DWARF GALAXY STARBURST STATISTICS IN THE LOCAL VOLUME

Ultimately, the 11HUGS sample is a composite of numerous catalogs with diverse selection criteria. For details on the construction of the sample, the reader is referred to Lee (2006) and Paper I. Here, we estimate the completeness limits of our sample by performing a statistical test described in Rauzy (2001), which is similar to the well known V/V_max test (Schmidt 1968), but does not require the galaxy distribution to be spatially homogeneous. We also qualitatively compare our number density distributions with independently established B-band luminosity and H i mass functions.

2.1.1. The Rauzy T_C Completeness Statistic

The Rauzy (2001) T_C completeness statistic is analogous to the V/V_max test (Schmidt 1968), but does not rely on assumptions about the spatial homogeneity of the sample. The method is based on the estimation of the uniform variate ζ, which is a ratio of number densities. Specifically, if F(M) is the normalized integral of the luminosity function (LF) from −∞ to M, and Z is the distance modulus m − M, then

$$\begin{equation} \zeta \equiv \frac{F(M)}{F[M_{\rm lim}(Z)]}, \end{equation} \tag{ 1 }$$

where M_lim(Z) is the faintest absolute magnitude that a galaxy at a given Z can have and still be visible, given a limiting magnitude m_lim.

An estimate of ζ is given by

$$\begin{equation} \zeta _i\sim \frac{r_i}{n_i+1}, \end{equation} \tag{ 2 }$$

where r_i is the number of galaxies with M ⩽ M_i and Z ⩽ Z_i, and n_i is the number of galaxies with M ⩽ Mⁱ_lim and Z ⩽ Z_i. If M and Z are uncorrelated, as should be the case for a complete local sample of galaxies, the expectation value of ζ is 0.5. The variance is given by

$$\begin{equation} V_i=\frac{1}{12}\frac{n_i-1}{n_i+1}. \end{equation} \tag{ 3 }$$

The T_C completeness statistic is then computed as

$$\begin{equation} T_C=\frac{\sum \nolimits _{i=1}^{N_{\rm gal}}\big(\zeta _i-\frac{1}{2}\big)}{\sqrt{\sum \nolimits _{i=1}^{N_{\rm gal}}V_i}}. \end{equation} \tag{ 4 }$$

The test is preformed by computing T_C for subsamples truncated at increasing apparent magnitude limits. For complete samples, the values of T_C will follow a Gaussian distribution, with an expectation value of 0 and variance of order unity. Systematically decreasing negative values of T_C indicate that the sample becomes incomplete. We use the T_C statistic to determine the completeness for our main survey volume outside of the zone of avoidance (|b|>20°) as functions of the B-band brightness, 21 cm (H i) flux, and the Hα+[N ii] flux (Figures 1(a), 2(a), and 3(a)).

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By construction, the 11HUGS primary sample has been limited at B ∼ 15 to avoid the severe incompleteness that is known to set in at fainter magnitudes in the original surveys that have provided the bulk of our knowledge about the local volume galaxy population (e.g., Tully 1988b). To select the sample, we primarily used the NASA Extragalactic Database (NED), with initial cuts based on the Local Group corrected recessional velocity and the "indicative optical magnitudes" provided in NED's "Basic Data" service. Direct distance estimates from the literature and information on group membership were then added. The photometry compilation was refined by adopting measurements from the following large, homogeneous catalogs in the following order of preference: (1) B_T from the collection of dwarf galaxy observations obtained by van Zee and collaborators (van Zee 2000; van Zee et al. 1996, 1997) and Binggeli and collaborators (Barazza et al. 2001; Bremnes et al. 1998, 2000; Parodi et al. 2002); (2) B_T from the RC3 (de Vaucouleurs et al. 1991) as reported on NED; and (3) B_T as compiled and reduced to the RC3 system from HyperLeda. For 31 galaxies, measurements are not available from one of these sources. In these cases, the literature was searched and a B magnitude was taken from smaller data sets of published photometry. As a last resort, the "indicative optical magnitude" given by NED was adopted in seven cases. The resulting collection of data is given in Table 1 of Paper I, and is used to calculate T_C as a function of B. Corrections for Galactic extinction, but not internal extinction, are applied. The results are plotted in Figure 1(a). T_C drops precipitously below zero for B ≳ 15.6. This limit of B = 15.6 for the 11HUGS main survey volume corresponds to a completeness in M_B = −14.6 at 11 Mpc. In Figure 1(b), a plot of M_B with the distance is also shown to illustrate the depth of the sample.

