spirals (the mean axial ratio for dwarfs is 0.80 and 0.71 for spirals). This was first observed in Holmberg II by Puche et al. (1992) and explained by Walter & Brinks (1999) and Brinks et al. (2002) as follows. Within R₂₅ the H i disk in a dwarf is intrinsically thicker than in spirals (see Table 3) which prevents holes from blowing out at an early stage. Holes will therefore remain intrinsically spherical and the apparent axial ratio of an intrinsically spherical hole is not affected by the inclination of the host galaxy. Holes in spiral galaxies will tend to break out of the disk, leaving an approximately circular feature in the plane of the galaxy which, when seen under an angle, will appear elliptical. As will be discussed below another possible explanation for this trend is the effect that shear plays on the appearance of holes.

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The trend that dwarf galaxies tend to have rounder holes on average is also evident in Figure 11 which presents the axial ratio versus the diameter of the holes. The last panel in this figure shows the entire sample of holes from which can be seen that there are few, if any, round holes larger than 1 kpc. For holes smaller than 1 kpc, we see that the axial ratio takes the whole range of values indicative of the different factors that determine the axial ratio. This is the first of the next series of figures where we search for correlations between two observed and/or derived properties among the holes detected in each of the 19 targets.

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Figure 12 is a scatter diagram of axial ratio versus galactocentric radius. Shear at a point R_o with respect to a neighboring point at distance ΔR in a differentially rotating disk is given by

$$\begin{equation} S(\mathrm{km\,s^{-1}\,kpc^{-1}})=\frac{1}{2}\left(\frac{V_o}{R_o}-\frac{\Delta V}{\Delta R}\right), \end{equation} \tag{ 19 }$$

where V_o is the velocity at R_o, ΔR is the distance between the points, and ΔV is the difference in velocity between these two points. The last term in Equation (18) is zero for a flat rotation curve (ΔV = 0). Using the rotation curves of galaxies (where applicable), we calculated values for shear between points separated by ∼0.05% R_max as illustrated in Figure 12 (red line). The shear gradient is typically 10–50 km s⁻¹ kpc⁻¹, the upper boundary being typical for Sa galaxies, later types reaching half this value. This is enough to turn an originally circular structure of 500 pc diameter, into an elliptical feature with axial ratio of ∼0.5 within 50 Myr. The range in shear gradient listed is what one expects to find between the outer and inner regions of a spiral galaxy, the inner regions having the highest values (not considering the very inner solid body part). Most dwarf galaxies are dominated by solid body rotation, so shear is absent. One would therefore naïvely expect holes at larger radii to be rounder than holes at smaller radii. However, no such trend is apparent in the galaxies shown in Figure 12. This null result does not imply, though, that shear is not important in spirals. Holes are supposedly continuously formed, and more so within R₂₅, which means that there will be many young, and therefore round H i holes populating the inner regions of galaxies. In the outer regions, some holes might be sufficiently long-lived for the more modest shear there to have an observable effect. One would therefore expect a relation of axial ratio with kinetic age in the sense that on average older holes in shear dominated, i.e., spiral galaxies, are less round than younger ones (Silich et al. 1996; Elmegreen et al. 2002). This is investigated in the next figure, Figure 13. Except for some of the dwarf galaxies, where shear is not that important, only 5% of holes older than 50 Myr are round (average axial ratio is 0.70). In the case of younger holes, the entire range of axial ratio is found, as expected, as one gets a mix of young, round holes in low shear environments with elliptical ones that find themselves in regions of high shear. The destructive role of shear is also illustrated in Figure 14 where the diameter of holes is plotted against shear for galaxies for which it could be measured. There is a clear negative correlation between shear and the diameter of holes, i.e., in environments where shear is higher than ∼15 km s⁻¹ kpc⁻¹ no holes larger than 1 kpc were found.

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It is not just shear which defines the observed axial ratio of an H i hole. The axial ratio of those holes which have blown out and those that have lost their initial spherical shape due to shear depends crucially on viewing angle. The inclination should cause the major axis of holes to be aligned with the major axis of the disk. We also know that holes do not live in isolation and that two neighboring spherical holes can merge to form a larger elliptical one. All these effects will tend to wash out whatever trend of axial ratio there might be with radius. However, there are some conclusions which can be drawn from Figures 10–13: (1) H i holes in dwarfs are rounder than in spirals (see Section 5.3), (2) blow-out holes in spirals are not circular as we do not observe any alignment between the major axis of the holes and the major axis of the disk (if anything, cursory inspection of the figures in the Appendix suggests the holes align with the spiral arm structure), and (3) there is a sharp cutoff to the axial ratio at a level of 0.4. Regarding (3), in the identification process there was no requirement for holes to be less flattened than a certain value. The sharp lower envelope at 0.4 then suggests that structures more elongated than that become difficult to identify as holes.

