THE KENNICUTT–SCHMIDT RELATION IN EXTREMELY METAL-POOR DWARF GALAXIES

From observations of the Milky Way, Schmidt (1959) proposed that the star formation rate (SFR) and total gas volume density were related through a power law, reflecting the gas as the driver of star formation. The index was estimated to be approximately 2 from observations in the solar neighborhood. More recently (Kennicutt 1989, 1998; Kennicutt & Evans 2012), a relation was found to hold for global, disk-averaged total atomic H i plus molecular H₂ gas surface densities (${{\rm{\Sigma }}}_{{\rm{gas}}}\equiv {{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}+{{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}$) and SFR surface densities (Σ_SFR):

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{\rm{SFR}}}\propto {{\rm{\Sigma }}}_{{\rm{gas}}}^{n},\end{eqnarray} \tag{ 1 }$$

where n = 1.4 ± 0.15 in nearby spiral and starburst galaxies (hereinafter the Kennicutt–Schmidt [KS] law). Subsequent observations of radial distributions and spatially resolved disks (individual star-forming regions) in nearby galaxies have demonstrated that the KS law is valid for Σ_gas down to a few M_⊙ pc⁻² (e.g., Bigiel et al. 2008, 2010; Daddi et al. 2010; Kennicutt & Evans 2012; Wong et al. 2013). However, when the relation is investigated on kiloparsec scales below this total gas surface density threshold, the data generally show a turnover (e.g., Bigiel et al. 2008; Elmegreen & Hunter 2015). Below the threshold, the relation between the Σ_SFR and Σ_gas steepens, with an index n ≃ 2–3. By plotting Σ_SFR as a function of ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ alone, it is found that a correlation persists down to the lowest measurable H₂ column densities, resulting in a "universal" molecular gas depletion timescale (${\tau }_{{{\rm{H}}}_{2}}\equiv {{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/{{\rm{\Sigma }}}_{{\rm{SFR}}}\quad \simeq$ 2.35 Gyr; e.g., Bigiel et al. 2008, 2010). Notwithstanding, although Σ_SFR varies by approximately three to four orders of magnitude, Σ_{H i} varies by approximately one order of magnitude, showing a saturation limit at Σ_{H i} ≃ 9 M_⊙ pc⁻², roughly at the position of the turnover seen in the data (e.g., Wong et al. 2013). It was proposed that this saturation limit signals a drop in the molecular gas fraction; below the threshold, the regime is H i dominated. In this regime, although the H i dominates the H₂ molecules in column density, the H₂ gas depletion timescale remains ${\tau }_{{{\rm{H}}}_{2}}\quad \simeq$ 2 Gyr (e.g., Schruba et al. 2011). Above the threshold, the molecular gas is shielded from photodissociation, allowing an efficient phase transition from atomic to molecular gas. The saturation limit was demonstrated to depend on the metallicity (Krumholz et al. 2008, 2009a, 2009b). Recently, Elmegreen & Hunter (2015) have modeled the relation between Σ_SFR and Σ_{H i} in the H i-dominated regime as a combination of three-dimensional gaseous gravitational processes and molecular hydrogen formation, with a 1% efficiency per unit free-fall time. Elmegreen (2015) also describes a simple gas collapse model, at the local dynamical speed, where, in the H i-dominated regime, ${{\rm{\Sigma }}}_{{\rm{SFR}}}\propto {{\rm{\Sigma }}}_{{\rm{gas}}}^{2}$. The Krumholz (2013; hereinafter KMT +) analytical model computes the H₂ fraction and SFR as a function of the metallicity, gas surface density, and the density of the stellar disk. Although this model assumes a, perhaps, unrealistic constant stellar density as the gas surface density varies, it reproduces the KS law above the threshold, the metallicity-dependent turnover, and the steep correlation between Σ_SFR and Σ_gas in the H i-dominated regime, with a total gas depletion timescale ceiling of τ_gas ≡ Σ_gas/Σ_SFR ≃ 100 Gyr.

This H i-dominated, low surface density region of the KS plot (i.e., Σ_SFR versus Σ_gas), below where the turnover occurs, is generally populated by H i-dominated outer regions of galaxies, low surface brightness (LSB) galaxies, and low-metallicity dwarf galaxies (e.g., Wyder et al. 2009; Fumagalli et al. 2010; Bolatto et al. 2011; Cormier et al. 2014; Shi et al. 2014; Elmegreen & Hunter 2015). At extremely low metallicities, molecule formation is affected (e.g., Glover & Clark 2012b). CO is no longer an efficient tracer of the H₂ gas, owing to possible dust depletion, a harder ionizing radiation field, and slower chemical reaction rates in a turbulent medium (e.g., Krumholz et al. 2008, 2009a, 2009b; Wolfire et al. 2010; Shetty et al. 2011; Krumholz 2013). Consequently, there is a large uncertainty associated with the CO-to-H₂ conversion factor, which has been suggested to depend on metallicity, and which can vary by over an order of magnitude (e.g., Leroy et al. 2011; Schruba et al. 2012; Bolatto et al. 2013; Elmegreen et al. 2013; Cormier et al. 2014; Amorín et al. 2015; Hunt et al. 2015; Shi et al. 2015). Indeed, efforts to detect CO (as a proxy for the H₂ gas) in low-metallicity dwarf galaxies have been largely unsuccessful; CO has only ever been detected in dwarfs with Z ≳ 0.1 Z_⊙ (e.g., Leroy et al. 2005; Bigiel et al. 2008; Schruba et al. 2012; Elmegreen et al. 2013; Cormier et al. 2014; Hunt et al. 2015; Shi et al. 2015). Dust, suggested as a better tracer of H₂ gas at low metallicities, also suffers from large uncertainties in the determination of the gas-to-dust ratio (e.g., Rémy-Ruyer et al. 2013, 2014; Cormier et al. 2014; Hunt et al. 2015; Shi et al. 2015). Moreover, dust-based determinations of the H₂ mass in some low-metallicity sources may differ by one to two orders of magnitude from that estimated from the CO emission (e.g., Cormier et al. 2014; Shi et al. 2014, 2015). In addition, observations of some of the lowest-metallicity sources (e.g., Madden et al. 2012; Rémy-Ruyer et al. 2013), and, in particular, recent observations of I Zw 18 (Fisher et al. 2014; Hunt et al. 2014), suggest that dust may be depleted in some extremely low metallicity galaxies. In such low-metallicity mediums, the [C ii] line becomes a potential effective coolant of the gas (e.g., Wolfire et al. 2010; Cormier et al. 2015). Observations of the [C ii] line in some dwarf galaxies show line fluxes in large excess relative to the CO line, which has been interpreted as the presence of large amounts of "dark" H₂ gas (e.g., Madden 2000; Leroy et al. 2005; Cormier et al. 2014, 2015). Indeed, such large molecular masses would imply that these galaxies are extremely inefficient at forming stars (e.g., Shi et al. 2014), or that a part of the gas is not involved in the star formation process (e.g., Cormier et al. 2014). It has also been suggested that, in extremely low metallicity galaxies, star formation will occur in collapsing cold atomic clouds rather than in molecular clouds, until the density reaches a high enough value that molecules begin to form in the cloud core, i.e., the star formation will proceed before the bulk of the atomic gas is converted into molecular gas (Glover & Clark 2012a; Krumholz 2012).