Table 1. Starburst Number and Star Formation Fractions

Quantity	−19.0 ⩽ MB < −17.0 (N = 59)	−17.0 ⩽ MB < −15.0 (N = 93)	−15.0 ⩽ MB < −13.0 (N = 104)
N(EW >100 Å)	0	6	5
%(EW >100 Å)	0%	6%	5%
%LHα(EW >100 Å)	0%	23%	16%
N(EW >80 Å)	1	6	8
%(EW >80 Å)	2%	6	8%
%LHα(EW >80 Å)	4%	23%	20%
	Logarithmic Hα Distribution Statistics
	〈lg(EW)〉 = 31 Å	〈lg(EW)〉 = 32 Å	〈lg(EW)〉 = 26 Å
1σ range	20 Å, 47 Å	22 Å, 48 Å	12 Å, 55 Å
N ⩽ 1σ, N ⩾ 1σ	12, 11	21, 19	26, 17
%⩽1σ, %⩾1σ	20%, 19%	23%, 20%	24%, 16%
%LHα ⩽ 1σ, %LHα ⩾ 1σ	11%, 33%	10%, 41%	5%, 32%
2σ range	13 Å, 71 Å	15 Å, 71 Å	6 Å, 115 Å
N ⩽ 2σ, N ⩾ 2σ	2,2	13, 7	9, 5
%⩽2σ, %⩾2σ	3%,3%	14%, 8%	8% 5%
%LHα ⩽ 2σ, %LHα ⩾ 2σ	<1%, 9%	4%, 25%	<1%, 16%
3σ range	9 Å, 107 Å	10 Å, 105 Å	3 Å, 242 Å
N ⩽ 3σ, N ⩾ 3σ	0, 0	8, 6	5, 3
%⩽3σ, %⩾3σ	0%, 0%	9%, 6%	5%, 3%
%LHα ⩽ 3σ, %LHα ⩾ 3σ	0%, 0%	2%, 23%	<1%, 13%

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We also check the completeness of the 11HUGS sample with respect to the H i gas mass. To do this, we have compiled 21 cm single-dish fluxes from the literature. Measurements are available for 96% of the galaxies in the overall 11HUGS target catalog. The data are primarily taken from the following three sources, in the following order of preference: the digital archive of Springob et al. (2005; N = 112), the H i Parkes All Sky Survey (HIPASS) catalog as published in Meyer et al. (2004; N = 75), and the homogenized H i compilation of Paturel et al. (2003) as made available through the Hyperleda database (N = 215). Finally, data for 16 galaxies are taken from the H i compilation in the Catalog of Neighboring Galaxies of Karachentsev et al. (2004) and other individual papers.

We evaluate T_C as a function of the H i flux and plot the results in Figure 2(a). Here, the T_C statistic begins to become systematically negative at integrated fluxes less than 6 Jy km s⁻¹. To find the corresponding completeness in the H i mass, we apply the standard relation M_H i[M_☉] =  2.36 ×  10⁵ D² F, where D is the distance in Mpc and F is the 21 cm line flux in Jy km s⁻¹. The gas is assumed to be optically thin, and corrections for the presumably small amount of H i self-absorption (≲ 10%; Haynes & Giovanelli 1984; Zwaan et al. 2003) are not applied. The H i mass is plotted against the distance in Figure 2(b). At the edge of the 11 Mpc target volume, a limit of 6 Jy km s⁻¹ corresponds to a completeness in M_H i down to 2× 10⁸ M_☉.

