OUTSIDE–IN SHRINKING OF THE STAR-FORMING DISK OF DWARF IRREGULAR GALAXIES^*

The 34 galaxies included in this work (listed in Table 1) are drawn from the LITTLE THINGS⁷ sample, which is a representative sample of 41 nearby dIrr galaxies spanning a large range in galactic parameters: integrated luminosity (M_B of −9 to −19), surface SFR density (0–1.3 M_☉ yr⁻¹ kpc⁻²), (atomic) gas richness (0.02–5 M_☉/L_B), and central surface brightness μ^V₀ (18.5–25.5 arcsec⁻²). The sample galaxies were selected to be relatively isolated. LITTLE THINGS is a large Very Large Array (VLA) project that was granted nearly 376 hr of VLA time in the B, C, and D array configurations to obtain deep H i-line emission maps of dIrr galaxies with high angular and velocity resolutions. The subsample of 34 galaxies studied here was mostly chosen because these galaxies have the full complement of data available, including GALEX FUV/NUV, UBV, and Hα and Spitzer 3.6 μm. The Two Micron All Sky Survey (2MASS) JHKs images are too shallow to be used in this work. However, UGC 8508, which cannot be observed by GALEX due to a nearby bright star, and DDO 165, which does not have FUV observations, are also included in the current sample. Another three galaxies without Hα detection (LGS 3, M81dwA, DDO 210) were also included.

Table 1. Galaxy Sample

Galaxy	Other Names	D	b/a	E(B − V)f	MB	log($\Sigma _{\tt SFR(H\alpha)}$)	Mbary	Mgas/M⋆	Θ
		(Mpc)				(M☉ yr−1 kpc−2)	(M☉)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
LGS 3	Pisces Dwarf	0.7	0.51	0.041	−9.11	⋅⋅⋅	5.68	0.76	1.7
DDO 210	Aquarius Dwarf	0.9	0.48	0.051	−10.38	⋅⋅⋅	6.60	4.90	−0.1
DDO 69	UGC 5364, Leo A	0.8	0.56	0.021	−11.38	−3.28	7.16	13.19	0.2
DDO 155	UGC 8091, GR 8	2.2	0.71	0.026	−12.25	−1.50	7.22	4.54	−1.2
DDO 216	UGC 12613, Pegasus Dwarf	1.1	0.45	0.066	−13.06	−4.15	7.22	0.10	1.2
M81dwA	KDG 052	3.6	0.73	0.021	−11.46	⋅⋅⋅	7.29	13.17	0.7
Mrk 178	UGC 6541	3.9	0.46	0.018	−13.76	−1.53	7.39	1.21	−0.7
DDO 187	UGC 9128	2.2	0.80	0.024	−12.38	−2.64	7.43	6.78	−1.3
UGC 8508	IZw 60	2.6	0.54	0.015	−13.24	−2.12	7.63	4.54	−1.0
NGC 4163	NGC 4167, UGC 7199	2.9	0.64	0.020	−13.95	−2.43	7.69	0.81	0.1
CVnIdwA	UGCA 292	3.6	0.78	0.016	−12.16	−2.64	7.83	15.66	−0.4
DDO 70	UGC 5373, Sextans B	1.3	0.59	0.032	−13.74	−2.86	7.86	2.68	−0.7
IC 1613	UGC 668, DDO 8	0.7	0.81	0.025	−14.17	−2.64	7.87	1.57	0.9
DDO 101	UGC 6900	6.4	0.69	0.022	−14.40	−2.99	7.89	0.19	⋅⋅⋅
WLM	UGCA 444, DDO 221	1.0	0.44	0.037	−13.98	−2.85	7.98	4.88	0.3
DDO75	UGCA 205, Sextans A	1.3	0.85	0.044	−13.72	−1.40	8.04	13.05	−0.6
VIIZW 403	UGC 6456, VV 574	4.4	0.85	0.036	−13.99	−1.82	8.05	5.37	−0.3
Haro 29	UGCA 281, Mrk 209, IZw 36	5.9	0.58	0.015	−14.39	−0.82	8.06	7.04	−1.0
DDO 133	UGC 7698	3.5	0.69	0.016	−14.36	−2.93	8.30	5.55	−1.1
DDO 126	UGC 7559	4.9	0.47	0.014	−14.56	−2.45	8.31	11.60	0.1
DDO 165	UGC 8201, IIZw 499	4.6	0.54	0.024	−15.38	−3.52	8.35	5.56	0.0
DDO 63	UGC 5139, Holmberg I	3.9	1.00	0.048	−14.58	−3.44	8.40	6.45	1.5
NGC 6822	IC 4895, DDO 209	0.5	0.79	0.240	−14.76	−1.96	8.40	2.28	0.6
DDO 53	UGC 4459, VIIZw 238	3.6	0.51	0.037	−13.43	−2.50	8.41	25.32	0.7
DDO 154	UGC 8024, NGC 4789A	3.7	0.50	0.009	−13.88	−2.60	8.50	36.97	−0.9
DDO 87	UGC 5918, KDG 072, VIIZw 347	7.7	0.58	0.011	−14.52	−3.16	8.53	9.37	−1.5
DDO 52	UGC 4426	10.3	0.67	0.037	−15.05	−3.27	8.63	7.04	−1.5
DDO 168	UGC 8320	4.3	0.63	0.015	−15.35	−2.33	8.71	7.77	0.0
NGC 1569	UGC 3056, Arp 210, VIIZw 16	3.4	0.55	0.604a	−17.35	0.11	8.78	0.66	−0.4
NGC 3738	UGC 6565, Arp 234	4.9	1.00	0.010	−16.70	−1.72	8.83	0.45	−1.0
DDO 50	UGC 4305, Holmberg II	3.4	0.72	0.032	−16.39	−1.83	9.03	9.11	0.6
NGC 2366	UGC 3851, DDO 42	3.4	0.42	0.036	−16.49	−1.73	9.04	14.81	1.0
NGC 4214	UGC 7278	3.0	0.91	0.022	−17.26	−1.10	9.11	1.69	−0.7
NGC 1156	UGC 2455, VV 531	7.8	0.86	0.224	−18.29	−0.87	9.42	1.19	−1.7

Notes. (1) Galaxy names adopted in this work. (2) The other commonly used names in the literature. (3) Distance from D. A. Hunter et al. (2012, in preparation and references therein). (4) Minor-to-major axis ratio measured on the V-band images. (5) The foreground reddening from Schlegel et al. (1998). (6) B-band absolute magnitude. (7) Integrated star formation rate (SFR) normalized to the area of the galaxy within one V-band disk scale length. The SFR is derived from Hα. (8) The baryonic (stellar plus atomic gas) mass. The stellar mass is derived from our SED modeling in this work. The atomic gas ($1.34\times M_{H_{\tt I}}$) mass is collected from single-dish observations in the literature (see Hunter & Elmegreen 2004 for the references). (9) The atomic gas-to-stellar mass ratio. (10) Tidal index Θ from Karachentsev et al. (2004). Θ quantifies the density enhancement (and thus the tidal action) caused by the main disturber. The larger the value of Θ is, the stronger the gravitational disturbance exerted by the neighboring galaxies. Galaxies with zero or negative values of Θ could be considered as isolated objects. ^aThe average of the E(B−V) derived by Burstein & Heiles (1982, 0.508) and by Schlegel et al. (1998, 0.700). See M. C. Johnson et al. (2012, in preparation).

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The acquisition and reduction of the UBV and Hα narrowband images of our sample were described by Hunter & Elmegreen (2004, 2006; hereafter HE04, HE06); the GALEX UV data for 21 galaxies included in the current sample were presented by Hunter et al. (2010).

There have been significant improvements in the flat fields and absolute calibration of the GALEX imaging data from the GR2/3 pipeline to the GR4/5 pipeline. The work of Hunter et al. (2010) used the GR2/3 pipeline data for some galaxies. Therefore, we obtained the GR4/5 imaging data of all of our galaxies from the GALEX archive. Most of the Spitzer 3.6 μm images were obtained and reduced by the Local Volume Legacy survey (LVL; Dale et al. 2009) and the Spitzer Infrared Nearby Galaxy Survey (SINGS; Kennicutt et al. 2003). Besides the standard post-pipeline processing, additional processing, including distortion corrections, rotation of the individual frames, bias structure, and bias drift corrections, has been done by the teams of these two surveys.