The diameter of a hole versus its position in the host galaxy is shown in Figure 15 where the horizontal axis represents the galactocentric distance as a fraction of the H i disk radius. In the case of dwarf galaxies, there is a tendency for large holes to be located at larger galactocentric radii. In spiral galaxies, no such trend is seen. This is somewhat contrary to expectations. As we have explained in Section 3, galaxy disks flare at large radius. One would therefore expect holes to be able to grow to larger diameters before breaking out and subsequently stalling, considering that the ambient density also drops with radius. The lack of finding a corresponding trend in the diameters of holes could be due to a variety of factors, one being that there is a growing incompleteness of finding holes in the very low H i surface brightness outer regions of spirals. Alternatively, we know that SF is inefficient at large radii (e.g., Kennicutt 1989; Martin et al. 2001; Schaye 2004; Thornley et al. 2006; Bigiel et al. 2008; Leroy et al. 2008) and consequently the number of stars more massive than 8 M_☉ which are formed cospatially decreases, reducing the total energy deposited after their demise as SNe, resulting in holes of more modest diameter. Another possible explanation is that the ambient density drops significantly perpendicular to the disk of the galaxy and therefore holes will tend to be elongated in this direction. Consequently, even though these chimney-like structures will be intrinsically larger in the direction perpendicular to the disk, when viewed inclined, they might appear to have a lower ellipticity and mimic spherical features.

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Figure 16 shows the diameter of a hole versus its expansion velocity for those holes for which we could measure this, i.e., types 2 and 3. Contrary to the findings by Stanimirović (2007), we do not find a trend of increasing expansion velocity with increasing size. However, Stanimirović (2007) was able to detect holes down to 15 pc where this trend is strongest. We should also like to note that their result is based on just one interacting system, the LMC/SMC. There are examples in Figure 16 of individual galaxies which show a similar trend, such as Ho II and NGC 4736. However, this trend does not dominate across galaxies. This is also illustrated in Figure 17 where we plot on a log–log scale expansion velocity versus hole diameter for all expanding holes across the entire sample of galaxies. In this figure, the straight lines correspond to lines of constant age, drawn at (0.5, 1, and 5) × 10⁷ yr. Assuming Chevalier's single blast approximation and, for simplicity a constant ambient density of n_○ = 1.0 cm⁻³, we can draw curves of constant required energy. There are two competing mechanisms which both contribute to the way the points are distributed in Figures 16 and 17. Once an SN has gone off, the ejecta which initially move at high speeds will slow down as they interact with the surrounding circumstellar medium and ISM, eventually entering the snowplough phase which is when we tend to identify them as H i holes. One would therefore expect larger holes to have lower expansion velocities. On the other hand, as long as stars from the parent OB association go off as SNe, energy is continuously deposited and, if the expanding shell has not yet suffered blow-out, will maintain an overpressure within the shell which in turn will keep up an accelerated flow. The latter mechanism is more likely to dominate in the smaller, younger holes. It is then interesting to note that the distribution of points in our Figure 17 agrees with the corresponding graph in Stanimirović (2007) if one restricts the comparison to holes larger than 100 pc. In other words, we hypothesize that the relation found by these authors is possibly a consequence of the smaller holes still harboring SN activity.

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In Figure 18, the energy requirement of each hole versus its radial distance is presented. The radial distribution is normalized by the radius of the H i disk, R_max. In 10, mostly spiral galaxies there appears to be a trend of higher energy holes to be located at smaller galactocentric radius. Further analysis showed that in all galaxies, apart from NGC 4214, the average energy requirement is higher for holes lying within R₂₅ (E_E = 2.1 × 10⁵² erg) than for those outside this radius (E_E = 1.0 × 10⁵² erg). This might be due to the fact that the external pressure is higher within R₂₅. First of all the gas density is higher near the center, as can be verified in the paper by Leroy et al. (2008). Second, the SFR is higher in the central regions than in the outskirts which translates into more holes being created (cf. Figure 9), increasing the velocity dispersion and hence pressure of the gas.

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We also investigated the following relations, but there do not seem to be any trends of expansion velocity with either of axial ratio, displaced mass, energy required to produce the hole, or with galactocentric radius. Neither is there any relation linking kinetic age with position from the center in a galaxy.