The main aim of this paper is to adequately sample the low surface density region of the KS diagram, using information gathered for a local extremely metal-poor dwarf galaxy data set (Morales-Luis et al. 2011; Sánchez Almeida et al. 2016). These data, in combination with auxiliary literature data for dwarf galaxies and LSB galaxies, will allow us to test empirical and theoretical models of metallicity-dependent star formation and empirical relations for molecular gas mass prediction.

The paper is organized as follows. Section 2 contains the detailed description of the extremely low metallicity dwarf galaxy and auxiliary data used in the analysis. This section contains the references and procedures used to compile the heterogeneous data sets used in the present study and has been laid out to allow the reader to reproduce the results; it can be skipped without interrupting the thread of the argumentation in the other sections. Section 3 includes a qualitative comparative analysis of the data with respect to empirical and theoretical star formation predictions. Section 4 contains a discussion on empirical molecular mass scaling relations and predicted molecular mass content for the extremely metal-poor dwarf galaxy data set. Sections 5 and 6, respectively, include a discussion of the results and conclusions. The Appendix contains supplementary information about some of the auxiliary and XMP data.

Throughout this paper, the cosmological parameters Ω_m = 0.27, Ω_Λ = 0.73, and H₀ = 73 km s⁻¹ Mpc⁻¹ have been adopted.

2.1. XMP Data

Using the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7; Abazajian et al. 2009) and data from literature, Morales-Luis et al. (2011) assembled a comprehensive 140-source sample of local extremely metal-poor dwarf galaxies (XMPs). The galaxies were chosen so that they presented metallicities below 1/10 solar; explicitly, 12 + log(O/H) ≲ 7.65 (see Morales-Luis et al. 2011 for details). Our present sample consists of 23 XMPs taken from this original XMP sample (hereinafter original XMP sample), which have both Hα-derived SFR measurements and published H i data. In addition, the original Morales-Luis et al. (2011) XMP sample has been recently updated with 165 new XMPs (hereinafter new XMP sample), also from the SDSS DR7 (Sánchez Almeida et al. 2016). The metallicity of the latter sample was estimated using the HII-CHI-mistry code by Pérez-Montero (2014), which provides metallicities consistent with the direct method. From this updated XMP sample, 20 new XMPs have been included, which have both Hα-derived SFR measurements and published H i data.

Skillman et al. (2013) and James et al. (2015) postulated the existence of two types of extremely metal-deficient galaxies: "starburst," whose star formation may be triggered by external causes, such as the infall of low-metallicity gas, and "quiescent," which may be metal-poor owing to their small masses and the existence of the mass–metallicity relationship (e.g., Lee et al. 2006). The XMPs in the present analysis correspond, mainly, to the Skillman et al. (2013) and James et al. (2015) "starburst" category, i.e., they are bright for their metallicity (Filho et al. 2013).

The SFRs that are used in the present analysis were obtained from the Max Planck–John Hopkins University (MPA-JHU) DR7 release of spectrum measurements.⁹ SFRs are inferred from extinction-corrected Hα emission-line luminosities and extrapolated to the whole galaxy, according to the procedure outlined in Brinchmann et al. (2004). The MPA-JHU SFR measurements are in good agreement with those resulting from dust-corrected Galaxy Evolution Explorer (GALEX) measurements (Salim et al. 2007).

The XMP optical radii (r_opt) used for the surface density estimations were obtained from the SDSS r-band images, or, when not available, the Digitized Sky Survey (DSS) II R-band images, measured up to the 25 mag arcsec⁻² isophote. Because XMPs generally show very diffuse, irregular optical morphology, the optical radii have not been corrected for projection effects. The optical radii were found to vary by approximately an order of magnitude across the XMP sample: r_opt ≃  0.2–5 kpc. Distances were corrected for Virgocentric infall and were obtained from the NASA/IPAC Extragalactic Database¹⁰ (NED).