We follow the procedures as laid out in HE04 for the surface photometry. First, the background-subtracted images were geometrically transformed and aligned with the V-band image. Background and foreground sources not belonging to the galaxy were masked. Then the surface photometry was carried out with the Image Reduction and Analysis Facility (IRAF⁸) task ELLIPSE, adopting the same geometrical parameters as derived by HE04. Briefly, the center of the galaxy, position angle, and the ellipticity were determined from a contour in the outer half of the V-band image that was block-averaged by factors of a few to increase the signal-to-noise (S/N). The center was fixed as the geometrical center of this isophote, and the major axis is the longest bisector that passes through the center that, as much as possible, symmetrically divides the galaxy. The surface photometry of the UBV and Hα images has been described in HE04 and HE06.

To perform photometry on the GALEX images, as described in Hunter et al. (2010), we masked foreground stars and background galaxies identified mainly in the higher resolution optical band images. For the removal of sky background, because the background was quite flat, we measured the sky level in tens of square regions (10 pixel by 10 pixel) sampled along an ellipse around the galaxy, but far enough from the galactic center to avoid emission from the target galaxy. The final sky level was determined as the average of the sky values in all sampled subregions. As to the uncertainties of the background removal, the standard deviation of the mean sky values σ_〈sky〉 among the surrounding sky regions is usually an order of magnitude smaller than the mean of the standard deviation in individual regions 〈σ_sky〉. Therefore, only the 〈σ_sky〉 were considered.

For the Spitzer 3.6 μm images, we first applied the same photometry mask as was used for the optical band images. Background galaxies can significantly affect the photometry on 3.6 μm images of faint dwarf galaxies. Whenever available, high-resolution archival Hubble Space Telescope (HST) imaging data were inspected to further remove the background galaxies based on morphology.

Finally, by dividing the luminosity differences between adjacent ellipses by the area of the annulus, we obtained azimuthally averaged surface brightness profiles. A value of 0.15 mag was taken as the calibration uncertainty for the GALEX UV data and 0.1 mag for Spitzer 3.6 μm. The calibration uncertainties of the UBV data for each galaxy were given in HE06. We adopt a 20% calibration uncertainty for all of the Hα photometry.

Routinely, one uses the average Milky Way (MW) extinction curve (R_V = A_V/E_{B − V} = 3.1; Cardelli et al. 1989) to correct the photometry for the Galactic extinction. However, there is considerable scatter of the extinction curve shape from sightline to sightline through the Galaxy (Fitzpatrick 1999), which could significantly influence de-reddening of the short-wavelength data (e.g., UV, U) when only an average extinction curve was adopted. Fitzpatrick (1999) gave the typical uncertainties (σ_{E(λ − V)/E(B − V)}, Table 1 in Fitzpatrick 1999) when correcting the data with the average extinction curve. In this work we included the uncertainty (E(B − V) × σ_{E(λ − V)/E(B − V)}) of adopting the average MW extinction curve. The final uncertainties assigned to each annulus are a quadratic sum of four contributions: 〈σ_sky〉, photometric calibration uncertainties, Poisson noise, and E(B − V) × σ_{E(λ − V)/E(B − V)}. The flux calibration uncertainties usually dominate all other contributions except in the outer disks.

Detailed studies of high-quality stellar CMDs for very nearby dIrr galaxies (e.g., Dolphin 2000; Skillman et al. 2003; Cole et al. 2007; Tolstoy et al. 2009) suggest that low-mass dIrr galaxies exhibit a wide variety of SFHs over their lifetime. Therefore, rather than adopting the commonly used two-component form of SFHs for larger galaxies (an exponentially declining component superimposed by random bursts, e.g., Kauffmann et al. 2003; da Cunha et al. 2008), we use multi-component population models to create our SFH library. Specifically, we divided logarithmically the lifetime from the present to 13.70 Gyr ago into six independent age bins (0–0.03 Gyr, 0.03–0.10 Gyr, 0.10–0.35 Gyr, 0.35–1.18 Gyr, 1.18–4.02 Gyr, 4.02–13.70 Gyr), where each age bin has a constant SFR. The way we choose these age bins reflects the fact that the separation between isochrones of different ages strongly decreases with age. Our choice of the first age bin (0–0.03 Gyr) is justified by the nearly constant UV colors in the first ∼0.03 Gyr of evolution (e.g., Leitherer et al. 1999). Similarly, the last age bin (from ∼4 Gyr to 13.7 Gyr) is justified by the nearly unevolving shape of the optical to NIR continuum spectra at ages older than ∼4 Gyr (e.g., Bruzual & Charlot 2003). In the Appendix, we show that the commonly assumed forms of SFHs tend to systematically bias the most probable estimates of the physical parameters. We make no assumptions here. Similarly, a common way to generate the SFH library is by Monte Carlo realizations of different SFHs. Instead, our library consists of a uniform, multi-dimensional grid of models without any prior assumption of the relevant physical parameters. This is necessary for exploring the complex SFHs of low-mass dIrr galaxies. When creating the SFH library, we allow the relative SFR among different age bins to logarithmically vary by an order of 1.5.

We use the S. Charlot & G. Bruzual (2012, in preparation) stellar population models, which have implemented several important improvements compared to the Bruzual & Charlot (2003) models. In particular, the new models include the recently improved treatment (Marigo et al. 2008) of the thermally pulsing asymptotic giant branch, which at maximum (from a few hundred Myr to ∼1 Gyr for a single stellar population) could dominate the NIR emission. We adopt the stellar initial mass function (IMF) parameterized by Chabrier (2003). We did not consider chemical evolution in the models, all the components of each model in the library have the same metallicity. The metallicities of the population models are allowed to vary among 0.02, 0.2, and 0.4 Z_☉, which are adequate for our dIrr galaxies.

To make use of the Hα data, the nebular hydrogen emission lines from H ii regions are also obtained for each model in the library. Collisionless Case B recombination for a 10,000 K gas is assumed. Using the recombination rates given by Hummer & Storey (1987), the Hα luminosity is calculated from the UV stellar photons shortward of the Lyman limit. A systematic decline of the ratio of the integrated Hα to FUV flux in low-luminosity dwarf galaxies has been shown (e.g., Sullivan et al. 2004; Meurer et al. 2009; Lee et al. 2009a). Three interpretations have been suggested in the literature, namely, the leakage of ionizing photons (e.g., Hunter et al. 2010), the stochasticity of massive SF at low SFR (e.g., Cervino & Luridiana 2004), and non-universal stellar IMF (especially the upper end; e.g., Pflamm-Altenburg et al. 2009). These possibilities all imply that the observed Hα emission only gives a lower limit to the current SFR, under the assumption of Case B recombination and a well-populated universal IMF, which is the case for our modeling. Therefore, our Hα data are used to set a lower limit on the Hα flux from the models. Considering the uncertainties in the observed Hα flux (HE04), we set the lower limit to be $0.8\times L_{\tt H\alpha, obs}$.

We use the two-phase dust attenuation recipes developed by Charlot & Fall (2000) as our standard dust extinction law. The effective absorption is proportional to λ^−0.7. Charlot & Fall (2000) found a typical value of ∼3 for the ratio μ of the extinction experienced by the young stellar populations (⩽10 Myr ) to that by the older stellar populations. We adopt this value when creating our SFH library. The V-band dust extinction affecting the young stellar populations is allowed to vary linearly between 0.0 and 0.5 mag. The gray extinction curve derived by Charlot & Fall (2000) is in agreement with recent results for a large sample of nearby galaxies (Johnson et al. 2007).