As was mentioned earlier, we selected 20 galaxies out of the 34 available. We ensured that we had two galaxies per Hubble type. Within each Hubble type, we tried to select galaxies which differed in some property, such as SFR, to explore as wide a parameter space as possible. If we look at the distribution of the kinetic age of the holes across the sample, we find a trend which becomes more obvious if we group the galaxies in terms of early-type, late-type, and dwarf galaxies as illustrated in Figure 19. The mean kinetic age shifts toward higher values when we move from early-type spirals ($\overline{t_\mathrm{kin}}=13.9\,\mathrm{Myr}$) to late-type ($\overline{t_\mathrm{kin}}=31.6\,\mathrm{Myr}$) to dwarf galaxies ($\overline{t_\mathrm{kin}}=32.5\,\mathrm{Myr}$). This can be understood as follows. The kinetic age is driven by the diameter distribution, as the expansion velocities cover a limited range, as mentioned earlier. Disks of earlier type spirals present shear and spiral density waves, both mechanisms actively rearranging the gas within the disks and limiting the time an H i hole may survive. Dwarf irregulars lack both shear and spiral arms. In spiral galaxies where shear is strong (NGC 2976, NGC 3031, NGC 3627, and NGC 4736), the maximum kinetic age is observed to be 40 Myr in contrast to the dwarf galaxies where we find holes of up to 140 Myr. Overall the kinetic age ranges from 10 to 140 Myr, the upper limit reflecting the fact that holes would blend and possibly be destroyed after one full rotation of the disk. A second explanation for the observed trend is the same argument that applies as used when discussing Figure 10, namely, that the scale heights of the disks in early-type spirals are small in comparison with those in dIrrs which means that shells will break out of the disk sooner in the former and the remnant shell will stall, leading to H i holes in spirals having on average smaller diameters, hence leading to systematically lower age estimates.

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In order to make this analysis more quantitative, we decided to split the sample up in spiral and dwarf galaxies. The subsamples consist of 12 and 7 members, respectively, with a total of 831 versus 183 H i holes. The comparison between the subsamples was carried out via Mann–Whitney (M–W) and Kolmogorov–Smirnov (K–S) tests. Both are non-parametric tests for assessing whether two samples can be drawn from the same parent distribution based on differences in the location of the medians (M–W) and differences in the shape of the two distributions (K–S). In order to accept or reject a hypothesis, we took a 3σ level as the minimum significance level on both tests. In the following, we compare the two subsamples of spirals and dwarfs in graphical form. The results of the M–W and K–S tests are summarized in Table 5.

Figure 20(a) shows the difference in the distribution of diameters of the holes between dwarf and spiral galaxies. It appears that dwarf galaxies have larger holes on average than spiral galaxies. An M–W test confirms this as well. This is of course not surprising in the light of the current discussion. What is surprising is that dwarf galaxies appear to have fewer small holes (d < 200 pc) even though the spatial resolution is better on average. This can be understood by realizing that whereas SF is a continuous process in spiral galaxies, it tends to occur in bursts in dwarfs. The time between bursts is longer than the duration of the SF episodes (e.g., Gerola et al. 1980) so it is likely that the dwarf galaxies whose ISM is currently riddled with H i holes are in a post-starburst phase. There will therefore be a relative lack of small holes, most holes having had the time to grow to typically ∼100 pc and beyond.

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In this comparative analysis, we included all holes. However, the galaxies are spread over distances which cover a factor of ∼3 between the closest and most distant one (in addition, the angular resolution varies slightly across the sample). Therefore, small holes in those galaxies that are further away will escape detection. If we exclude NGC 2841 and NGC 3521 from our sample then for the rest of the galaxies there is a minimum common spatial resolution of 280 pc, i.e., we should have a complete sample of holes larger than 280 pc in 17 galaxies.

Figure 20(b) illustrates the distribution of diameters of holes larger than 280 pc between dwarf and spiral galaxies. We now see that spiral galaxies have slightly larger holes than dwarf galaxies. We can understand this as follows: the energy requirement of expanding holes larger than 280 pc, Figure 20(c), shows a clear differentiation in the distribution of the two types of galaxies (K–S test). Holes in spiral galaxies appear to be significantly more energetic. Based on the way the energy is derived this difference could be due to the fact that holes in spiral galaxies are intrinsically more energetic, for example, because OB associations in spirals are more massive. The effect of spirals having a larger energy input, on average, translates into spiral galaxies having larger H i holes.

As was mentioned earlier, it appears that dwarf galaxies have rounder holes than spirals. This trend is apparent in Figure 20(d), where we plot the distributions of the axial ratio for the two types of galaxies. It is again clear that the two distributions differ (M–W test).

The spatial distribution of the holes appears to be dependent on the type of the host galaxy as illustrated in Figure 20(e). In spiral galaxies, holes appear to be concentrated within a smaller area of the disk, whereas in dwarf galaxies holes are distributed over a larger area. This is confirmed by an M–W test. Even though the difference between the two distributions is subtle the result can be understood as being due to the fact that SF in spiral galaxies mostly happens in the spiral arms which are located within the R₂₅ radius. In dwarf galaxies, SF is spread across the entire disk.