Finally, we also compute T_C as a function of the Hα+[N ii] flux and show the results in Figure 3(a). At fluxes below 4 × 10⁻¹⁴ erg s⁻¹ cm⁻², T_C becomes systematically negative. This corresponds to a completeness of L(Hα)= 6 × 10³⁸ erg s⁻¹ at 11 Mpc. To translate this into a limiting SFR, two issues must be kept in mind. Most standard conversion recipes (e.g., Kennicutt 1998, which results in an SFR of 0.005 M_☉ yr⁻¹) are based on expectations for a solar metallicity population, so using this conversion for metal-poor (∼ Z_☉/5) dwarf galaxies would result in an overestimate of the SFR—a relative deficiency of metals should cause a greater number of ionizing photons to be produced per unit stellar mass formed (e.g., Lee et al. 2002). We compute a calibration that is based on a Z_☉/5 population using the Bruzual & Charlot (2003) stellar population synthesis models, assuming Case B recombination and no leakage of Lyman continuum photons from the galaxy. Relative to a solar metallicity conversion derived from the same models, SFR/L(Hα) is a factor of ∼ 0.7 lower for the Z_☉/5 population. This SFR scale is shown on the right-hand side of Figure 3(b). Another issue is that stochasticity in the formation of high-mass ionizing stars may begin to become important for the ultra-low SFRs in this regime. However, note that a simple calculation, treating the initial mass function (IMF) as a probability distribution function (e.g., Oey & Clarke 2005), shows that such sampling issues become important only at SFRs less than ∼10⁻⁴—a constant SFR of 0.0002 M_☉ yr⁻¹ over timescales greater than 10 Myr will yield roughly 10 O stars at any given time, for a Salpeter IMF and mass limits of 0.1 and 1 M_☉.

2.1.2. Luminosity and Mass Function Comparisons

The goal of this paper is to directly tally the number of currently bursting dwarf galaxies in an approximately volume-limited sample and to compute the fraction of star formation that is concentrated in such systems. These statistics help provide constraints on the average amplitudes and durations of starburst episodes. To proceed, however, we require criteria that distinguish bursting galaxies from the rest of the population. This is a nontrivial issue, and many different (and sometimes subjective) criteria have been previously applied (e.g., see overviews given in the proceedings of the 2004 Cambridge Starbursts Conference by Gallagher 2005, Heckman 2005, and Kennicutt et al. 2005). To some extent, this variation has been driven by the type of data that are available to gage the star formation state of the systems under consideration. For example, while perhaps the most physically fundamental approach is to define starbursts as those galaxies that form stars near the maximum possible rate set by causality (i.e., the rate that results when all of the gas in a system is consumed in one dynamical time; Heckman 2005), measurements of the total gas masses and global velocity dispersions that are required to compute the limiting SFRs may not be available, particularly for large samples. An alternate approach, which is less absolute but can be reasonably applied to present-day galaxies, is to develop a characterization of normal star formation activity based on the overall population of galaxies, and then to identify starbursts as the subset of galaxies that form stars at an anomalously prodigious rate with respect to this fiducial (e.g., Gallagher 2005; Kennicutt et al. 2005, and references therein). We follow such strategies here.

One frequently used definition of a starburst is a galaxy whose current SFR exceeds its average past value by a factor of 2–3 (e.g., Hunter & Gallagher 1986; Salzer 1989; Gallagher 2005; Brinchmann et al. 2004; Kennicutt et al. 2005). The ratio of the current SFR to the past average is commonly referred to as the stellar birthrate (i.e., the b parameter) and can be observationally traced by the integrated, galaxy-wide Hα EW (i.e., the Hα flux divided by the continuum flux density under the line). The integrated EW is thus also related to the specific SFR, the current SFR normalized by the total mass of stars. Using the model grids computed in Kennicutt et al. (1994), and updated in Lee (2006) by using the population synthesis code of Bruzual & Charlot (2003), an approximate mapping between b and EW may be constructed. Assuming a Kennicutt IMF (Γ = −1.5; Kennicutt 1983), Case B recombination, no leakage of Lyman continuum photons from the galaxy, and no internal extinction, galaxies with b > 2–3 should have EW(Hα) ≳ 80–110 Å. Thus, one way of operationally identifying starbursts is by using such thresholds in the Hα EW.

Given the close relationship between the EW and b, it is interesting to study the EW distribution of the galaxies in our sample and ask whether EW values of ∼ 80–110 Å coincide with any particular features of the distribution. In Lee et al. (2007), we initially examined trends in the M_B–EW plane by using the measurements given in Table 2, Paper I, and showed that local star-forming galaxies trace a continuous sequence in EW as well as in the morphological type. The sequence exhibits two characteristic transitions. At M_B ∼ −15, a narrowing of the galaxy locus occurs as the luminosities increase and morphologies shift from predominantly irregular to late-type spiral (termed the "main sequence of star-forming galaxies" by Noeske et al. 2007). Above M_B ∼ −19, the high luminosity end of the sequence turns off toward lower EWs and becomes mostly populated by intermediate and early-type bulge-prominent spirals. In the current analysis, we focus on the low-luminosity galaxies below the upper transition at M_B ∼ −19 and divide the galaxies into three equal sized bins: two in the narrow EW "waist" (−19 ⩽ M_B < −17 and −17 ⩽ M_B < −15) and one containing the most extreme dwarf galaxies, which lie below the lower transition where the EW distribution broadens (−15 ⩽ M_B < −13). Histograms of the logarithmic EW in these three bins are shown in Figure 4. A histogram of the higher luminosity galaxies (−22 ⩽ M_B < −19) is also provided for completeness.