Our final library consists of ∼4 ×10⁶ different SFHs, which are created by allowing all the relevant parameters (i.e., dust extinction, metallicity, and relative SFR among different age bins) to vary uniformly (either logarithmically or linearly) among physically reasonable ranges. For each SFH in the library, we integrate, respectively, back to the past 0.03 Gyr, 0.1 Gyr, 1 Gyr, and the galaxy lifetime (here we adopt a Hubble time of 13.7 Gyr) in order to get the SFR averaged over these different timescales. In the following we denote the SFR averaged over the past 0.03 Gyr, 0.1 Gyr, 1 Gyr, and the whole lifetime as SFR_0.03, SFR_0.1, SFR₁, and SFR_13.7, respectively. The accumulated present-day stellar mass is also derived. From our experiments, the broadband SED modeling is not expected to be very sensitive to the SFR averaged over the past 0.03 Gyr (see the Appendix) and that averaged over the past 4 Gyr. So we concentrate on the interpretation of SFR_0.1, SFR₁, SFR_13.7, and the stellar mass in this work. However, as mentioned above, our choice of six age bins guarantees that we have a relatively complete SFH library adequate for broadband SED modeling.

We use the Bayesian technique (e.g., Kauffmann et al. 2003; Kong et al. 2004) to find the most probable estimates of parameters related to each of the observed broadband SEDs. Specifically, given an observed SED, we construct the probability density function (PDF) and the related cumulative distribution function (CDF) for each parameter defining our SFH library by calculating the likelihood exp (− χ²/2) that an observed SED corresponds to each SFH model in our library. As described in the previous subsection, all the models with an emergent Hα luminosity (scaled by a factor determined from the weighted least squares) smaller than $0.8\,{\times}\, L_{\tt H\alpha, obs}$ are excluded from the final determination of the parameters. The most probable value for a specific parameter is taken as the median of the corresponding PDF. The confidence interval is defined as the central 68% of the CDF. Simply adopting the single best-fitting value as the most probable estimate for the parameter is blind to degeneracies among relevant parameters. Our method is very robust, and the possible degeneracies are factored into the related confidence intervals. In the Appendix, we thoroughly discuss the validity of these parameter estimates from the broadband SED modeling.

Before feeding the data into the SED modeling, we correct the data for the foreground MW extinction (Schlegel et al. 1998), adopting the Cardelli et al. (1989) extinction curve with R_V = A_V/E(B − V) = 3.1. While there are nearly fixed, linear relations between A_λ and E(B − V) for most bands, this is not the case for GALEX NUV (e.g., Wyder et al. 2007), due to the finite bandwidth (∼1000 Å) and the presence of the 2175 Å bump in the MW extinction curve. The ratio $A_{\tt NUV}/E(B-V)$ is sensitive to both the emergent extragalactic SED and the foreground MW reddening E(B − V). For example, the SED emergent from a constant SF and the SED from a 7 Gyr old single stellar population have $A_{\tt NUV}/E(B-V) \sim 8.2$ and ∼7.5, respectively, for 0.05 mag of MW reddening. Also, for the SED emergent from a constant SF, $A_{\tt NUV}/E(B-V)$ would be 7.9 if the Galactic reddening was 0.5 mag. Therefore, unlike the other bands, the NUV data are not corrected for MW extinction with a fixed ratio of $A_{\tt NUV}/E(B-V)$. Before comparing the data with each model in the library, the NUV is corrected for the Galactic extinction using the ratio of $A_{\tt NUV}/E(B-V)$ predetermined from the emergent SED of the model and the Galactic E(B − V) toward the specific galaxy. In other words, the MW extinction correction for NUV is consistently applied before comparing each model with a given galaxy.

Before proceeding to the interpretation of our SED modeling results, we compare our results with those derived from existing stellar CMD-based analysis. Half of our galaxies have stellar CMD-based SFHs obtained from HST data and available in the literature. Among these galaxies, 14 galaxies have CMD analysis for a significant part (the observing field covers more than 20% of the area enclosed by the Holmberg radius) of the stellar disk. The availability of GALEX FUV and Spitzer 3.6 μm data makes our results for SFR$_{\tt 0.1}$ and $M_{\tt \star }$ particularly reliable. Therefore, we compare our results of the fraction of the SF accumulated during the recent 1 Gyr $f_{\tt 1\,Gyr}$ ((SFR$_{\tt 1}$/SFR_13.7)/13.7) with that derived from CMD modeling. Among the 14 galaxies, the $f_{\tt 1\,Gyr}$ for 11 galaxies (denoted as black circles in Figure 4) are from Weisz et al. (2011), and those of the other 3 galaxies (denoted as triangles in Figure 3) are from Orban et al. (2008). From the comparison of these 14 galaxies (Figure 4), our globally integrated results from the SED modeling are broadly consistent with those from the CMD-based analysis.

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Generally speaking, studying the resolved stellar populations is the best way to derive a detailed SFH, provided the data are deep enough. We note that most of the above CMD analyses are carried out on relatively shallow data sets (M_I ≲ 0). This means that the SFH at older times (≳ 2 Gyr) is mainly constrained by the age-insensitive red giant branch (RGB). At best, the RGB alone could provide a good estimate of the total SF, from its formation to a few Gyr ago (Greggio 2002) to within a factor of ∼2. Nevertheless, the results from the current CMD fitting are also sensitive to the modeling techniques used by different works. For example, using the same HST data sets of WLM, Weisz et al. (2008) derived a $f_{\tt 1\,Gyr}$ two times higher than Dolphin et al. (2005). Even more, adopting different stellar evolution libraries can lead to differences of at least a factor of two in the derived SFR (Gallart et al. 2005). Our library of SFHs is created with stellar evolution tracks combining those of the Padova 1994 model (see Bruzual & Charlot 2003) and Marigo et al. (2008) for the evolution of AGB, whereas the above CMD modeling adopted the tracks of the Padova 2000 model (Girardi et al. 2002) and Marigo et al. (2008). This undoubtedly leads to some differences in the SFHs. We also noticed that the CMD modeling works listed above adopted different IMFs from ours. Finally, the CMD modeling often covers a smaller spatial area of the galaxy than our SED fitting. Given these uncertainties and differences, the results from our SED modeling are in reasonable agreement with those from CMD modeling.

In the past, SFHs derived from broadband SED modeling have been thought to be biased toward younger, luminous populations. The above comparison shows that the average SFR during the past ∼ Gyr is well constrained by modeling the broadband SED with a relatively complete library of SFHs. We point out that the availability of U- and B-band data is essential in SED modeling because the two bands straddle the age-sensitive 4000 Å break.

By integrating the relevant surface density profiles times the circumference over radius, we derived (in Table 3) the asymptotic total stellar masses M_⋆ (See Section 5.4), SFR_13.7, b_0.1 Gyr (SFR_0.1/SFR_13.7), and b_1 Gyr (SFR₁/SFR_13.7). The galaxies are listed in order of increasing total baryonic mass in Table 3. A factor of 2–3 (e.g., Hunter & Gallagher 1986; Kennicutt et al. 2005; Lee et al. 2009b) enhancement of current SF compared to the averaged past rate has been used as one way to define a starburst in the literature. Compared to SFR_13.7, 38% (13) of the sample galaxies have SFR_0.1 enhanced at least by a factor of two, and 47% (16) have SFR_0.1 consistent with SFR_13.7 within a factor of two. For SFR₁, 12% (4 galaxies) have been enhanced at least by a factor of two, and 71% (24 galaxies) have SFR₁ consistent with SFR_13.7 within a factor of two. Among the five galaxies with SFR_0.1/SFR_13.7 < 0.5, four (LGS 3, DDO 216, DDO 101, WLM) have baryonic mass (here approximated to be the stellar mass plus atomic gas mass, see below) smaller than 10⁸ M_☉. Similarly, all six galaxies with SFR₁/SFR_13.7 < 0.5 have baryonic mass smaller than 10⁸ M_☉. The total atomic gas mass was derived by multiplying the H i gas mass by 1.34 to account for He. The H i masses used here were all from single-dish H i emission line observations reported in the literature (see HE04 for the references), and the total stellar mass is derived from our SED modeling (see Section 5.4).