To investigate this further, we divided our sample of holes based on R₂₅, i.e., holes located inside and outside of this radius. We find that regardless of the type of the galaxy, holes outside R₂₅ are larger and older, as illustrated in Figures 21(a) and (b). This is confirmed by the relevant M–W test. This is in agreement with holes outside R₂₅ being less affected by processes such as shear. When we look at the sample of holes larger than 280 pc, no significant differences are found in the diameter and kinetic age distributions within and outside R₂₅.

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There is no discernible difference in expansion velocity (Figure 21(c)) even after we remove the stalled holes from our sample (Figure 21(d)). Looking at the distribution of the energy requirement of holes larger than 280 pc with respect to R₂₅, we find that on average holes have higher energies when located inside this radius (Figure 21(e)). This holds true even when we break down our sample according to galaxy type (dwarf or spiral). Given that the diameter and expansion velocity distributions are similar in these two regions we conclude that this is driven by the difference in volume density.

Regarding the types of the holes, we found that in spiral galaxies there are more partially and fully contained holes (types 2 and 3) within R₂₅ (Figure 21(f)). This is in agreement with the fact that holes inside R₂₅ are smaller and younger, on average.

Because we are essentially complete (to our detection limit) for holes larger than 280 pc in diameter, we looked for relations of this subset of holes across our galaxy sample (excluding NGC 2841 and NGC 3521). The number per kpc² of these holes is presented in Figure 22 where this number density is plotted on a log scale against (1) optical diameter of the host galaxy and (2) against galaxy (Hubble) type. It is clear that despite the scatter there is a clear trend in the sense that the number density of holes decreases with increasing galaxy size.

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In the plot of number density against Hubble type there is a weak trend, the number density remaining more or less constant except for the class of dwarf galaxies (which are dominated by large holes). This trend is also apparent in Figure 23 which shows the average porosity of all the holes in a galaxy, versus the type of the host galaxy. The porosity parameter, Q (Cox & Smith 1974), is the ratio of total volume (or area) occupied by holes to the total volume (or area) of the host galaxy. For the total volume of the galaxy, we used R_max and the radial scale height as the thickness of the galaxy. The dwarf galaxies stand out as being on average more porous than spirals. Silk (1997) predicted that the porosity should anti-correlate with SF surface density. As illustrated in Figure 23, later Hubble types have higher porosity, and these same late-type galaxies form stars at a lower rate per unit mass than earlier types (Leroy et al. 2008). Figure 24 shows the two-dimensional porosity as a function of radius. Although similar to the radial H i hole surface density distribution (Figure 9), it is not identical as the two-dimensional porosity distribution takes into account not just the number of holes, but also their size. What we find is that the porosity generally decreases as a function of radius with, as an exception perhaps, several of the dwarf galaxies such as Ho I and Ho II, where the histograms stay flat. The low porosity in the outskirts is likely a reflection of the generally lower SFR density in these regions, i.e., well beyond the optical radius of a galaxy.

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The ISM of galaxies is dominated by structures which are thought to originate from local energetic events. Even though OB associations, through the effects of stellar winds and SNe, seem to be the obvious mechanism behind the creation of these structures (Oey & Clarke 1997), a direct correlation between OB associations and H i holes is often lacking (e.g., Kamphuis 1993; van der Hulst 1996; Rhode et al. 1999; Pasquali et al. 2008; Weisz et al. 2009a). Furthermore, there have been cases of holes with energy requirements that surpass the energy input from OB associations, or of holes which are found in regions, such as the Magellanic Bridge (Stanimirović 2007) where SF is essentially absent. This has often been used as an argument to question the SN origin leading to the many alternative explanations listed in the introduction. There are cases, notably the LMC/SMC where a clear link is found between massive stars and the location of H i holes (Stanimirović 2007), although even there in some 10% of the cases no such link is evident (Hatzidimitriou et al. 2005).

As Figure 8 illustrates, in all but three galaxies many holes are found beyond the optical disk (R₂₅) of the parent galaxy. If the holes are indeed due to OB associations this picture may be problematic as the SFR density drops steeply beyond R₂₅. It has been suggested that a "galactic fountain" (Bregman 1980), i.e., material expelled into the halo of the galaxy by SN explosions, could be invoked. Hot fountain gas would eventually cool and fall back on to the disk, creating holes away from the original location of the OB association. If this is indeed the case then one would expect to find extraplanar gas in abundance surrounding the H i disk. Evidence of extraplanar gas expelled from galaxy disks, as expected in the "galactic fountain" model has recently been found by Oosterloo et al. (2007) and Boomsma et al. (2008). In addition, models of extraplanar gas by Fraternali & Binney (2006) and deep H i studies by Sancisi et al. (2008) do show a substantial supply of gas within a thick H i disk and halo, but their analysis seems to favor an external origin for this gas.