Filho et al. (2013) presented new single-dish Effelsberg H i measurements and low-resolution/single-dish and/or interferometric H i data from literature for the original XMP sample. The H i data for the new XMP sample are contained in Table 1. The total H i masses for the new XMP sample were calculated utilizing the same procedure as in Filho et al. (2013). When multiple H i entries are present, the most appropriate total H i measurement was chosen for the subsequent data analysis. It is found that the total H i masses of the XMPs are at least an order of magnitude larger than the stellar masses, while the H i radii are at least three times the optical radii (see Filho et al. 2013 and references therein). H i masses and H i surface densities, within the optical radius, have been derived in a very simplified manner, as follows. Interferometric studies of star-forming dwarf galaxies have found that the H i gas generally follows an exponential profile, Σ_{H i}(x) ∝ exp (−x/h), where h is the scale length and x is the galactocentric radius (e.g., van Zee et al. 1998, 2001; Roychowdhury et al. 2011, 2014; Lelli et al. 2014a, 2014b; Elmegreen & Hunter 2015). In addition, it has been demonstrated that the H i gas is more centrally concentrated in blue compact dwarf galaxies (BCDs; which are also more compact in the optical: r_opt < 3 kpc) than in dwarf irregular galaxies (dIrrs; which are also more extended in the optical: r_opt ≳ 3 kpc). According to this size criterion, only four of the sample sources are XMP "dIrrs." Assuming that the H i-to-optical radius is constant for each "BCD"/"dIrr" subgroup, h = 3/2 r_opt for XMP "BCDs" and h = 2 r_opt for XMP "dIrrs" have been adopted (Figure 11 in van Zee et al. 1998, and Figure 10 in van Zee et al. 2001). H i gas surface densities and H i masses, within the optical radius, have been derived from the total H i mass assuming the exponential profile for the H i gas and the scale length values as defined above. H₂ masses, within the optical radius, have been derived from the metallicity, SFR, and/or H i mass, using the empirical mass scaling relations given in Section 4, and subsequently used to compute the H₂ surface gas densities, within the optical radius. Total (H₂+H i) gas surface densities, within the optical radius, have also been derived. These values do not include helium; including helium would increase Σ_gas by ≃0.134 in log and would not significantly alter the results (Section 5). The errors in Σ_{H i} and Σ_SFR were estimated considering the errors in H i mass (see Filho et al. 2013 for details), a 20% error in the determination of the galaxy optical radius, and a 1σ error in the SFR from the MPA-JHU data set. When a quoted error in the H i integrated flux density (S_{H i}) is not available (Table 1), a 5% error has been assumed, comparable to the errors quoted for the other XMPs.

Table 1.  Summary of the H i Data for the 20 New XMPs in the Local Universe, with Hα-derived SFR Measurements and Published H i Data

Source	R.A. (J2000)	decl. (J2000)	vhel	vpeak	w50	SH i	log MH i	D	References
	(h m s)	(° ' '')	(km s−1)	(km s−1)	(km s−1)	(Jy km s−1)	(M⊙)	(Mpc)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
J093840.27+080809.8	09 38 40.29	08 08 9.86	3366	...	103 ± 10	1.70 ± 0.06	8.94	46.7	1
...	09 38 40.29	08 08 9.86	3366	...	103 ± 10	...	8.60	46.7	2
SBS 0943+543	09 47 5.78	54 05 40.40	1594 ± 20	3 ± 0.8	46	0.21	7.55	26.7	3
UGC 05347	09 57 16.52	04 31 37.22	2153	...	211 ± 3	6.24 ± 0.09	9.15	30.8	1
J100642.44+511623.9	10 06 42.45	51 16 23.95	4901	...	119	0.34	8.59	69.9	4
J102344.95+270639.8	10 23 44.96	27 06 39.85	540	...	39 ± 4	0.94 ± 0.04	6.89	5.92	1
LSBC D640-13	10 56 13.92	12 00 40.66	989	...	30 ± 4	1.84 ± 0.04	7.89	13.3	1
...	10 56 13.92	12 00 40.66	990	...	42	1.22	7.71	13.3	5
SBS 1119+586	11 22 37.77	58 19 42.68	1583 ± 6	...	52 ± 13	1.32 ± 0.36	8.34	26.6	6
SBS 1137+589	11 40 32.09	58 38 32.03	1856 ± 14	4 ± 1.0	300	0.73	8.26	32.5	3
...	11 40 32.09	58 38 32.03	2032	...	110 ± 17	0.85 ± 0.39	8.33	32.5	6
J121046.30+235454.9	12 10 46.31	23 54 54.94	2541	...	129 ± 9	1.58 ± 0.07	8.74	38.3	1
SBS 1208+531	12 11 0.68	52 49 56.91	973 ± 6	11 ± 3	...	1.3	7.94	16.8	7
J121413.78+085429.7	12 14 13.80	08 54 29.73	1933	...	95 ± 3	2.07 ± 0.08	7.92	13.1	1
...	12 14 13.80	08 54 29.73	1933	...	95 ± 3	1.89 ± 0.07	7.88	13.1	8
J122025.78+331431.7	12 20 25.79	33 14 31.75	...	12	33.9	0.72	7.82	19.8	9
J122712.80+073820.4	12 27 12.81	07 38 20.49	1170	...	44 ± 3	1.28 ± 0.05	7.71	13.1	1
MCG +07-26-024	12 33 52.74	39 37 33.36	...	113	32.5	5.94	8.11	9.60	9
KUG 1243+265	12 46 10.75	26 15 0.82	1878	...	54 ± 6	1.11 ± 0.06	8.39	30.6	1
J140802.53+251507.2	14 08 2.53	25 15 7.30	3104	...	23 ± 11	0.27 ± 0.04	8.14	46.4	1
J141323.49+130443.9	14 13 23.50	13 04 43.99	4982	...	113 ± 6	0.84 ± 0.06	9.01	71.7	1
J141710.66+104412.6	14 17 10.66	10 44 12.64	9737	...	238 ± 11	0.94 ± 0.10	9.61	136	1
J143238.20+121909.0	14 32 38.21	12 19 9.02	6703	...	164 ± 44	0.70 ± 0.08	9.17	95	1
J144038.18+035616.9	14 40 38.19	03 56 16.98	8549	...	128 ± 32	0.27 ± 0.06	8.95	118.6	1

Note. Column (1): Source name. Column (2): R.A. (J2000). Column (3): decl. (J2000). Column (4): Heliocentric velocity (and error). Column (5): H i velocity profile peak (and error). Column (6): H i line width at 50% of the peak flux density (and error). Column (7): H i integrated flux density (and error). Column (8): Logarithm of the derived (total) H i mass. Column (9): Virgocentric infall-corrected Hubble flow distance from the NED. Column (10): Reference for the H i data.

References and instrument/survey. Code in Column (10), and instrument/survey: (1) Haynes et al. (2011)—Arecibo Legacy Fast ALFA Survey (ALFALFA). (2) Stierwalt et al. (2009)—ALFALFA. (3) Huchtmeier et al. (2005)—Effelsberg Radio Telescope. (4) Kreckel et al. (2011)—Westerbork Synthesis Radio Telescope (WSRT). (5) Eder & Schombert (2000)—Arecibo Radio Telescope. (6) Thuan et al. (1999)—Very Large Array (VLA). (7) Huchtmeier et al. (2007)—Effelsberg Radio Telescope. (8) Kent et al. (2008)—ALFALFA. (9) Kovač et al. (2009)—WSRT.