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To characterize the distributions plotted in Figure 4, we first checked whether simple Gaussian functions with means and standard deviations directly computed from the logarithmic EWs in a given luminosity bin would yield an adequate description of the data. The gray dashed curves plotted in each panel represent these functions. For galaxies with −19 ⩽ M_B < −17, the gray curve provides a good fit, but for the lower luminosity galaxies (bottom panels), the curves appear somewhat too broad; this is due to the outliers appearing at the tails of the distributions. Thus, to attempt to place more weight on the central components of the distributions, Gaussian fits to the histograms minimizing χ² were also performed. This second set of Gaussian functions is overplotted in yellow, and appears to provide an improved fit to the majority of the data with −17 ⩽ M_B < −15 and −15 ⩽ M_B < −13. This suggests that one way to describe the distribution of EWs is as a log-normal function with non-Gaussian excesses at the tails. In order to check this, we performed Anderson–Darling (A–D) tests for normality (Anderson & Darling 1952) on both the complete set of logarithmic EWs in each luminosity bin and the sets that were clipped at 3σ, where σ was provided by the fits to the histograms (see Table 1). The A–D statistic is similar to the Kolmogorov–Smirnov (K–S) statistic in that it examines differences between two cumulative distribution functions to determine if one sample is consistent with being drawn from the other (the null hypothesis). On the other hand, whereas the K–S test is most sensitive near the median values of the distribution and does not perform as well at detecting statistically significant deviations at the ends of distributions, the A–D test is "stabilized" by accounting for the variance in the calculated differences between the distributions and is thus more robust at the tails (Press et al. 1992). Applying the A–D test to our data, we find that the logarithmic EWs for galaxies with −19 ⩽ M_B < −17 are consistent with a Gaussian function insofar as the null hypothesis cannot be rejected. As would be expected from the consistency of the gray and yellow curves, clipping removes no galaxies from the sample in this bin. However, for the subsets of EWs in the −17 ⩽ M_B < −15 and −15 ⩽ M_B < −13 bins, the A–D test rejects the null hypothesis at the 99% and 90% levels, respectively. However, after these subsets are clipped at 3σ, the null hypothesis can no longer be rejected with any confidence. We note that clipping with different thresholds (e.g., 2σ, 4σ) yields distributions that also violate the assumption of normality. Thus, this exercise provides some support that the central components of the EW distributions can be adequately represented by log-normal distributions (as described by the yellow curves in Figure 4), although, of course, it by no means proves that the underlying logarithmic distributions are Gaussian.

What is interesting about this characterization of the EW distributions is the coincidence between their upper 3σ points to the frequently adopted starburst birthrate thresholds discussed above. In the two luminosity bins that are bounded by the EW transitions (−19 ⩽ M_B < −17, −17 ⩽ M_B < −15; note that the sample, as discussed in Section 2, is complete within this regime), the means and dispersions of the yellow Gaussian curves (Table 1) are essentially the same. Here, the EW values of ∼ 80–110 Å, which correspond to starburst-like birthrates of 2–3, occur where the distributions drop steeply, and the upper value of this range is close to the upper 3σ points of the curves (107 Å and 105 Å for −19 ⩽ M_B < −17 and −17 ⩽ M_B < −15, respectively). Although our sample is not large enough to cleanly define the 3σ ranges, the correspondence suggests that for a complete sample of field galaxies with −19 ⩽ M_B < −15, the upper 3σ point of the central component of the logarithmic EW distribution may provide one empirical criterion for identifying dwarf galaxies currently undergoing global starbursts, at least in the local universe. Such a definition will not nominally predetermine the starburst number fraction to be 0.135% (the > 3σ probability for a normal distribution) since it appears that a sufficiently populated distribution should show significant non-Gaussian excesses at the tails, as has already been discussed. These excesses are also illustrated in Figure 5(a) where the observed logarithmic EW cumulative number distributions are compared with those of the best-fit Gaussians. We speculate that the low EW outliers are postburst galaxies.