Recent results from the GALEX NGS (Gil de Paz et al. 2007; Thilker et al. 2007) have obviated the need for these alternative explanations. Many galaxies show extended UV (XUV) disks. HST ACS observations resolve these XUV-disk sources into loosely clustered complexes, likely evolved OB associations (Thilker et al. 2008) and it thus seems safe to assume for the moment that the majority of H i holes, even those well beyond R₂₅ are the result of massive SF. If this is the case we can derive an independent estimate of the SFR of the host galaxy based on the energy requirements of the holes in a manner similar to how H ii regions are related to the SFR based on measurements of their Hα flux; for an instantaneous burst and as long as the SF region is younger than 20 Myr, the Lyman-continuum output is constant and scales linearly with the SFR. An instantaneous burst does not have an SF rate but taking an ensemble average over an entire galaxy, this becomes equivalent to seeing a continuous SFR. The conversion between Ly-cont flux and SFR thus becomes independent of burst age. In the case of H i holes we are looking at individual and instantaneous events which, if averaged over an entire galaxy can similarly be linked to an SFR, under the assumption that every SF event will result in a hole. This assumption is likely to lead to an underestimate of the true SFR, of course.

If we assume that the SNe that create H i holes are all of Type II and that the kinetic energy released by a Type II SN is 10⁵¹ erg (McCray & Kafatos 1987), we can estimate for each hole the number of stars in the parent stellar association with masses above 8 M_☉ (the minimum mass a star must have to go off as a Type II SN; Biermann & Tinsley 1974).

Figure 25 shows the number of SN per 10 Myr bin, i.e., the SF history of each galaxy by plotting a histogram of the number of SN needed per hole against age of the hole for all holes in a galaxy. The letters E, L, and D in the upper right corner stand for early-type, late-type, and dwarf galaxy. For most of the galaxies (and specially all early types), there is a cutoff at ∼60 Myr indicating there is large incompleteness due to shear and diffusion. The lack of SNe younger than ∼10 Myr in some of the galaxies is a reflection of the lower resolution obtained for these galaxies. So, overall, holes trace the SN history from 10 to 60 Myr, a time frame of 50 Myr. From Figure 25, we learn that for most galaxies the SN history as derived from the H i holes is more or less constant. What this means is that by taking an ensemble average over a galaxy, and assuming a typical lifetime for an H i hole of 60 Myr, we can convert the SN history into a supernova rate (SNR). The values for the calculated SN rates are listed in Table 6.

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Table 6. Supernova and Star Formation Rates

Galaxy	log(SFRHα)	log(SFRHoles)	log(SFRcorr)	log(EE)	log(νSN)	log(νcorrSN)	aν	Q2D	Q3D
	$({M_{\odot }}\rm \,yr^{-1})$	(${M_{\odot }}\rm \,yr^{-1})$	$({M_{\odot }}\rm \,yr^{-1})$	(1050 erg)	(SN yr-1)	(SN yr-1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
NGC 628	0.1	−2.0	1.3	4.9	−4.2	−0.8	−3.6	3.1%	4.8%
NGC 2366	−1.0	−3.2	−1.2	3.4	−5.3	−3.3	−2.6	2.6%	0.8%
NGC 2403	−0.1	−2.8	−0.6	4.0	−4.9	−2.7	−2.7	2.7%	2.9%
Holmberg II	−1.1	−2.7	−2.2	4.4	−4.8	−4.3	−0.8	12.8%	16.3%
DDO 53	−2.1	−4.2	...	2.0	−6.1	...	...	...	...
NGC 2841	−0.7	−3.0	...	3.8	−5.1	...	...	0.3%	0.3%
Holmberg I	−2.2	−4.3	...	2.7	−6.4	...	...	1.9%	0.8%
NGC 2976	−1.0	−5.1	...	2.3	−7.2	...	...	1.5%	0.9%
NGC 3031	0.0	−3.5	0.3	3.3	−5.6	−1.8	−3.9	0.8%	0.4%
NGC 3184	0.2	−2.2	−0.4	4.9	−4.3	−2.5	−2.7	3.0%	2.0%
IC 2574	−0.9	−3.1	−1.8	4.3	−5.3	−4.0	−1.8	5.1%	7.0%
NGC 3521	0.5	−2.3	...	4.3	−4.4	...	...	0.4%	0.4%
NGC 3627	0.4	−2.7	0.6	4.1	−4.8	−1.6	−3.5	1.9%	2.8%
NGC 4214	−1.3	−3.1	−0.2	3.7	−5.2	−2.3	−3.1	11.8%	19.4%
NGC 4449	−0.3	−2.4	−0.1	4.3	−4.5	−2.2	−3.1	7.4%	2.5%
NGC 4736	−0.4	−2.7	0.7	4.2	−4.8	−1.4	−3.8	0.7%	0.1%
DDO 154	−2.4	−4.1	...	2.5	−6.2	...	...	1.1%	0.3%
NGC 5194	0.8	−1.7	0.4	4.9	−3.8	−1.7	−3.1	3.9%	6.1%
NGC 6946	0.7	−2.1	0.3	5.0	−4.2	−1.8	−2.8	3.4%	3.0%
NGC 7793	−0.3	−2.6	−1.2	4.0	−4.8	−3.3	−2.4	2.9%	2.5%