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Table 2 provides the relevant auxiliary and SFR, H i, and total gas surface density data for the sample of 43 XMPs, 23 from the original sample and 20 from the new sample.

Table 2.  Summary of the Auxiliary and SFR, H i, and Total Gas Surface Density Data for the 43 XMPs in the Local Universe, with Hα-derived SFR Measurements and Published H i Data

Source	ropt	log SFR	log ΣSFR	log ΣH i	log ${M}_{{{\rm{H}}}_{2}}^{{\rm{Shi}}}$	log ${{\rm{\Sigma }}}_{{\rm{gas}}}^{{\rm{Shi}}}$	log ${M}_{{{\rm{H}}}_{2}}^{{\rm{Amorin}}}$	log ${{\rm{\Sigma }}}_{{\rm{gas}}}^{{\rm{Amorin}}}$	log ${M}_{{{\rm{H}}}_{2}}^{{\rm{universal}}}$	log ${{\rm{\Sigma }}}_{{\rm{gas}}}^{{\rm{universal}}}$	log ${M}_{{{\rm{H}}}_{2}}^{{\rm{short}}}$	log ${{\rm{\Sigma }}}_{{\rm{gas}}}^{{\rm{short}}}$
	(kpc)	(M⊙ yr−1)	(M⊙ yr−1 kpc−2)	(M⊙ pc−2)	(M⊙)	(M⊙ pc−2)	(M⊙)	(M⊙ pc−2)	(M⊙)	(M⊙ pc−2)	(M⊙)	(M⊙ pc−2)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
Original XMP Sample
J0119-0935	1.31	−2.14	−2.61	1.03	8.54	1.88	9.20	2.49	7.49	1.22	7.13	1.12
HS 0122+0743	6.54	−1.17	−3.01	0.68	9.59	1.53	10.02	1.92	8.48	0.85	8.11	0.76
J0126-0038	1.86	−2.93	−3.71	1.22	9.03	2.06	9.49	2.48	6.68	1.23	6.32	1.22
J0133+1342	0.88	−1.39	−1.49	0.73	7.89	1.58	8.05	1.71	8.25	1.90	7.89	1.58
J0204-1009	1.93	−1.54	−2.33	1.73	9.57	2.57	10.36	3.31	8.10	1.81	7.74	1.76
J0301-0052	0.67	−1.99	−1.87	2.10	9.02	2.94	9.46	3.34	7.64	2.19	7.28	2.14
HS 0822+03542	0.39	−1.99	−1.41	0.88	7.34	1.72	7.70	2.05	7.64	1.99	7.28	1.67
I Zw 18	0.62	−1.74	−1.55	1.66	8.52	2.50	9.39	3.31	7.90	2.04	7.53	1.87
J0940+2935	0.76	−2.09	−2.09	0.76	7.80	1.61	7.80	1.61	7.54	1.39	7.18	1.15
KUG 1013+381	1.38	−1.46	−1.96	1.00	8.55	1.85	8.81	2.07	8.18	1.55	7.82	1.32
UGCA 211	1.89	−2.20	−2.99	0.80	8.63	1.65	8.94	1.92	7.43	0.94	7.06	0.87
HS 1033+4757	0.89	−1.89	−2.02	1.54	8.71	2.38	8.90	2.54	7.74	1.75	7.38	1.64
J1105+6022	1.99	−1.53	−2.35	1.03	8.90	1.87	9.14	2.08	8.11	1.32	7.75	1.18
J1201+0211	0.45	−1.99	−1.52	1.05	7.62	1.89	7.83	2.08	7.64	1.91	7.28	1.62
SBS 1211+540	0.68	−1.94	−1.83	1.11	8.04	1.95	8.11	2.01	7.69	1.67	7.33	1.44
J1215+5223	0.43	−2.82	−2.33	0.96	7.49	1.80	7.76	2.04	6.80	1.30	6.43	1.14
VCC 0428	0.49	−1.92	−1.53	1.17	7.82	2.01	7.84	2.03	7.71	1.92	7.35	1.65
J1230+1202	0.55	−1.69	−1.40	1.26	8.02	2.11	7.95	2.04	7.95	2.04	7.58	1.76
GR 8	0.27	−3.16	−2.27	0.90	7.04	1.75	6.98	1.62	6.45	1.31	6.09	1.13
HS 1442+4250	2.36	−2.08	−3.05	0.80	8.82	1.65	9.19	1.98	7.55	0.92	7.19	0.86
J2053+0039	1.89	−1.73	−2.51	1.65	9.48	2.50	10.30	3.26	7.91	1.72	7.54	1.68
J2104-0035	0.64	−2.01	−1.86	1.67	8.56	2.52	9.62	3.51	7.62	1.90	7.26	1.78
PHL 293B	0.74	−1.52	−1.48	1.32	8.33	2.16	8.49	2.30	8.12	1.99	7.76	1.73
J093840.27+080809.8	1.30	−1.13	−1.85	1.84	9.34	2.68	10.15	3.43	8.23	2.00	7.87	1.92
SBS 0943+543	0.81	−1.59	−1.91	0.86	7.95	1.71	8.21	1.94	7.77	1.55	7.41	1.29
UGC 05347	5.61	−0.93	−2.92	0.63	9.41	1.48	9.93	1.95	8.44	0.85	8.07	0.74
J100642.44+511623.9	1.86	−0.75	−1.79	1.18	8.99	2.02	9.38	2.37	8.61	1.72	8.25	1.50
J102344.95+270639.8	0.34	−2.60	−2.15	0.96	7.29	1.81	7.60	2.08	6.76	1.40	6.40	1.21
LSBC D640-13	0.90	−1.93	−2.34	1.11	8.29	1.95	8.38	2.03	7.43	1.37	7.07	1.24
SBS 1119+586	1.29	−1.43	−2.14	1.25	8.74	2.09	8.96	2.29	7.94	1.53	7.57	1.39
SBS 1137+589	0.77	−2.68	−2.95	1.61	8.66	2.46	8.94	2.71	6.69	1.64	6.32	1.62
J121046.30+235454.9	1.76	−1.66	−2.65	1.37	9.14	2.22	9.61	2.64	7.70	1.46	7.34	1.41
SBS 1208+531	0.49	−2.20	−2.08	1.69	8.34	2.53	8.39	2.58	7.16	1.83	6.80	1.76
J121413.78+085429.7	0.65	−1.74	−1.87	1.42	8.32	2.27	8.76	2.66	7.62	1.76	7.26	1.60
J122025.78+331431.7	0.60	−2.12	−2.17	1.39	8.22	2.24	9.04	3.00	7.24	1.60	6.88	1.50
J122712.80+073820.4	0.57	−3.24	−3.25	1.32	8.11	2.17	8.34	2.37	6.12	1.35	5.76	1.33
MCG +07-26-024	1.12	−2.14	−2.73	1.14	8.51	1.99	8.79	2.24	7.23	1.26	6.86	1.20
KUG 1243+265	1.67	−1.18	−2.12	1.07	8.79	1.92	8.95	2.05	8.18	1.46	7.82	1.29
J140802.53+251507.2	1.18	−1.59	−2.23	1.12	8.54	1.97	9.05	2.43	7.78	1.43	7.41	1.28
J141323.49+130443.9	2.00	−0.94	−2.04	1.54	9.41	2.38	9.69	2.63	8.42	1.74	8.06	1.64
J141710.66+104412.6	3.30	−0.73	−2.26	1.56	9.87	2.40	10.40	2.89	8.63	1.69	8.27	1.62
J143238.20+121909.0	2.76	−0.73	−2.11	1.41	9.57	2.26	10.09	2.74	8.63	1.64	8.27	1.53
J144038.18+035616.9	3.74	−1.49	−3.14	0.79	9.21	1.63	9.61	2.00	7.87	0.89	7.51	0.84