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In the analysis that follows, we compute starburst number and star formation fractions for a range of EW thresholds, which roughly span stellar birthrates of 2–3, using both fixed EW values and those determined by σ limits (Table 1). In the interest of clarity in the discussion, however, we will primarily adopt results based on an intermediate EW cut of 100 Å, which maps to a birthrate value of 2.5. As will be shown, the results are robust to variations in the exact choice of the threshold between the EW values of ∼ 80–110 Å since, as already noted, such values occur where the distributions fall off sharply.

With the criteria established in the previous section, it is straightforward to determine the number fraction of starbursting dwarf galaxies in the local 11 Mpc volume using data from the Hα component of our survey. In Table 1, we list the fractions and numbers of galaxies with EW > 80 Å (b ∼ 2), EW > 100 Å (b ∼ 2.5), as well as above and below the 1σ, 2σ, and 3σ ranges of EW values in each luminosity bin. The 3σ point for galaxies with −19 < M_B < −15 maps to b ∼ 3, so the selected EW thresholds roughly span stellar birthrates of 2–3. Cumulative number distributions as a function of the EW are shown in Figure 5(a). Here, we quote statistics based on the observed Hα+[N ii] fluxes and EWs, which have not been corrected for internal extinction. Corrections for internal extinction and the contribution of the [N ii] lines to the observed flux (as described in Paper I) have a negligible effect on these results. This is reasonable given that the low-luminosity population being considered is metal poor and has low dust content, and that the [N ii] and internal extinction corrections tend to cancel each other out. Again, we focus the discussion on results based on an intermediate EW cut of 100 Å, which maps to a birthrate value of 2.5, and then describe the effects of varying the threshold about this value.

Our most robust fractions are based on the −17 ⩽ M_B < −15 luminosity bin since this is where the number statistics are tolerable (N = 93), and the sample in our main survey volume (|b|>20°) is complete (Section 2.1). Dwarf galaxies in this bin should also be outside of the regime where stochasticity in the formation of high-mass ionizing stars may affect the interpretation of Hα luminosity as an SFR indicator. As noted at the end of Section 2.1.1, such effects may become important for SFR ≲10⁻³ M_☉ yr⁻¹, but as shown in Figure 6, galaxies with M_B = −15 generally have an average SFR of 0.01 M_☉ yr⁻¹.¹⁰ Therefore, the subsequent discussion will be based on the statistics from this bin. Within this luminosity range, systems with EW > 100 Å are rare—there are only six galaxies with EWs this high, and this represents 6% of the population. (A listing of these starbursting dwarf galaxies along with some basic properties is given in Table 2, and images are shown in Figure 7.) The dominant component of the uncertainty in the starburst number fraction is likely due to Poisson noise, so we quote the primary result as 6⁺⁴₋₂%. All the uncertainties given here and below span the 68% confidence interval as tabulated in Feldman & Cousins (1998). Again, it is important to note that the shape of the distribution ensures that this result is not critically dependent on the exact EW threshold that is adopted, for those values that map to b ∼ 2–3. If the threshold is dropped down to 80 Å, the results do not change since there are no galaxies between 80 Å and 100 Å within this luminosity range in the sample. If the 3σ point of 105 Å is instead used, the results remain the same as well. Finally, if the threshold is dropped down to 2σ above the mean (EW = 71 Å), the number count increases to seven systems (8%).

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For the higher luminosity galaxies (−19 ⩽ M_B < −17), the tails of the EW distribution do not show excesses and there are no global starbursts based on a EW ≳ 100 Å cut. This can partly be attributed to small number statistics, since there are ∼ 35% fewer galaxies in this bin as compared with the −17 ⩽ M_B < −15 bin. If we assume that the starburst number fraction is also 6% here, then there should be 4⁺³₋₂ galaxies with EW > 100 Å for a sample of 59, so it is reasonable that no galaxies with such high EWs are observed. A Poisson upper limit for the observation of zero events may also be computed, which yields a fraction of ≲2% in this luminosity regime. If an EW > 80 Å threshold is instead applied, the fraction is 2% (N = 1). Dropping further to the upper 2σ point (EW = 71 Å), the fraction becomes 3% (N = 2). Comparison of these numbers with those computed for the −17 ⩽ M_B < −15 bin shows that it is possible that the starburst number fraction decreases for more massive systems.