Notes. (1) Galaxy name. (2) SFR based on Hα. (3) SFR based on observed holes. (4) Corrected SFR derived from hole distribution function. (5) Total energy of holes. (6) SN rate based on observed holes. (7) Corrected SN rate derived from hole distribution function. (8) The slope of the hole distribution function. (9) Surface porosity. (10) Volume porosity.

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The SNR is proportional to SFR(M>8 M_☉), the SFR for stars more massive than 8 M_☉. We can derive the total SFR by assuming a Salpeter initial mass function (IMF) and integrating this between upper and lower stellar mass limits of 120 and 0.1 M_☉. Given the fairly large uncertainty in the energy requirements of the holes the choice of IMF is not particularly critical. Using a Salpeter IMF we find that SFR ∼134 × SFR(M>8 M_☉). The calculated SFR over 50 Myr, hereafter referred to as SFR_Holes, correlates very well (Spearman's rho = 0.78) with the current (Hα) SFR as illustrated in Figure 26 and Table 2 but our values fall short by 2–3 orders of magnitude. The error bars are formal errors using the uncertainties quoted in the observed and derived quantities.

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The SFR based on the H i holes is a lower limit and depends critically on the energies we estimated. Various factors conspire to make this a lower limit. Our estimate for the kinetic age is an upper limit therefore the SFR derived here is a lower limit by default. In addition, our limited spatial resolution means that we are missing holes smaller than 100–200 pc in size. Given that these dominate the H i distribution (compare, e.g., the higher linear resolution data on the LMC/SMC), they account for a large fraction of the SF. From theoretical models (Oey & Clarke 1997) and from studies of the Magellanic Clouds, we know that the size distribution of holes follows a power law. This is also evident from our data for the galaxies with the highest resolution (e.g., NGC 3031, NGC 2403, and Holmberg II). We derived an H i hole distribution function for each of the galaxies and fitted a power law to the high end part of each galaxy. On average, we find an exponent of a_ν = −2.9, in good agreement with Oey & Clarke (1997) who predict −4 < a_ν < −2. This is illustrated in Figure 27 for the integrated distribution function over all galaxies, for holes with sizes larger than 280 pc. The average error of a_ν is of the order of 25% although in cases where the sample of holes is small the error is of the order of 50%. As Figure 28 illustrates there is a correlation between a_ν and the type of the galaxy. Early-type spirals exhibit a steeper slope than late-type spirals and dwarf galaxies. This is a consequence of what we discussed above, namely, that early-type spirals are dominated by a large number of small holes whereas dwarf galaxies have proportionally larger holes. We can extrapolate this power law to the smallest size hole that has been discovered (Milky Way and SMC). Integrating from 10 pc to the spatial resolution in a particular galaxy, we can derive corrected SN and SF rates, ν^corr_SN and SFR_corr for all galaxies, assuming that every star-forming region will create a hole. These values (tabulated again in Table 6) are likely to be representing the current SF since the vast majority of holes will now be small and young. We plot SFR_corr in Figure 29 and find that our rather bold extrapolation indeed brings the corrected SFR to within the range found based on other SFR indicators, although the scatter has increased considerably as well. We do not wish to push this exercise too far but if anything, the hypothesis that holes are a consequence of massive SF is compatible with our observations. Moreover, future studies at higher linear resolution and increased sensitivity should allow us to probe the size distribution down to H i holes created by a single SN event, leading to a near complete census of such features and a thorough testing of this assertion. Deep optical and UV observations that are able to resolve individual stars will be instrumental in characterizing the expected remnant stellar population inside the holes.

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NGC 628 is a grand-design Sc spiral galaxy located at a distance of 7.3 Mpc and is seen almost completely face-on. We detected 104 holes in its H i layer, with diameters ranging from 0.24 kpc (the resolution limit) to 2 kpc. Only a small fraction of these holes are type 3 (≈5%). Several supershells were detected, however four cases (holes number 24, 45, 47, and 99) are probably superpositions of two or more holes.