We plot b_0.1 Gyr and b_1 Gyr for the galaxies (Table 3) in Figure 5. The galaxies with baryonic mass smaller and higher than 10⁸ M_☉ are denoted as different symbols. A mass of 10⁸ M_☉ is about the median baryonic mass of our whole sample. There is almost no relationship between SFHs and total baryonic mass for our sample, except that all galaxies that exhibit extremely declining SFHs are relatively low-mass systems. Among the whole sample, five galaxies (LGS 3, DDO 216, NGC 4163, DDO 101, and WLM) have experienced a significant (>30%) decline in SF activity (both SFR_0.1 and SFR₁) over the past ∼ Gyr. We note that all of these five galaxies have baryonic masses smaller than 10⁸ M_☉, and, except for WLM, the (atomic) gas-to-stellar mass ratios are smaller than 1.

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The tidal indices (Table 1) of LGS 3 and DDO 216 are larger than 1, which indicates that their evolution has likely been significantly influenced by environment. Most of the gas in these two galaxies could have been removed by either ram pressure stripping or tidal disturbance. In particular, based on the morphology of the H i distribution, McConnachie et al. (2007) concluded that DDO 216 is currently undergoing ram pressure stripping caused by the intragroup medium. The tidal index (Karachentsev et al. 2004) of a galaxy is defined as the maximum density enhancement caused by any neighboring galaxies, and higher values correspond to the likelihood of more significant tidal interaction with neighboring galaxies. This is consistent with the expectation that lower mass systems (below a few times 10⁸ M_☉) are more susceptible to significant influence of both the environment (Gunn & Gott 1972) and stellar feedback (e.g., winds and supernova (SN) explosions), which could even induce blow-out of interstellar gas from the galaxy (e.g., Mac Low & Ferrara 1999).

A few galaxies deserve special mention. M81dwA, DDO 75, Haro 29, and NGC 1569 hold the largest ratio (>5) of SFR_0.1/SFR_13.7 among our sample. The recent starburst in NGC 1569, which has the highest current SF intensity in our sample, may be ascribed to an interaction of some kind (e.g., Stil & Israel 1998; Muhle et al. 2005; Johnson et al. 2012, in preparation). Similarly, the recent rise of the SFR in M81dwA may be due to the recent interaction within the M81 group, which has several dramatic galaxy–galaxy interactions in progress. Haro 29 is a typical BCD galaxy undergoing an intense burst of SF in the central regions (Thuan & Martin 1981). DDO 75 and Haro 29 both have negative tidal indices (−0.6 for DDO 75, −1 for Haro 29), indicating that the recent enhancement of SF may be ascribed to internal processes. For DDO 154, Kennicutt & Skillman (2001) reported a factor of ∼2–4 times lower recent SF compared to the past based on Hα imaging, which is inconsistent with our results for SFR_0.1. We interpret the disagreement as being due to the fact that Hα is not a robust SFR indicator in LSB dwarf galaxies (e.g., Meurer et al. 2009; Lee et al. 2009a; Hunter et al. 2010).

The sample as a whole exhibits a diversity of radial variations of SFHs (Figure 1). However, some systematic trends do exist.

Figures 6 and 7 show the radial variations of log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) and of log($\Sigma _{\tt SFR_{1}}/\Sigma _{\star }$). The radius has been normalized by the V-band disk scale length. The galaxies in Figures 6 and 7 are plotted in order of increasing total baryonic mass from the upper left panel to the lower right panel, and the galaxy names are listed in order of increasing baryonic mass in each panel. The dashed line in each panel marks a constant SFH over a Hubble time. Deep stellar CMD analysis of nearby dIrr galaxies (e.g., IC 1613; Skillman et al. 2003; Leo A: Cole et al. 2007) suggests that most of the SF in dIrr galaxies may take place at intermediate ages, rather than persisting constantly over a Hubble time. In contrast, LGS 3 (Hidalgo et al. 2011) consists of mostly old populations. Therefore, the data points lying above the dashed line in Figures 6 and 7 do not necessarily mean elevated SFR compared to the past.

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To quantify the radial trends of the SFH variation, we did linear least-squares fitting to the radial variations of log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) and log($\Sigma _{\tt SFR_{1}}/\Sigma _{\star }$). If there is an obvious break in the stellar mass surface density profiles (Figure 1 and Table 3), we fit the slope for the inner and the outer parts separately. The GALEX data (thus the measure of recent SFR) of several galaxies (LGS 3, DDO 210, DDO 216, and NGC 1569) have high S/N ratios only in the inner disk, in this case we only fit the inner disk. We also fit the relevant ratios as a function of normalized (by the V-band scale length) radius. The slopes (or gradients) from the fitting are listed in Table 4. The weighted averages of the relevant slopes for the galaxies below and above a baryonic mass 10⁸ M_☉ are also listed in Table 4. The slopes are further plotted in Figures 8 and 9. Also shown in Figures 8 and 9 are the weighted averages (large diamonds) of the whole sample. Negative slopes indicate that the SFR averaged over the relevant timescale becomes more centrally concentrated compared to the past. As shown, on average, our sample galaxies have more negative radial variations of log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) and log($\Sigma _{\tt SFR_{1}}/\Sigma _{\star }$) in the outer disks than the inner disks, and log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) has steeper radial declining than log($\Sigma _{\tt SFR_{1}}/\Sigma _{\star }$). In the inner disks, 44% (15 galaxies) of the sample show negative radial slopes for both log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) and log($\Sigma _{\tt SFR_{1}}/\Sigma _{\star }$), and 9 of the 15 galaxies have M_bary < 10⁸ M_☉. A few galaxies with M_bary > 10⁸ M_☉ (i.e., DDO 165, DDO 63, DDO 154, DDO 168, NGC 2366, and NGC 4214) have almost zero slopes (|slope| < 0.2) in the inner disks. In the outer disks, 80% (27 galaxies) have negative slopes for both log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) and log($\Sigma _{\tt SFR_{1}}/\Sigma _{\star }$), and 14 of the 27 galaxies have M_bary < 10⁸ M_☉. Particularly, 35% (12 galaxies, 7 with M_bary ≲ 10⁸ M_☉) of the galaxies exhibit negative slopes across the observed disk. In NGC 4163 and NGC 3738, the recent SF has been suppressed by more than an order of magnitude in the outer regions, even though these two galaxies have almost constant ratios of log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) and log($\Sigma _{\tt SFR_{1}}/\Sigma _{\star }$) at large radii. These results are in line with Figure 2, which shows the comparison between the scale length of FUV, B, and 3.6 μm.

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Table 4. Slopes of the SFH Variations with Radius