If we again apply a threshold value of 100 Å to the most extreme dwarfs (for −15 ⩽ M_B < −13), the starburst number fraction is 5% (N = 5). However, the situation is more complicated for this lowest luminosity bin. First, as discussed in Section 2, the completeness steeply drops at M_B = −14.6, so statistics reported for this population must be interpreted with caution. However, it is likely that starburst mass and star formation fractions computed in this bin represent upper limits since the low surface brightness dwarfs are more likely to be missing from the sample than the higher surface brightness starbursting systems. Second, the EW starburst criterion changes because the distribution widens in this regime, so the 3σ threshold is higher than for the more luminous dwarfs. This may be partly due to stochastic sampling of the upper part of the IMF as discussed above and/or burstier star formation in the lowest mass systems. If the 2σ and 3σ values of this bin are instead used (i.e., 115 Å and 242 Å), the number fractions are 5% and 3%, respectively.

Finally, the results from all three bins (for EW > 100 Å) may be compared. Beginning with the most luminous bin, the starburst number fractions are ≲2%, 6⁺⁴₋₂%, and 5⁺⁵₋₃%, respectively. The local starburst number fraction may thus only be a weak function of luminosity.

Arguably, however, a more important statistic is the fraction of star formation that occurs in starburst episodes. If the star formation fraction is high, this could be evidence that the burst mode dominates the evolution of dwarfs, even if the number fraction is low. This statistic is also more robustly measured, as it is relatively immune to incompleteness in the less active galaxies that are most likely to be missed, as such systems would not add considerably to the Hα volume density.

To estimate the star formation fraction, we sum the observed Hα+[N ii] luminosities (L_{Hα+[N  ii]}) as a function of the EW. Again, applying corrections for internal extinction and the contribution of the [N ii] lines to the observed flux as described in Paper I does not significantly impact the results.¹¹ The percentages of L_{Hα+[N  ii]} contained in galaxies with EW > 100 Å (b ∼ 2.5), EW > 80 Å (b ∼ 2), as well as above and below the 1σ, 2σ, and 3σ ranges of the EW are reported in Table 1, and the cumulative L_Hα+[N ii] distributions as functions of the EW are plotted in Figure 5(b).

For galaxies with −17 ⩽ M_B < −15, the six systems with EW > 100 Å are responsible for 23% of all of the star formation occurring in this luminosity bin. A simple translation of the Poisson error in the number fraction results in an error of ⁺¹⁴₋₉% in the L_α fraction. From Figure 5(b), it can be seen that the starburst system that has the largest SFR contributes ∼ 10% (NGC 3125), so our simple estimation of the error appears to be reasonable. This quantity is also not highly sensitive to the exact EW value used to classify starbursts. A lower EW > 80 Å cut or a slightly higher 3σ point of 107 Å does not change the result, while applying a lower 2σ threshold only causes a small increase of the fraction to 25⁺¹⁶₋₈%.

For −19 ⩽ M_B < −17, no systems are observed with EW > 100 Å as described in the previous section. However, the two galaxies with EWs higher than the 2σ threshold of 71 Å produce 9% of the Hα luminosity, and this is consistent with the ratio of seven galaxies with −17 ⩽ M_B < −15 and EW > 71 Å producing 25% of the Hα luminosity in that bin. Therefore, estimating that each starburst galaxy is on average responsible for ∼ 4% of the total star formation occurring in the population, we infer that the fraction of star formation due to EW > 100 Å galaxies in this luminosity range must be less than ∼ 5%. Application of an EW > 80 Å cut yields a star formation fraction of 4%, and a further drop to the 2σ point results in 9%.

Finally, the fraction of star formation occurring in the lowest luminosity starbursts is computed, keeping in mind the caveats on incompleteness and stochasticity in the formation of ionizing stars that affect the sample in this bin. Systems exceeding EWs of 100 Å are responsible for 16% of the star formation. Dropping the cut to an EW of 80 Å results in a small increase of the starburst star formation fraction to 20%. If the σ thresholds defined by the broader EW distribution for −15 ⩽ M_B < −13 are instead used, galaxies with EW greater than 115 Å (2σ) and 242 Å (3σ) are responsible for 16% and 13% of the star formation, respectively.