NGC 2366 is a dwarf galaxy member of the M 81 group. It is considered to be a typical irregular galaxy and is characterized by a supergiant H ii complex (NGC 2363) in the southwest (Youngblood & Hunter 1999). We detected 26 holes in NGC 2366 but since this is a highly inclined system (i = 638) we might be missing out on some features. All of the detected holes appear to be relatively young as their kinetic age was found to be between 6 and 38 Myr. In a previous H i study, Hunter et al. (2001) searched for but did not detect any holes, however their spatial resolution was ≈260 pc, almost 2.5 times worse than our resolution.

Another member of the M 81 group, NGC 2403, is a late-type spiral galaxy located at a distance of 3.2 Mpc. It has previously been studied by Thilker et al. (1998) and by Mashchenko et al. (1999) using an automated object recognition package who detected 50 and 601 holes, respectively. Even though this galaxy is highly inclined (i = 629), its close proximity and good resolution allowed us to detect 173 holes in its H i distribution. A few supershells were detected but the majority (81%) of the holes are smaller than 350 pc in diameter and younger than 20 Myr. There are also several special cases: some of the holes are most likely a superposition of two or more holes (hole number 51, 157, and 173); holes number 10 and 43 have a peculiar signature in pV space.

Holmberg II is a gas-rich dwarf galaxy located at a distance of 3.4 Mpc and is a member of the M 81 group of galaxies as well. An H i hole analysis was carried out by Puche et al. (1992) in which they found 51 holes across the entire face of this galaxy. We detected 39 holes in Holmberg II the majority of which were also detected by Puche et al. (1992), all with very good agreement with respect to the position and size of the holes. The difference in the number of holes detected is due to our strict criteria of what constitutes a genuine hole as well as to the fact that several holes in our catalog were classified as a superposition of two (or more) holes from the Puche et al. catalog. As panel (C4) in the online version of Figure 30 illustrates, these holes dominate the H i distribution of Holmberg II. In addition, several supershells were detected, the largest one (no. 17) being over 2 kpc, one of the largest and oldest (∼150 Myr) across the entire sample of holes. Note that hole no. 37 is probably a superposition of two or more holes.

The dwarf irregular galaxy DDO 53 lies at a distance of 3.6 Mpc and also belongs to the M 81 group. It is the smallest galaxy in our sample with an H i radius of only 2 kpc. It is therefore not surprising that we detected only three holes in its H i distribution, one of each type, with diameters in the range of 180–360 pc.

NGC 2841 is an early-type Sc spiral and is the most distant galaxy in our sample located at 14.1 Mpc. The limited spatial resolution and the high inclination of this galaxy meant that we were only able to detect supershells of type 1. We found 15 H i holes and only one of these shows signs of expansion (number 10). In addition, holes number 4, 12, and 13 are possibly superpositions of holes. Hole no. 5 is probably not a genuine hole but the result of the warp in the H i disk.

Holmberg I is a dwarf irregular galaxy which forms part of the M 81 group of galaxies. Its H i distribution is dominated by a supergiant shell as was noted by Ott et al. (2001) which does not coincide with the kinetic center of the galaxy. We detected six holes in this galaxy including the one detected by Ott et al. (2001). There is good agreement with respect to the position of this hole between both studies.

The dwarf galaxy NGC 2976 is located at 3.6 Mpc and belongs to the M 81 group of galaxies. In NGC 2976, we found only 10 holes, which is not surprising given that this is one of the smallest galaxies in our sample with an H i disk radius of R_max = 2.5 kpc. No supershells were detected.

NGC 3031, also known as M 81, is a grand-design spiral galaxy located at a distance of 3.6 Mpc. The analysis of NGC 3031 revealed 321 H i holes, the largest number of holes detected in any of the galaxies in this sample. This is due to various factors such as its relatively large size, close proximity, and high spatial resolution of the observations. We were able to detect holes down to ∼80 pc which led to the detection of a wealth of small holes; almost 90% of the total holes detected have a size less than 200 pc. Only a handful of supershells were detected. The mean kinetic age is 7.8 Myr with 80% of the holes being younger than 9 Myr. Several features detected (holes nos.: 34, 72, 88, 93, 94, 97, 99, 102, 106, 109, 113, 114, 170, 255, and 273) are probably not genuine holes but the result of the warp in the H i disk. Due to its large angular size, M 81 was divided into six areas (panels (C9)–(C15) in the online version of Figure 30) to be able to show all detected holes.

NGC 3184 is a spiral galaxy located at D = 11.1 Mpc and is viewed almost face-on. Its analysis revealed 40 H i holes, the majority of those being supershells (80%). This could be attributed to the large distance of this galaxy which only allowed us to detect holes with diameters larger than 400 pc. We failed to detect any type 3 holes something which is expected since the scale height of the disk was found to be 250 pc, significantly smaller than our spatial resolution. Nevertheless, this is the largest number of supershells detected in any one galaxy. The mean kinetic age of the holes is also high (≈64 Myr). Note that hole number 13 is a superposition of two holes.