Galaxy	$\frac{\Delta (\log (\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }))}{\Delta ({R[{\rm kpc}])}}$		$\frac{\Delta (\log (\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }))}{\Delta ({R/R_{{D}}^{\tt {\it V}})}}$		$\frac{\Delta (\log (\Sigma _{\tt SFR_{1}}/\Sigma _{\star }))}{\Delta ({R[{\rm kpc}])}}$		$\frac{\Delta (\log (\Sigma _{\tt SFR_{1}}/\Sigma _{\star }))}{\Delta ({R/R_{D}^{\tt {\it V}})}}$
	Inner	Outer	Inner	Outer	Inner	Outer	Inner	Outer
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
Galaxies with Mbary < 108 M☉
LGS 3	−0.10 ± 0.16	...	−0.02 ± 0.04	...	−1.25 ± 0.19	...	−0.29 ± 0.04	...
DDO 210	−2.59 ± 0.59	...	−0.43 ± 0.10	...	−0.85 ± 0.22	...	−0.14 ± 0.04	...
DDO 69	−0.50 ± 0.06	−2.29 ± 0.28	−0.09 ± 0.01	−0.43 ± 0.05	1.01 ± 0.15	−0.95 ± 0.05	0.19 ± 0.03	−0.18 ± 0.01
DDO 155	−1.41 ± 0.06	−2.53 ± 0.24	−0.21 ± 0.01	−0.37 ± 0.03	−0.65 ± 0.15	−1.19 ± 0.26	−0.10 ± 0.02	−0.18 ± 0.04
DDO 216	−0.44 ± 0.04	...	−0.24 ± 0.02	...	−0.52 ± 0.04	...	−0.28 ± 0.02	...
M81dwA	0.58 ± 0.03	−0.75 ± 0.02	0.15 ± 0.01	−0.20 ± 0.01	0.56 ± 0.03	−0.65 ± 0.01	0.14 ± 0.01	−0.17 ± 0.01
Mrk 178	−2.53 ± 0.12	−1.86 ± 0.02	−0.66 ± 0.03	−0.48 ± 0.01	−1.28 ± 0.09	−0.35 ± 0.02	−0.33 ± 0.02	−0.09 ± 0.01
DDO 187	−1.01 ± 0.08	−2.09 ± 0.05	−0.18 ± 0.01	−0.37 ± 0.01	−0.64 ± 0.01	−1.42 ± 0.06	−0.11 ± 0.01	−0.26 ± 0.01
UGC 8508	−0.58 ± 0.02	−2.56 ± 0.07	−0.15 ± 0.01	−0.68 ± 0.02	0.07 ± 0.02	−0.72 ± 0.02	0.02 ± 0.01	−0.19 ± 0.01
NGC 4163	−0.74 ± 0.02	−0.06 ± 0.02	−0.74 ± 0.02	−0.06 ± 0.02	−0.55 ± 0.01	0.00 ± 0.04	−0.55 ± 0.01	0.00 ± 0.04
CVnIdwA	0.24 ± 0.07	−2.50 ± 0.15	0.14 ± 0.04	−1.42 ± 0.09	0.07 ± 0.03	−1.40 ± 0.18	0.04 ± 0.02	−0.79 ± 0.10
DDO 70	−1.58 ± 0.03	−0.82 ± 0.11	−0.75 ± 0.01	−0.39 ± 0.05	0.00 ± 0.05	−1.42 ± 0.11	0.00 ± 0.02	−0.68 ± 0.05
IC 1613	0.11 ± 0.05	−1.22 ± 0.03	0.06 ± 0.03	−0.72 ± 0.02	0.04 ± 0.01	−0.67 ± 0.02	0.02 ± 0.01	−0.39 ± 0.01
DDO 101	0.52 ± 0.01	−0.24 ± 0.07	0.48 ± 0.01	−0.23 ± 0.06	0.68 ± 0.03	−0.18 ± 0.08	0.63 ± 0.03	−0.16 ± 0.07
WLM	−0.46 ± 0.02	−0.90 ± 0.04	−0.26 ± 0.01	−0.52 ± 0.02	−0.26 ± 0.01	−0.39 ± 0.06	−0.15 ± 0.01	−0.23 ± 0.03
Averages	0.01 ± 0.72	−1.26 ± 0.68	−0.10 ± 0.31	−0.37 ± 0.17	−0.29 ± 0.65	−0.71 ± 0.35	−0.14 ± 0.16	−0.19 ± 0.07
Galaxies with Mbary > 108 M☉
DDO 75	0.76 ± 0.03	−2.01 ± 0.06	0.17 ± 0.01	−0.45 ± 0.01	0.82 ± 0.03	−0.55 ± 0.05	0.18 ± 0.01	−0.12 ± 0.01
VIIZW 403	−1.42 ± 0.06	−0.95 ± 0.03	−0.75 ± 0.03	−0.50 ± 0.02	−0.52 ± 0.05	−0.71 ± 0.06	−0.27 ± 0.02	−0.37 ± 0.03
Haro 29	−0.60 ± 0.09	−1.09 ± 0.05	−0.18 ± 0.02	−0.32 ± 0.01	0.63 ± 0.06	−0.13 ± 0.03	0.18 ± 0.02	−0.04 ± 0.01
DDO 133	0.46 ± 0.01	−0.23 ± 0.05	0.52 ± 0.01	−0.26 ± 0.06	0.15 ± 0.01	0.02 ± 0.02	0.17 ± 0.01	0.02 ± 0.02
DDO 126	−0.23 ± 0.02	−0.02 ± 0.02	−0.20 ± 0.01	−0.02 ± 0.02	0.42 ± 0.03	−0.02 ± 0.01	0.36 ± 0.03	−0.01 ± 0.01
DDO 165	0.05 ± 0.01	−1.00 ± 0.02	0.12 ± 0.01	−2.27 ± 0.04	0.12 ± 0.01	−0.16 ± 0.01	0.26 ± 0.02	−0.35 ± 0.02
DDO 63	−0.07 ± 0.02	−1.02 ± 0.03	−0.03 ± 0.01	−0.51 ± 0.02	0.19 ± 0.03	−0.39 ± 0.02	0.09 ± 0.02	−0.19 ± 0.01
NGC 6822	−0.07 ± 0.02	1.75 ± 0.14	−0.04 ± 0.01	1.00 ± 0.08	−0.67 ± 0.02	0.44 ± 0.13	−0.38 ± 0.01	0.25 ± 0.07
DDO 53	−0.48 ± 0.02	−1.56 ± 0.05	−0.35 ± 0.01	−1.13 ± 0.04	−0.54 ± 0.02	−1.32 ± 0.03	−0.39 ± 0.02	−0.95 ± 0.02
DDO 154	0.03 ± 0.03	−0.45 ± 0.02	0.02 ± 0.02	−0.27 ± 0.01	−0.02 ± 0.01	0.03 ± 0.02	−0.01 ± 0.01	0.02 ± 0.01
DDO 87	0.20 ± 0.03	−0.04 ± 0.01	0.29 ± 0.04	−0.06 ± 0.02	−0.05 ± 0.01	0.12 ± 0.01	−0.07 ± 0.01	0.17 ± 0.01
DDO 52	0.19 ± 0.02	0.13 ± 0.01	0.25 ± 0.02	0.17 ± 0.02	0.31 ± 0.01	−0.10 ± 0.01	0.40 ± 0.02	−0.14 ± 0.01
DDO 168	−0.06 ± 0.01	−0.88 ± 0.02	−0.05 ± 0.01	−0.73 ± 0.02	0.01 ± 0.02	−0.40 ± 0.02	0.01 ± 0.02	−0.33 ± 0.02
NGC 1569	−0.64 ± 0.03	...	−0.25 ± 0.01	...	−0.46 ± 0.03	...	−0.18 ± 0.01	...
NGC 3738	−1.31 ± 0.04	−0.27 ± 0.01	−1.31 ± 0.04	−0.27 ± 0.01	−1.22 ± 0.02	−0.25 ± 0.02	−1.22 ± 0.02	−0.25 ± 0.02
DDO 50	0.35 ± 0.01	−0.31 ± 0.02	0.38 ± 0.01	−0.34 ± 0.03	0.08 ± 0.01	0.00 ± 0.01	0.09 ± 0.01	0.00 ± 0.01
NGC 2366	0.04 ± 0.01	−0.07 ± 0.01	0.06 ± 0.01	−0.10 ± 0.02	−0.14 ± 0.01	−0.09 ± 0.01	−0.19 ± 0.02	−0.12 ± 0.02
NGC 4214	0.15 ± 0.01	...	0.11 ± 0.01	...	−0.01 ± 0.01	...	0.00 ± 0.01	...
NGC 1156	−0.30 ± 0.01	−0.41 ± 0.01	−0.24 ± 0.01	−0.33 ± 0.01	−0.05 ± 0.01	−0.25 ± 0.02	−0.04 ± 0.01	−0.20 ± 0.02
Averages	0.12 ± 0.29	−0.59 ± 0.44	0.08 ± 0.25	−0.44 ± 0.65	0.04 ± 0.21	−0.08 ± 0.20	0.02 ± 0.20	−0.09 ± 0.16

Notes. (1) Galaxy names. (2–5) Slopes of the radial variations of $\Delta (\log (\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }))/\Delta ({\rm \textit R[kpc]})$ and $\Delta (\log (\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }))/\Delta ({\rm \textit R/\textit R_{D}^{\tt {\it V}}})$, for the inner disks and the outer disks, respectively. (6–9) Slopes of the radial variations of $\Delta (\log (\Sigma _{\tt SFR_{1}}/\Sigma _{\star }))/\Delta ({\rm \textit R[kpc]})$ and $\Delta (\log (\Sigma _{\tt SFR_{1}}/\Sigma _{\star }))/\Delta ({\rm \textit R/\textit R_{D}^{\tt {\it V}}})$, for the inner disks and the outer disks, respectively.