Note. Column (1): Source name. Column (2): Optical radius from the SDSS r-band image, or, when not available, the DSS-II R-band image, and measured at 25 mag arcsec⁻². Column (3): Median estimate of the logarithm of the total SFR from the MPA-JHU DR7 release of spectrum measurements, obtained from the extinction-corrected Hα emission line luminosities and integrated over the whole galaxy, according to the procedure outlined in Brinchmann et al. (2004). Column (4): Logarithm of the SFR surface density within the optical radius. Column (5): Logarithm of the H i gas surface density within the optical radius. Column (6): Derived logarithm of the H₂ mass, within the optical radius, assuming the scaling relation of Shi et al. (2014; Section 4). Column (7): Derived logarithm of the total (H₂+H i) gas surface density, within the optical radius, assuming the scaling relation of Shi et al. (2014; Section 4). Column (8): Derived logarithm of the H₂ mass, within the optical radius, assuming the scaling relation of Amorín et al. (2015; Section 4). Column (9): Derived logarithm of the total (H₂+H i) gas surface density, within the optical radius, assuming the scaling relation of Amorín et al. (2015; Section 4). Column (10): Derived logarithm of the H₂ mass, within the optical radius, assuming the "universal" H₂ depletion timescale of 2.35 Gyr (Bigiel et al. 2008, 2010; Section 4). Column (11): Derived logarithm of the total (H₂+H i) gas surface density, within the optical radius, assuming the "universal" H₂ depletion timescale of 2.35 Gyr (Bigiel et al. 2008, 2010; Section 4). Column (12): Derived logarithm of the H₂ mass, within the optical radius, assuming the "short" H₂ depletion timescale of 1 Gyr (Section 4). Column (13): Derived logarithm of the total (H₂+H i) gas surface density, within the optical radius, assuming the "short" H₂ depletion timescale of 1 Gyr (Section 4).

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Several XMPs with other literature data have also been included in the analysis. Sextans A and ESO 146-G14 are XMPs belonging to the original XMP sample. However, because MPA-JHU SFR estimates are not available, their integrated H i gas and SFR surface densities have been derived as follows: an integrated SFR surface density value from Dohm-Palmer et al. (2002; their Figure 8; Σ_SFR ≃ 0.015 M_⊙ yr⁻¹ kpc⁻²) has been adopted for Sextans A, and an average (integrated over the galaxy and over the galaxy lifetime) SFR value from Bergvall & Rönnback (1995; SFR ≃ 0.3 M_⊙ yr⁻¹) has been adopted for ESO 146-G14. Atomic gas masses are from Filho et al. (2013), and optical source sizes are from the present work. Also included are resolved measurements for individual star-forming regions in Sextans A and ESO 146-G14 (Shi et al. 2014), where ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ (Spitzer and Herschel data), Σ_{H i} (Very Large Array [VLA] and Australia Telescope Compact Array [ATCA] data), and Σ_SFR (Herschel, Spitzer, and GALEX data) are sampled within r_dust ≃ 0.2–1 kpc (the dust aperture).

The Dwarf Galaxy Survey (DGS; Madden et al. 2013) contains 18 sources in common with the original XMP sample, only three of which possess CO observations that yield H₂ mass upper limits (Cormier et al. 2014): SBS 0335-052, I Zw 18, and VII Zw 403 (UGC 6456). The integrated measurements for these XMPs are included in the subsequent analysis, where ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ (Spitzer, Atacama Pathfinder Experiment [APEX], Mopra 22 m, and Institut de Radioastronomie Millimétrique [IRAM] 30 m data), Σ_{H i} (see Madden et al. 2013 for details), and Σ_SFR (Spitzer data) are sampled within r_dust ≃ 50''–90'' or r_CO ≃ 10''–30'' (the dust and CO aperture, respectively).

We caution that the use of standard calibrations to infer the SFR from the integrated light of a stellar population, generally applied at higher SFRs, may lead to systematic errors when applied at low SFRs (SFR ≲ 10⁻³ M_⊙ yr⁻¹), as is the case in some of the XMP data. The use of the Hα emission to infer the SFR can be particularly problematic, owing to the stochastic nature of the star formation and the short timescales over which the integration is performed (da Silva et al. 2014). Note that this may also affect some of the auxiliary data (Section 2.2).