From these calculations, it is clear that a significant amount of the overall star formation in present-day dwarf galaxies takes place in starbursts. However, the results also clearly show that a more continuous, steady state of star formation dominates in the present epoch, both in terms of being the mode that operates during the vast majority of the time and in which most of the stars are being created. The average burst amplitude does not appear to strongly vary with luminosity, with 4% of the total star formation density in a given luminosity bin being concentrated in individual starbursting systems.

Based on the 11HUGS sample, 6⁺⁴₋₂% of late-type dwarf galaxies are starbursting and 23⁺¹⁴₋₉% of the overall star formation in dwarfs occurs in the starburst mode. The numbers are quoted from the luminosity bin where the sample is both complete and has tolerable number statistics (for −17 ⩽ M_B < −15), but the flanking bins (for −19 ⩽ M_B < −17 and −15 ⩽ M_B < −13) do appear to have fractions that are consistent within the Poisson errors. In this section, we compare these results with previous estimates.

With respect to the starburst number fraction, there is good consistency between our result and previous estimates, which is notable because the studies cover a range of independent approaches to the problem and use different starburst selection criteria. Early work compared the space densities of low-luminosity UV-selected Markarian galaxies with those of field galaxies (Sargent 1972; Huchra 1977a) in order to constrain the fraction of "flashing" systems. These studies produced tentative estimates of 7% at M_p = −17 and 10% at M_p = −14, based on very small samples containing about a dozen objects each and large incompleteness corrections. Later, Salzer (1989) also examined the space densities of active galaxies, using the University of Michigan [O iii]λ5007 emission-line selected survey. The comparison set of normal galaxies was derived from the magnitude-selected Catalog of Galaxies and Clusters of Galaxies (Zwicky 1961–1968) and the UGC (Nilson 1973). The number statistics of the Salzer (1989) samples were improved relative to the earlier Markarian studies, with N ∼ 40 and N ∼ 30 in the emission-line selected and general population samples, respectively. Using the numbers reported there, we calculate that the low-luminosity Michigan emission-line galaxies (−17 ⩽ M_B < −15) represent ∼ 10% of the general population in that luminosity range.

Kauffmann et al. (2003a, 2003b) have used an entirely different method to calculate the fraction of bursty galaxies as a function of stellar mass in the Sloan Digital Sky Survey (SDSS). In this work, an extensive grid of stellar population synthesis models, spanning a range of metallicities and SFHs, is constructed. A Bayesian technique is then used to compare the observed and modeled Hδ absorption and 4000 Å break to generate a likelihood distribution of the fraction of stellar mass formed in bursts for each galaxy in their sample. Two statistics are reported: (1) "F_burst(50%)>0," the fraction of galaxies whose likelihood distribution of burst masses has median values greater than zero, and (2) "F_burst(2.5%)>0," the fraction of "high confidence bursty galaxies" whose likelihood distributions have their lower 2.5 percentile point above zero. For galaxies with 8.0 < logM_*< 8.5 (which corresponds to −16 ≲ M_B ≲ −14.5 assuming an approximate M/L_B = 1, which is typical for dIrr galaxies; e.g. Miller & Hodge 1994; Lee 2006), Kauffmann et al. found that the fraction with F_burst(50%)>0 is over 50%, while that for F_burst(2.5%)>0 is 9%. For higher mass dwarfs with 8.5 < logM_*< 9.0 (−17 ≲ M_B ≲ −16), the values instead are 36% and 4%, respectively. Clearly, the number fractions estimated using F_burst(50%)>0 to discriminate whether a galaxy has undergone a burst in the recent past are too large to be consistent with any of the other estimates discussed above. This is perhaps not too surprising since half of the models that are consistent with the observations for the F_burst(50%)>0 galaxies have SFHs in which there have not been any bursts at all, and the burst number fractions computed in this way are probably overestimates. However, the fractions of "high confidence bursty galaxies" are in good agreement with the other measurements. Thus, it appears that the starburst number fraction for dwarf galaxies with M_B ≳ −15 is well determined and, moreover, relatively robust to the method used to pick out starbursts. Past estimates consistently lie between 4% and 10%, and our 11HUGS measurement of 6⁺⁴₋₂% is representative of these values.

Kauffmann et al. (2003a, 2003b) have also reported a decrease of the burst number fractions with increasing stellar mass. When their F_burst(50%)>0 criterion is used to identify galaxies with recent bursts, the number fraction of starbursts drops from 54% at ∼ 10⁸ M_☉ to 12% at ∼10¹⁰M_☉. However, when the analysis is restricted to the "high confidence bursty galaxies," the absolute decline is much smaller, changing from 9% to 1%. The latter result is more consistent with our finding that the number fraction is ≲ 10% over 7 mag in M_B.