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Among the whole sample, the weighted averages of the slopes of $\Delta (\log (\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }))/\Delta ({\rm \it R})$ are 0.10 ± 0.42 and −0.64 ± 0.48 for the inner and the outer disks, respectively. The weighted averages of $\Delta (\log (\Sigma _{\tt SFR_{1}}/\Sigma _{\star }))/\Delta ({\rm \it R})$ are −0.05 ± 0.37 and −0.28 ± 0.39 for the inner and the outer disks, respectively. Furthermore, as listed in Table 4, the inner disks tend to have much shallower radial slopes of $\Delta (\log (\Sigma _{\tt SFR}/\Sigma _{\star }))/\Delta ({\rm \it R})$ than the outer disks. The SF activity in dIrr galaxies was suggested to be a random percolating process across the stellar disk (e.g., van Zee 2001). According to our results, this randomly percolating scenario may only be appropriate for the inner disks of some galaxies (especially those relatively massive galaxies). As far as the whole disk of the low-mass dIrr galaxy is concerned, the star-forming disk has been shrinking from the outer part.

The stellar mass surface density Σ_⋆ profiles of the galaxies are shown in the last column of Figure 1. The stellar mass surface density profiles are well fitted with piece-wise exponentials, which are overplotted on the profiles. For some relatively faint galaxies, the high-quality IRAC 3.6 μm data reach further radii than the other bands. In this case, considering the relative insensitivity of the $M_{\star }/L_{\tt NIR}$ to the underlying stellar populations (Bell & de Jong 2001), we extend stellar mass surface density profiles derived from our multi-band fitting to the radius reached by the 3.6 μm surface photometry, adopting the M_⋆/L_3.6 μm ratio from the radius where our multi-band SED fitting ends. The 3.6 μm photometry for some galaxies with large angular size does not reach as far as the other bands due to the limited field of view of the Spitzer observations. In this case, we constrained the stellar mass profiles at the outer part by modeling the UBV (and GALEX FUV/NUV if available) data. We also derived the asymptotic total stellar mass by applying the "growth curve" method to the accumulated stellar mass profiles with radius (e.g., Muñoz-Mateos et al. 2009). Briefly, we obtained the radial gradient of the accumulated stellar mass at each observed radius, then the accumulated stellar mass was linearly fit as a function of the gradient, and the y-intercept (zero gradient) of the fit was adopted as the asymptotic stellar mass. Table 3 lists the central stellar mass surface density Σ_{⋆, center} extrapolated from the inner exponential disk, stellar mass concentration index C₃₁, the inner/outer scale lengths of the stellar mass surface density profiles, and the asymptotic total stellar mass for each galaxy. C₃₁ is defined here as the ratio of the radii that encompass 75% and 25% of the total stellar mass (de Vaucouleurs 1977).

Like the radial light profiles, the stellar mass surface density profile of dIrr galaxies usually can be described by either a single or piece-wise exponential profile over most of the disk. The most striking feature about our galaxies is that most of them have broken stellar mass surface density profiles. Specifically, ∼80% (27) of the galaxies exhibit obvious down-bending profiles, in the sense that the outer disks have steeper stellar mass surface density profiles than the inner disks. Whereas three (DDO 70, Haro 29, NGC 1569) galaxies have obvious up-bending (flatter outer disk) profiles, in the sense that the outer disks exhibit shallower profiles than the inner disks. Both Haro 29 and NGC 1569 have experienced intense, extended starbursts in their inner regions. It has been shown that down-bending surface brightness profiles are very common in spiral galaxies (e.g., Ferguson & Clarke 2001; Pohlen & Trujillo 2006). Nevertheless, Bakos et al. (2008) found a pure exponential stellar mass profile for the spiral galaxies with down-bending surface brightness profiles. This is different from our dIrr galaxies which tend to have both down-bending stellar mass surface density profiles and down-bending surface brightness profiles. Amorín et al. (2007) found that the LSB stellar hosts in BCD galaxies have near-exponential profiles. From our results for BCD galaxies, not only the LSB stellar hosts, but also the inner starburst regions have exponential stellar mass surface density profiles, although the scale length is different for the inner starburst region and the LSB stellar host (Figure 1).

The present-day stellar mass surface density profile of dIrr galaxies should be primarily determined (see Section 6.1.1) by the radial variations of SFH. As shown above (Table 4), on average, the SF in the outer disks has been decreasing significantly more, both over time (Figures 6 and 7) and over radius (Figures 8 and 9), compared to the inner disks, which would naturally lead to down-bending stellar mass surface density profiles, seen in most of our galaxies. This scenario of outside–in depression of SF is different from what may be happening in typical BCD (e.g., Haro 29, NGC 1569; see Figure 1) galaxies. The central starburst of BCD galaxies results in faster buildup of the inner stellar disk than the outer part, which could lead to up-bending stellar mass surface density profiles.

As can be seen, a local enhancement of the recent SF may affect the stellar mass density profile in these low-mass dIrr galaxies. For example, in NGC 2366, the supergiant H ii complex (NGC 2363; Youngblood & Hunter 1999) ∼1 kpc away from the center corresponds to the remarkable spike in the stellar mass surface density profile. The concentration index of the stellar mass distribution is generally close to, but lower than, that of a single exponential (C₃₁ = 2.81).

Lower mass systems tend to have lower stellar mass concentrations (Table 3). However, the BCD galaxy Haro 29 is as centrally concentrated (C₃₁ = 5.3) as a de Vaucouleurs r^1/4 bulge profile, which suggests that the centralized starburst has lasted an extended period of time. Assuming the SFR_0.1 in the central regions is representative of the recent episode of central starburst, and the central stellar mass is dominantly contributed by the burst, the starburst would have lasted more than 0.6 Gyr.

The "inside–out" growth mode has long been suggested for the formation of galaxy disks (e.g., Larson 1976; Chiappini et al. 1997; Mo et al. 1998; Naab & Ostriker 2006). The "inside–out" scenario reproduces many observations of spiral galaxies, such as the radial gradients of both broadband colors (bluer outward) and metallicity (lower outward; de Jong 1996; Bell & de Jong 2000; MacArthur et al. 2004; Wang et al. 2011), and the extended UV emission discovered in outer spiral disks (e.g., Thilker et al. 2007; Boissier et al. 2008). Muñoz-Mateos et al. (2007) found a moderate inside–out disk formation by studying the radial profiles of GALEX FUV and 2MASS Ks for a sample of relatively face-on nearby spiral galaxies. Recently, the "inside–out" growth mode has been confirmed for M 33 (Williams et al. 2009) and NGC 300 (Gogarten et al. 2010) from analysis of CMDs obtained with HST. For M 33, it was shown (Williams et al. 2009; Barker et al. 2011) that the "inside–out" scenario only applies to the inner disk, and the region beyond the surface brightness break exhibits positive age gradient.

However, unlike luminous spiral galaxies, late-type dIrr galaxies exhibit a variety of behaviors in terms of color profiles (Kormendy & Djorgovski 1989; Tully et al. 1996; Jansen et al. 2000; Taylor et al. 2005; HE06). Nevertheless, Tully et al. (1996) found that galaxies fainter than M_B ∼ −17 become redder with radius in the Ursa Major cluster. Later, with a larger sample, Jansen et al. (2000) found that the reddening (with radius) trend may occur at a fainter absolute magnitude than that found by Tully et al. (1996). Of a sample of 94 dIrr galaxies studied by HE06, 64% of the galaxies with a gradient in B − V become redder with radius. In particular, all the galaxies with M_B > −14 become redder at least in one color (either B − V or U − B) at outer radii. Recently, Tortora et al. (2010) analyzed the optical color gradients of ∼50,000 nearby Sloan Digital Sky Survey (SDSS) galaxies. They found a good correspondence between the color gradients and the stellar mass of the galaxies. In particular, below a stellar mass M_⋆ ≲ 10^8.7 M_☉, the color gradient slopes become positive, which were mainly attributed to the positive metallicity gradients. The lowest mass galaxies in the sample of Tortora et al. (2010) have M_⋆ ∼ 10^8.2 M_☉, compared to the median stellar mass ∼10^7.2 M_☉ of our sample. All the above studies are based on optical broadband data alone, which make the interpretation of the radial color gradients ambiguous, due to the degeneracy between age, metallicity, and extinction.