In the following figures (Figures 1–5), it is apparent that there are a few XMPs with low specific SFRs (sSFR ≡ SFR/M_*; sSFR ≲ 10⁻¹⁰ yr⁻¹), although they appear to be a continuation of other general XMP properties. These XMPs also generally possess the lowest SFRs (see above) and SFR surface densities. It is to be noted that, like the higher sSFR XMPs, these XMPs are "starbursts" (Skillman et al. 2013; James et al. 2015), according to the metallicity–luminosity relation (Filho et al. 2013). Closer inspection of the SDSS and MPA-JHU data reveals that these low sSFR XMPs are generally multiknot star-forming sources, where the automated estimation of the (MPA-JHU) integrated SFR has likely been unsuccessful; the real integrated SFR is presumably higher. These (few) low sSFR XMPs are included in the following plots for completeness, but no conclusions are drawn regarding their SFR properties.

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2.2. Auxiliary Data

Figure 1(a) contains the SFR surface density as a function of the total (filled symbols) or H i (open symbols) gas surface density for the auxiliary data (Section 2.2), plus the resolved observations for the XMPs Sextans A (red open and filled squares; Shi et al. 2014) and ESO 146-G14 (red open and filled circles; Shi et al. 2014), and the integrated measurements for the XMPs also contained in the DGS: SBS 0335-052, I Zw 18, and VII Zw 403 (encircled brown open and filled triangles; Cormier et al. 2014). Figure 1(b) contains the SFR surface density as a function of the H i gas surface density, integrated over the optical radii, for the 43 sample XMP galaxies (blue open circles; Section 2.1), plus the integrated values for the XMPs Sextans A (red open square; Section 2.1) and ESO 146-G14 (red open circle; Section 2.1), and for the metal-poor dIrr WLM (dark green open triangle; Section 2.2). In both figures, the KS law is plotted (magenta solid line; Kennicutt & Evans 2012), as well as the observed H i relation for the Faint Irregular Galaxies GMRT Survey (FIGGS; yellow solid line; Roychowdhury et al. 2014), and the observed (H₂+H i) relation for starburst galaxies (purple solid line; Daddi et al. 2010).

Figures 1(a) and (b) also include an analytical KMT+ model (black solid line; Krumholz 2013), corresponding to a model with a clumping factor of f_c = 5, a volume density of stars and dark matter within the gas disk of ρ_sd = 0.1, and a metallicity of Z = 0.1 Z_⊙, which is representative of the conditions expected in XMP galaxies. Lines of constant (total or H i) gas depletion timescale, $\tau \equiv {\tau }_{{\rm{gas}}}\equiv {\tau }_{{{\rm{H}}}_{2}+{\rm{HI}}}$ or τ ≡ τ_{H i}, respectively, are also included (gray solid and dotted lines).

3.1. DGS Data

3.2. XMP Data

For the XMPs, the H i star formation efficiency,

$$\begin{eqnarray}&&{\mathrm{SFE}}_{\mathrm{HI}}\equiv {{\rm{\Sigma }}}_{{\rm{SFR}}}/{{\rm{\Sigma }}}_{{\rm{HI}}},\end{eqnarray} \tag{ 2 }$$

or equivalently, its inverse, the H I gas depletion timescale,

$$\begin{eqnarray}&&{\tau }_{{\rm{HI}}}\equiv {{\rm{\Sigma }}}_{{\rm{HI}}}/{{\rm{\Sigma }}}_{{\rm{SFR}}},\end{eqnarray} \tag{ 3 }$$

shows a broad range of values: 10⁻¹¹–10⁻⁸ yr⁻¹ and 0.1–100 Gyr, respectively (Figure 1(b)). The XMP H i data (blue open circles, and red open square and circle; Figure 1(b)) are roughly consistent with the position of many of the DGS sources (brown open triangles; Figure 1(a)) and FIGGS galaxies (Roychowdhury et al. 2014) on the KS plot and extend, toward higher Σ_SFR and Σ_{H i} values, the position of the LSB galaxies (light green open triangles; Figure 1(a)). The XMP H i data are also roughly consistent with the range in H i depletion timescales observed for FIGGS galaxies at 400 pc resolution (Roychowdhury et al. 2015), with the average value for all H i-dominated regions in the H i Nearby Galaxy Survey (THINGS; Walter et al. 2008) 400 pc data set (Roychowdhury et al. 2015), and with the range observed in dIrr galaxies (Elmegreen & Hunter 2015).

Even considering the H i gas alone, XMPs are generally not consistent with the position of galaxies going through a merger process (purple solid line; Figure 1(b)).

Although there is scatter due to uncertainties in the observations, a significant fraction of the XMPs already fall below the empirical KS star formation law (magenta solid line; Figure 1(b)), implying that these XMPs are systems of low star-forming efficiency; they possess star formation surface densities too low for the values expected from the KS law for their gas masses. Nevertheless, these XMPs may still be consistent with the predictions of the Elmegreen (2015) (for log(Σ_gas/M_⊙ pc⁻²) = 1, log(Σ_SFR/M_⊙ yr⁻¹ kpc⁻²) = −2.8) and KMT+ (black solid line; Figure 1(b)), models, in the H i-dominated regime, if the H₂ masses are not large (Section 4).

In order to explore possible trends in the variation of the H i gas depletion timescales with the global properties, the XMPs have been plotted on the KS diagram (Figure 2), where the sample was divided into sets of high (blue open circles) and low (red open circles) total H i mass, stellar mass, metallicity, and optical source size. The limits between the high and low parameter range correspond, approximately, to the median values (except for the optical source size): log(M_{H i}/M_⊙) = 8, log(M_*/M_⊙) = 7, 12 + log(O/H) = 7.4, and r_opt = 3 kpc (which roughly corresponds to the "dIrr"/"BCD" size limit; Section 2.1), respectively. No strong trend is observed for the metallicity (Figure 2(c)) or size (Figure 2(d)). In terms of the total H i mass (Figure 2(a)) and stellar mass (Figure 2(b)), it is observed that the more massive XMPs (blue open circles; Figures 2(a) and (b)) tend to have the longest H i gas depletion timescales, and the least massive tend to have the shortest H i gas depletion timescales.