Another nearby (D = 4.0 Mpc) dwarf galaxy member of the M 81 group, IC 2574 hosts a relatively large number of H i holes (Walter & Brinks 1999). One particular hole (our hole no 21) has been extensively studied at different wavelengths from radio to X-ray (Stewart & Walter 2000; Cannon et al. 2005; Weisz et al. 2009b). These studies showed that there is a remnant star cluster in the center of this hole and revealed evidence of propagating SF along its rim. Here, we report on 29 H i holes compared to 48 holes detected by Walter & Brinks (1999) and 22 detected by Rich et al. (2008) the largest one being larger than 2 kpc in diameter. Almost all of the 29 holes we present here have been detected by Walter & Brinks (1999). The difference in the total number of holes detected is due to the strict criteria we used to classify a structure as a genuine hole. Almost half of the holes (45%) show signs of expansion.

Another distant galaxy in our sample, NGC 3521 is a spiral galaxy with a prominent bar located at a distance of 10.7 Mpc and viewed almost edge-on (i = 727). The analysis of NGC 3521 revealed 13 holes in the H i distribution. The small number of holes detected and the lack of type 3 holes can be attributed to the large distance of this galaxy as well as its high inclination. All of the holes detected were consequently classified as supershells.

NGC 3627 is a spiral galaxy at an adopted distance of D = 9.3 Mpc and is a member of the Leo Triplet. Its high inclination (i = 618) meant we were only able to detect 18 holes in NGC 3627, two of them being type 3. One holes worth noting is 7 which has unusually high expansion velocities of 65 km s⁻¹ respectively the highest observed in the entire sample.

NGC 4214 is a nearby (D = 2.9 Mpc) dwarf irregular galaxy classified as a starburst galaxy. We found 56 H i holes in NGC 4214 with a wide range of sizes, from 0.11 to 1.2 kpc. Given the close proximity of this galaxy and consequently the high spatial resolution, it is surprising that only about 7% of the holes detected are type 3. One possible explanation is that NGC 4214 has a low scale height and therefore holes can break out of the disk faster.

Another dwarf irregular galaxy, NGC 4449, is located at a distance of D = 4.2 Mpc. It has a very extended H i structure reaching out to six times the Holmberg radius. The R_max radius for this galaxy was taken at the edge of the H i disk thus excluding this structure. What is also unusual about NGC 4449 is that it has two counterrotating gas systems (Hunter et al. 1998). We detected 20 H i holes in NGC 4449, most of them located within R₂₅ and six located in the extended structure outside the R_max radius.

NGC 4736 is a ringed spiral galaxy located at D = 4.7 Mpc in which we encountered 23 holes in its H i distribution. The range of their diameters is quite small (0.21–0.45 kpc) and they are young holes (t_kin < 22 Myr). As panel (C22) in the online version of Figure 30 illustrates, the majority of the holes are located at small galactocentric radii and only type 1 holes lie at larger galactocentric radii. Finally, note that hole number 11 is probably a superposition of two holes.

The gas-rich dwarf galaxy DDO 154 is a nearby galaxy with a very extended and warped H i disk. Hoffman et al. (2001) found two holes in the H i distribution which were also detected in our analysis. We report here on a total of nine holes. The scale height of DDO 154 was found to be ≈ 500 pc which explains why holes can grow to a relatively large diameter before breaking out of the disk, as is the case with shell number 6 (≈400 pc).

NGC 5194 is a grand-design spiral galaxy at a distance of D = 8.0 Mpc which is undergoing interaction with NGC 5195. We report 47 holes in NGC 5194 with only one type 3 hole detected. Most of holes are in fact supershells, with one of those (no. 5) being superposition of two or more holes.

Located at a distance of 5.9 Mpc, NGC 6946 is a large spiral galaxy seen almost face-on. Previous H i studies by Kamphuis (1993) and Boomsma (2007) revealed 19 and 121 H i holes, respectively. We detected 56 holes, a subset of those detected by Boomsma (2007). There is excellent agreement for holes larger than 700 pc between our samples but not for smaller ones. This is due to the strict criteria (Q>5) we applied on holes whose size is comparable to the resolution limit. NGC 6946 hosts the largest hole in our sample (no 58) with a size of 2.1 kpc and energy requirements in excess of 4 × 10⁵⁴ erg, which corresponds to ∼4000 SNe. Most of the holes detected (84%) are type 1, with two holes (nos. 15 and 16) appearing as a superposition of more than one holes.

A member of the Sculptor group, NGC 7793 is an Sd spiral galaxy at D = 3.9 Mpc. A detailed analysis revealed 27 H i holes distributed almost uniformly across the H i disk with sizes ranging from 0.19 to 1.05 kpc.