Additionally, from stellar CMD analysis of two different (HST/WFPC2) fields in IC1613, Skillman et al. (2003; also Bernard et al. 2007) found that the SF activity in the outer field has been significantly depressed during the last Gyr, which is in contrast to the "inside–out" growth scenario. Similarly, by analyzing resolved stellar populations, the characteristics of more spatially extended older populations have been reported both for transition-type dwarf galaxies (e.g., DDO 210, McConnachie et al. 2006; Phoenix, Hidalgo et al. 2009) and dwarf spheroidal (dSph) galaxies (e.g., Sculptor dSph, Tolstoy et al. 2004; Fornax dSph; Battaglia et al. 2006). A clear trend of "outside–in" quenching of recent SF was also found (Gallart et al. 2008; Indu & Subramaniam 2011) in the Large Magellanic Cloud, which has a mass comparable to that of the most massive galaxy (i.e., NGC 1156) in our sample.

These studies are in agreement with our findings for our sample of dIrr galaxies. We point out that our analysis here is only sensitive to the SF during the recent ∼ Gyr and the SF averaged over the whole lifetime. The radial variations of SF at the intermediate ages could be different from the recent past. For instance, in the late-type spiral galaxy NGC 2976, Williams et al. (2010) found a deficit of stellar populations younger than ∼ Gyr beyond the break of the disk surface brightness profile, with similar ancient stellar populations at all radii. Nevertheless, since the suppression of SF in the outer disk was found in the majority of our sample galaxies, the trend we found here must reflect the evolutionary process of the stellar disks of low-mass dIrr galaxies in general. In the following, we discuss possible interpretations of the observed radial variations of the stellar populations in dIrr galaxies.

6.1.1. In Situ SF versus Secular Redistribution?

6.1.2. External Influences

External factors, such as interactions (both with the intragroup/intracluster medium and with neighboring galaxies) and the cosmic UV background (e.g., Gnedin 2000), can also lead to more centrally concentrated recent SF or depressed SF in outer disks. The tidal index (Table 1) suggests that less than five of the galaxies studied here could be noticeably affected (Θ ≳ 1) by neighboring galaxies. The slopes of the radial variations of log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) and log($\Sigma _{\tt SFR_{1}}/\Sigma _{\star }$) for the outer disks are plotted against the tidal index in Figure 10. (There is no calculation of Θ for DDO 101 in Karachentsev et al. 2004, so we arbitrarily set Θ as 0 when creating the plot.) There is no obvious correlation between the radial variations of the SFHs and the environment. However, most galaxies fall between an upper envelope and a lower envelope, as indicated by the dashed lines in the plot. NGC 6822 has such a large angular size projected on the sky that the observations only covered the central regions. Therefore, the slopes measured here for NGC 6822 do not reflect the SFH variations across the main star-forming disk. In fact, based on HST images of five fields in NGC 6822, Wyder (2001) found a higher ratio of the recent SFR to the average past rate in the inner bar regions than the outer regions, consistent with the trend we found in this study. The galaxies with larger Θ prefer lower (zero or negative) slopes of the radial variations of log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) and log($\Sigma _{\tt SFR_{1}}/\Sigma _{\star }$).

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Two of the most obvious environmental effects studied in the literature are ram pressure stripping of the gas component caused by a hot gaseous intragroup/intracluster medium (Gunn & Gott 1972; Lin & Faber 1983) and tidal disturbance from neighboring galaxies (e.g., Mayer et al. 2001). By studying the dynamical properties of the dwarf galaxies in the Fornax cluster, Drinkwater et al. (2001) show that the dwarf galaxies are still falling into the cluster, and the fraction of dwarfs with active SF drops rapidly toward the cluster center. This is a clear indication of the environmental effect on the evolution of dwarf galaxies. Similarly, Pustilnik et al. (2002) found that the blue compact galaxies in higher density environment have on average less H i which could be attributed to either ram pressure stripping of the gas or tidal interactions.

Presumably, both ram pressure stripping and tidal disturbance should be more effective on lower mass galaxies. The outer disk, especially in relatively low-mass systems where the gravitational well and the gas surface density are relatively low, is more prone to gas removal due to ram pressure. Furthermore, significant tidal forces from neighboring galaxies could induce a bar instability and ensuing gas inflows toward the inner regions (e.g., Mayer et al. 2001), which leads to a more centrally concentrated SF. In our sample, lower mass galaxies tend to have more centrally concentrated SF than relatively higher mass galaxies, which is in general agreement with the expectation of the above two environmental effects.

By studying the H i distribution of spiral galaxies in the Virgo cluster, Cayatte et al. (1994) found that some galaxies close to the cluster core have normal gas densities (compared to field counterparts) in the inner disks but show a strong decline or cutoff of the H i intensity starting within the half-light radius of the stellar disk. These galaxies should have been undergoing remarkable ram pressure stripping. Actually, the radial H i distributions (D. A. Hunter et al. 2012, in preparation) for some of our galaxies do show a strong decline starting well within the stellar disk. Numerical simulations (e.g., Vollmer 2003; Kapferer et al. 2008) suggest that both ram pressure stripping and tidal disturbance are needed to reproduce the observations. Figure 10 suggests that the tidal disturbance from neighboring galaxies should play some role in the inward movement of SF activity.

The presence of a UV background can heat the gas in low-mass halos, which prevents SF at least in the outer disk where the gas is not dense enough for self-shielding (Susa & Umemura 2004). By solving the radiative transfer equation for the diffuse UV background in a pre-galactic cloud, Tajiri & Umemura (1998) showed that, above a critical number density $n_{\tt crit} \sim 1.4 \times 10^{-2}$ cm⁻³ ($N_{\tt H_{I}} \sim 1.3 \times$ 10¹⁹ for a gas disk ∼0.3 kpc thick), which is almost independent of the UV background intensity, self-shielding of the ISM against the UV background is prominent. Based on our H i emission line maps of these galaxies, and according to this critical density, the typical gas column density (∼10²⁰ cm⁻²) across the observed stellar disks of our galaxies should be sufficient for them to be self-shielded. Therefore, the cosmic UV background is unlikely to be the main driver of the centrally concentrated SF.

6.1.3. Regulation of the SFR through Stellar Feedback

6.1.4. Pressure Support in the Gas Disk

H98 found that the radial profiles of current SF activity correlate better with those of the older stellar emission (e.g., V band) than with the atomic gas in dIrr galaxies. Since star-forming molecular clouds form from the atomic gas, this lack of a correlation between SF and gas is unexpected. Similarly, Ryder & Dopita (1994; see also James et al. 2009) discovered a good correlation between the surface brightness in I-band (a proxy for the mass-dominant old stellar populations) and Hα for a sample of nearby spiral galaxies. Recently, a good correspondence between the surface brightness in SDSS r-band and in FUV was also shown for a sample of LSB galaxies (Wyder et al. 2009). If the optical broadband emission traces the underlying mass-dominant old stellar populations, the above correlations would be equivalent to a relationship between SF and the stellar mass surface density. Is this correlation between $\Sigma _{\tt SFR}$ and Σ_⋆ causal or casual?

In spiral galaxies, the stellar component usually dominates the inner disk in terms of both surface density and volume density, so we might expect that the stellar component plays an important role in regulating the ISM and thus the SF process (e.g., Wong & Blitz 2002; Blitz & Rosolowsky 2006; Shi et al. 2011). However, that is not necessarily the case in dIrr galaxies, where the atomic gas usually dominates the baryonic component. In Figure 11, $\Sigma _{\tt SFR_{0.1}}$ is plotted against Σ_⋆ for the whole sample. The data points are extracted from the azimuthal averages of the relevant surface density profiles. Although there is a rough correspondence between $\Sigma _{\tt SFR_{0.1}}$ and Σ_⋆, the scatter is substantial. The Spearman rank correlation coefficient between Σ_⋆ and $\Sigma _{\tt SFR_{0.1}}$ is ∼0.6.