Figure 3 contains scatter plots of the relations between the total H i mass (blue filled circles), the H i mass within the optical radius (red filled circles), the H i mass fraction (M_{H i}/(M_{H i} + M_*)), the H i star formation efficiency, the stellar mass, the SFR, and the sSFR. All the plots show a large dispersion, although general trends may be pointed out. There is some tendency for the sSFR to be higher in the lower H i mass XMPs (Figure 3(a)). The sSFR also shows a possible tendency to increase as the H i mass fraction of the H i mass within the optical radius (red filled circles) increases (Figure 3(b)). However, the H i mass fraction of the total H i mass (blue filled circles) saturates to become of the order 1, and independent of the sSFR, for sSFR ≳ 10⁻⁹ yr⁻¹ (Figure 3(b)). It is further observed that the smaller the H i mass, the faster the XMP consumes its H i gas (Figure 3(c)). The SFR also shows a possible tendency to increase with H i mass (Figure 3(d)). However, disregarding the lower SFR points (see caveat on low SFRs in Section 2), for a fixed SFR, the mass of the H i reservoir can vary by approximately three orders of magnitude (Figure 3(d)).

Figure 4(a) shows the SFR as a function of the stellar mass and contains two observed relations extrapolated down to low stellar masses: the so-called main sequence of star formation for the Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey (CANDELS) at z ≃ 0.8 (broken power law, fit to lower stellar masses; dark green solid line; Whitaker et al. 2014) and for the SDSS blue galaxies at z ≃ 0 (magenta solid line; Elbaz et al. 2007). When compared to the extrapolated predictions, a significant fraction of the XMPs show elevated SFRs relative to their stellar masses at z ≃ 0 (magenta solid line; Figure 4(a)); they appear to possess sSFRs similar to galaxies at z ≃ 1 (dark green solid line; Figure 4(a)). The fact that XMPs possess sSFRs similar to higher-redshift galaxies is further evidenced in Figure 4(b), where the H i star formation efficiency is plotted against the sSFR. The figure includes the sSFRs expected from the linear extrapolation to low stellar masses of CANDELS galaxies at 0.5 < z < 2.5 (magenta, cyan, purple, and light green solid lines; Whitaker et al. 2014) and SDSS galaxies at z ≃ 0 (orange solid line; Brinchmann et al. 2004). For reference, Figure 4(b) also includes an average H i gas depletion timescale for GALEX-Arecibo-SDSS Survey (GASS) galaxies at z ≃ 0 (τ_{H i} = 3.4 Gyr; dark green solid line; Schiminovich et al. 2010), the total gas depletion timescale corresponding to the KS prediction at z ≃ 0 (τ_gas = 3.2 Gyr; dark green dotted line; Kennicutt & Evans 2012), and the total gas depletion timescale corresponding to the observed KS law at z ≃ 1.2 (τ_gas = 1.9 Gyr; dark green dashed line; Freundlich et al. 2013). It is to be noted that XMPs tend to have sSFRs in excess of the typical values observed in the local universe (orange solid line; Figure 4(b)). It is also observed that, generally, the higher the sSFR (and the higher the SFR surface densities), the faster the XMP depletes its H i gas (Figure 4(b)).

Figure 4(b) further demonstrates that the H i gas depletion timescales, i.e., τ_{H i}, can be almost an order of magnitude longer than the timescale to form the observed stellar mass at the current SFR, i.e., 1/sSFR. However, the H i gas depletion timescales inferred from the H i mass are only conservative upper limits if galaxy winds are important. A significant part of the (H i) gas may be expelled into the circumgalactic medium, so that the "effective" H i gas depletion timescales (${\tau }_{{\rm{HI}}}^{{\rm{eff}}}$) may be much shorter:

$$\begin{eqnarray}&&{\tau }_{{\rm{H}}\quad {\rm{I}}}^{{\rm{eff}}}={\tau }_{{\rm{H}}{\rm{I}}}/(1+W-R),\end{eqnarray} \tag{ 4 }$$

where R is the fraction of the H i gas returned to the interstellar medium by stellar winds and supernova explosions and W is the mass-loading factor (e.g., Sánchez Almeida et al. 2014). W is the ratio between the SFR and the mass-loss rate through winds and can be very large in dwarf galaxies; for example, Dayal et al. (2013) use W in excess of 10 to reproduce the mass–metallicity–SFR relation. For typical values of R = 0.2 and W = 5, the "effective" H i gas depletion timescales are shortened by a factor of ~6, increasing the "effective" H i star formation efficiencies to become of the same order as the sSFRs, i.e., ${\mathrm{SFE}}_{{\rm{HI}}}^{{\rm{eff}}}\quad \equiv$ 1/${\tau }_{{\rm{HI}}}^{{\rm{eff}}}\simeq$ sSFR. The one-to-one relation between the ${\mathrm{SFE}}_{{\rm{HI}}}^{{\rm{eff}}}$ and sSFR is shown in Figure 5 (gray solid line), which is similar to Figure 4(b), but where the sSFR has been compared with the "effective" H i star formation efficiency (red filled circles).

The above results highlight the fact that (extreme) low metallicity alone cannot account for the plethora of XMP dust and gas properties and, therefore, star-forming efficiency (e.g., Roychowdhury et al. 2015); star formation triggering, timescales, gas accretion, mass ratios, high gas mass and density, and mass loading are likely playing significant and interdependent roles (e.g., Hunt et al. 2014; Rubio et al. 2015).

Because there are no available H₂ measurements for the XMPs, with the exception of upper limits to three sources (Cormier et al. 2014), molecular gas masses for the 43 sample XMP galaxies have been derived (Table 2) using published empirical scaling relations. For the scaling between metallicity, M_{H i}, SFR, and ${M}_{{{\rm{H}}}_{2}}$, various recipes, existing in literature, have been considered:

• 1.  
  ${M}_{{{\rm{H}}}_{2}}$/M_{H i} = 6 (Shi et al. 2014);
• 2.  
  log(${M}_{{{\rm{H}}}_{2}}$) = 1.2 log(M_{H i}) − 1.5 (12 + log(O/H) − 8.7) − 2.2 (Amorín et al. 2015);
• 3.  
  ${M}_{{{\rm{H}}}_{2}}$ = SFR $\times {\tau }_{{{\rm{H}}}_{2}}$, where ${\tau }_{{{\rm{H}}}_{2}}$ = 2.35 Gyr (the "universal" H₂ depletion timescale; Bigiel et al. 2008, 2010);
• 4.  
  ${M}_{{{\rm{H}}}_{2}}$ = SFR $\times {\tau }_{{{\rm{H}}}_{2}}$, where ${\tau }_{{{\rm{H}}}_{2}}$ = 1 Gyr (the "short" H₂ depletion timescale).