As for the fraction of star formation that takes place in bursting systems, there appear to be few prior studies that have attempted to place constraints on this quantity for dwarf galaxies per se. Based on the SDSS, Brinchmann et al. (2004) have found that starbursts (which they define as galaxies with b ⩾ 2–3) are responsible for 20% of the local star formation density. Although this is consistent with our 11HUGS estimate of 23⁺¹⁴₋₉% for dwarf galaxies, the Brinchmann measurement refers to the total galaxy population.

As another check on the starburst star formation fraction, we perform a coarse calculation using the KPNO International Spectroscopic Survey (KISS; Salzer et al. 2000), a second-generation CCD-based, emission-line selected, objective-prism survey. We use the sample of Hα-selected galaxies cataloged in List 1 (Salzer et al. 2001), which contains 1128 candidates identified from the objective-prism images. Follow-up slit spectroscopy has been completed for all of these candidates and 907 of them are classified as star-forming galaxies based on their [O iii]λ5007/Hβ and [N ii]λ6584/Hα line ratios. Completeness of the sample has been assessed through the standard V/V_max test (Schmidt 1968), and limiting volumes have been calculated for each object (J. J. Salzer 2006, private communication; also see Lee et al. 2002). We use these volumes to calculate SFR densities as a function of the Hα EW. The Hα fluxes and EWs that are measured from the objective-prism spectra are used in this calculation, since they are less likely to suffer from aperture effects and will be closer to the integrated values than those measured from the slit spectra. For −17 < M_B < −15 (N = 61), we find that KISS galaxies with EW > 100 Å (N = 21) and EW > 70 Å (N = 34) are responsible for 22% and 38% of the total L_Hα output in this bin. We can also compute the starburst number fractions comparing the number densities of high EW KISS galaxies with those from a sample that better represents the overall population of dIrrs. Using the B-band LF determined by Zwaan et al. (2001), which is based on the H i selected sample from the Arecibo H i Strip Survey (AHISS; Zwaan et al. 1997) to provide the denominator of the fraction, we find that galaxies with EW > 100 Å comprise 5% of the dwarf population, while it is 12% for the lower threshold of EW > 70 Å. These results, which are based on better number statistics for the high EW galaxies, are in good agreement with the fractions estimated from 11HUGS and elsewhere.

Our star formation census within 11 Mpc robustly shows that dwarfs that are currently experiencing massive global bursts (as identified as galaxies with integrated Hα EWs exceeding 100 Å) are just the ∼ 6% tip of a low-mass galaxy iceberg, and that the bulk (∼ 70%) of star formation in the overall population takes place in a more continuous mode. To make more detailed inferences regarding the characteristic values of burst parameters, some assumptions must be made.

The starburst number and star formation fractions of a volume-limited galaxy sample provide constraints on the duty cycle since the relative number of galaxies observed in various phases of the cycle will scale with the relative durations of those phases. The statistics can be interpreted most directly in the simplest scenario where (1) there are only two phases, an "on" (burst) phase and an "off" (quiescent) phase, and (2) the objects share a common average SFH such that bursts can occur with equal probability in all galaxies on which the statistic is based. Under such "equal probability" assumptions, the fraction of star formation due to starbursts is equivalent to the fraction of stellar mass formed in the burst mode, the starburst number fraction is equivalent to the fraction of time spent in the burst mode, and the ratio of the two reflects the average amplitude during the burst. Based on the 11HUGS sample then, the duty cycle is 6⁺⁴₋₂%, the fraction of stars formed in the bursts is 23⁺¹⁴₋₉%, and the SFR in the burst mode is on average ∼ 4 times greater than in the quiescent mode. With the additional assumption that the typical duration of a global starburst is ∼100 Myr (based on the CMD-based SFHs of N1569, Vallenari & Bomans 1996; Sextans A, Dohm-Palmer et al. 1998; DDO 165, Weisz et al. 2008), such events can roughly be estimated to occur every 1–2 Gyr. If the equal probability assumptions are incorrect, and instead the starburst mode only operates in a particular subset of the population, then the statistics represent lower limits on the duty cycle and frequency.

4.2.1. Nondetections—Inferences Regarding the Interburst Phase

4.2.2. The "Equal Probability" Assumption