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As shown in Figure 6, the inner regions of our galaxies tend to have higher ratios of $\Sigma _{\tt SFR_{0.1}}$ to Σ_⋆ than outer parts, which is especially significant for relatively lower mass systems. In other words, $\Sigma _{\tt SFR_{0.1}}$ usually has a steeper radial dependence than Σ_⋆. This is consistent with our finding that shorter wavelengths tend to have shorter scale lengths than longer wavelengths (Figure 2). Within the inner disks, our sample galaxies, on average, have almost zero radial gradients of $\Sigma _{\tt SFR}/\Sigma _{\star }$ (Figures 8 and 9). Therefore, the correlation between $\Sigma _{\tt SFR}$ and Σ_⋆ is only good within the inner disk.

Recently, Leroy et al. (2008) found that, in dIrr galaxies and outer disks of spiral galaxies, the ratio of SFR surface density to atomic gas surface density exhibits an almost linear relationship to the stellar mass surface density. Given the narrow range of the H i surface density (5–10 M_☉ pc⁻²; Leroy et al. 2008) within the optical disk, the above correlation actually reflects the correlation between the SFR and the stellar mass surface density. We note that all of the dIrr galaxies studied by Leroy et al. (2008) have baryonic masses well above 10⁸ M_☉. For the four galaxies in common with the sample of Leroy et al. (i.e., DDO 154, DDO 63, DDO 50, and NGC 4214), except for DDO 50, all the other three have shallow ($|\Delta (\log (\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }))/\Delta ({\rm \it R})| < 0.2$, Table 4) radial slopes of log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) across the inner disks, which extend to more than ∼3 disk scale lengths, in agreement with Leroy et al. (2008). Nevertheless, some galaxies (DDO 52, NGC 1569, NGC 3738, and NGC 1156) in our sample with baryonic masses larger than 10⁸ M_☉ show obvious gradients (either positive or negative, see Table 4) of log($\Sigma _{\tt SFR_{0.1}}/\Sigma _{\star }$) across the whole disks.

The SFE in dIrr galaxies is generally very low (e.g., van Zee 2001; Lee et al. 2011), as evidenced by high gas-to-stellar mass ratios (e.g., see Table 1). Therefore, the stellar disk may have always been a small perturbation on the gas-dominated baryonic disk. The constant average ratios of $\Sigma _{\tt SFR_{0.1}}$ to Σ_⋆ and $\Sigma _{\tt SFR_{1}}$ to Σ_⋆ seen in the inner disks of dIrr galaxies could imply that the SFE at a given radius in the inner disk may have been nearly the same for most of the galaxy's lifetime. This would naturally lead to similar radial profiles of the SFR to the accumulated stellar mass. In agreement with this evolutionary picture, Elmegreen & Hunter (2006) considered multi-component SF triggering processes, including turbulent compression, to interpret SF in outer spiral disks and in dIrr galaxies. They suggested that the SF profile should be about the same as the accumulated stellar mass profile when the baryonic component of the disk is still dominated by atomic gas. In this scenario, the stellar mass distribution naturally follows the inefficient, patchy SF. On the other hand, for the outer disks, the SF process has been significantly affected by stellar feedback (e.g., SNe) and the external disturbance. As discussed in the previous section, feedback-regulated evolution or environmental effects may result in shrinking of the star-forming gas disk toward the inner part, and thus more and more centrally concentrated SF, without a remarkable correlation between SFR and stellar mass. We noticed that the "inner disk" of galaxies with M_bary >10⁸ M_☉ could extend over more than ∼3 disk scale lengths.

Here we explore the reliability of multi-band SED modeling in recovering the rough SFH of dIrr galaxies. In these tests, we first logarithmically divided the Hubble time (13.7 Gyr) into 14 independent age bins and took the SF as constant within each age bin. Then a series of 3000 mock SFHs were generated by randomly changing the SFR for each independent age bin. The internal extinction was fixed at A_V = 0.1 when generating the SEDs related to each mock SFH. We then modeled the SEDs (GALEX FUV/NUV, U, B, V, 3.6 μm) related to the mock SFHs with our library of 6 component superposition (6-C) SFHs used in this work. The calibration uncertainties of the real data were taken as the photometric uncertainties of the SED related to each mock SFH.

Figure 12 presents a comparison of the modeling results, including M_⋆, SFR₁, SFR_0.1, and SFR_0.03, and relevant integrated quantities of the mock input SFHs. We fit each histogram in Figure 12 with a Gaussian curve, which is overplotted as a red solid line in each panel. The standard deviation (σ) and mean value (μ) of the Gaussian curve is also indicated. As can be seen, except for SFR_0.03, all the parameters are well recovered in our SED modeling, with acceptable uncertainties related to the varying SFHs. Figure 13 further shows ratios of SFR averaged over different timescales. Again, except for SFR_0.03, the modeling with the 6-C library recovers the SFHs with negligible, if any, bias.

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FUV has been extensively used to estimate the recent (∼0.1 Gyr) SFR in galaxies (e.g., Kennicutt 1998; Salim et al. 2007; Hunter et al. 2010). The most commonly used formula (e.g., Kennicutt 1998) assumes constant SF during the past 0.1 Gyr, which is comparable to a galactic dynamical timescale. The assumption of more or less constant recent SF is generally true for a high-luminosity galaxy as a whole, as evidenced by the consistency between the Hα-based SFR and the FUV-based SFR (e.g., Sullivan et al. 2004). However, this is not necessarily the case in low-luminosity dIrr galaxies that may have more fluctuating SFHs.

Figure 14 shows how well the GALEX FUV passband can recover the SFR averaged over the past 0.1 Gyr from our mock SFHs. The formula for transforming FUV to SFR$_{\tt FUV}$ used here was from Hunter et al. (2010), where the Kennicutt (1998) formula was modified to account for the low metallicity of dwarf galaxies. Here we further divided their proportionality constant by 1.59 to account for the difference between the IMF assumed here (Chabrier 2003) and the Salpeter IMF that they used. Figure 15 justifies the practice of using FUV as an unbiased estimator of the recent SFR, with an uncertainty of less than a factor of two incurred by different SFHs. One should keep in mind that here we assumed low dust extinction, which is reasonable for dwarf galaxies in general.

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The incompleteness of the SFH library used in the SED modeling could significantly affect or bias the estimation of physical parameters (e.g., Kauffmann et al. 2003). To demonstrate this, we create another SFH library by assuming that the real SFH can be approximated as an underlying component that varies exponentially with time plus a single random burst of finite length. Here we denote this SFH library as "E+B." E+B is now the most commonly used SFH library in the literature. The exponential component is described with two parameters: the SF timescale τ and the age. The burst component is described with three parameters: the age, strength, and length of the burst. The Charlot & Fall (2000) extinction recipe was adopted here. All the parameters, including the extinction, are allowed to vary uniformly within physically reasonable ranges when creating the library. We model the same SEDs as related to the mock SFHs generated above. We obtained the most probable physical parameters, including M_⋆, SFR₁, SFR_0.1, and SFR_0.03, with the same method as described in Section 4.2.

Figures 15 and 16 show the modeling results of using the E+B library. Compared to the modeling results from our standard 6-C superposition method, the estimate of the relevant parameters exhibits a larger uncertainty and a considerable bias. For instance, the stellar mass M_⋆ has a tendency to be slightly underestimated, and the recent SFR, i.e., SFR_0.1 and SFR₁, tend to be overestimated. The problem of using the E+B library in modeling can be more clearly seen in Figure 16. The E+B library tends to overestimate the SFR averaged over recent times, e.g., SFR_0.1, SFR₁. As expected for the more or less continuous nature of the SFHs in the E+B library, there is a very strong bias toward almost constant recent SF. The above bias could be ascribed to the fact that, while the SF averaged over a timescale of ∼ Gyr may be relevant to the overall evolution histories of the galaxies, the SF over shorter timescales (i.e., 0.1 Gyr) is not necessarily as important.

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