The different derived molecular gas masses (Table 2), as a function of the measured H i masses, are included in Figure 6(a). In Figures 6(b) and (c), the KS plot of Figure 1(b) is reproduced with the 43 sample XMP H i data points (blue open circles) and the derived total gas surface densities contained in Table 2 (crosses). Using the SFR and H i data to predict the H₂ content recovers a broad range (2–3 orders of magnitude) in derived molecular gas masses (Figure 6).

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It has been shown that gas depletion timescales shorten with mass (Figure 3(c); Leroy et al. 2005; Hunt et al. 2015); smaller, less massive dwarf galaxies tend to deplete their gas faster. Because the sample sources are generally low mass, ${\tau }_{{{\rm{H}}}_{2}}$ = 1 Gyr may be an appropriate value for the H₂ gas depletion timescale in XMPs (green crosses; Figures 6(a) and (b)). In this case, the derived H₂ masses are generally low, with ${M}_{{{\rm{H}}}_{2}}\quad \lesssim$ M_{H i} (green crosses; Figure 6(a)). The resulting Σ_gas values (green crosses; Figure 6(b)) are roughly consistent with the predictions of the KMT+ model (black solid line; Figure 6(b)).

However, the molecular gas may be far more significant, as implied by Sextans A and ESO 146-G14 (red filled squares and circles, respectively; Figure 1(a)), some DGS sources (brown filled triangles; Figure 1(a)), and dwarf galaxies (Amorín et al. 2015), and as suggested by recent [C ii] observations of metal-poor environments (e.g., Madden 2000; Cormier et al. 2014, 2015). If the molecular gas is over five times more abundant than the H i gas (Figure 6(a)), then a significant fraction of the XMPs may possess large total gas surface densities (red and orange crosses; Figures 6(b) and (c)), accompanied by extremely low star formation efficiencies (τ_gas ≳ 10 Gyr). These (total and H₂) gas depletion timescales can be compared to the average values provided for the resolved observations of Sextans A and ESO 146-G14 (${\tau }_{{{\rm{H}}}_{2}}\quad \simeq$ 60 Gyr; Shi et al. 2014), subparsec-scale values in local spirals (${\tau }_{{{\rm{H}}}_{2}}\quad \simeq$ 2 Gyr; Bigiel et al. 2008; Leroy et al. 2008), and integrated values for dwarf galaxies (${\tau }_{{{\rm{H}}}_{2}}\quad \simeq$ 0.6 Gyr and τ_gas ≃ 5 Gyr; Cormier et al. 2014; see also Amorín et al. 2015). If, indeed, such large amounts of (as yet undetected) H₂ gas exist in XMPs, associated with large reservoirs of H i gas and an extremely low-star-formation efficiency, then that would necessarily imply a mechanism that efficiently converts H i gas into H₂, while greatly suppressing the star formation in these galaxies.

In light of the available auxiliary and XMP data, the location of a source on the KS plot may be dependent on the role of the H i gas in the star formation process, and, in some cases, the position on the KS diagram may be transient.

It is known that merging/starburst galaxies do not follow the "classical" KS law (Kennicutt & Evans 2012), but a displaced star formation law; mergers/starbursts exhibit an enhancement of the SFR for their total gas (purple solid line; Figure 1(a)), which is rapidly depleted by cooling and compression to form stars (e.g., Daddi et al. 2010). Several dwarf galaxies (Cormier et al. 2014) occupy this region of the KS plot and may represent merging systems in an advanced stage of interaction (Figure 1(a)).

Galaxies and galaxy regions (e.g., centers of galaxies) that follow the KS law are generally sources in which the star formation mode is secular or long-lasting, i.e., the observed H i reservoir is involved in the star formation process and is being appropriately depleted (e.g., Kennicutt 1998; Daddi et al. 2010; Kennicutt & Evans 2012).

At the other extreme, there are sources with longer total gas depletion timescales than predicted by the KS law (Figures 1(a) and (b)), which are suggested to be either interacting sources/sources with companions or sources whose global H i reservoir is involved, in a different manner, in the star formation process (e.g., Cormier et al. 2014). In the former case, the proximity with a close companion may cause an overestimation of the H i mass (within the large beam) of the main source. Alternatively, the interaction/presence of a companion may provide the main source with fresh H i gas, but this gas has not yet begun forming stars at a high rate. In a more advanced stage of the interaction, the source may move to a region of the KS plot corresponding to larger SFR surface densities.

Although some notable XMPs possess companions (e.g., I Zw 18 and SBS 0335-052; see Filho et al. 2015 and references therein for details; see also the Appendix), the majority appear to be relatively isolated (Filho et al. 2015). Therefore, the latter scenario, that of a different participation of the H i reservoir in the star formation process, relative to main spiral galaxy disks, is the scenario favored for the XMPs (and perhaps in some other "isolated" metal-poor dwarf galaxies). In the particular case of low-metallicity conditions, efficient gas cooling and H₂ formation, owing to the lack of metals and dust, are strongly constrained. It has also been hypothesized that, in low-metallicity environments, star formation may proceed even before most of the atomic gas is converted into molecular form (e.g., Glover & Clark 2012a; Krumholz 2012). However, dust grains may be present in some XMP star-forming regions, from previous star formation episodes or from contamination of neighboring regions, partly catalyzing H₂ formation and shielding the molecular gas from the ionizing radiation. This appears to be the case for Sextans A and ESO 146-G14 (Shi et al. 2014), which show, on star-forming region scales, significant amounts of dust and, therefore, dust-determined H₂ masses, associated with a low star formation efficiency (Figure 1(a)). Large quantities of "dark" H₂ mass are also inferred from the large [C ii]-to-CO line fluxes observed, on kiloparcsec scales, in several low-metallicity dwarf galaxies (Cormier et al. 2014, 2015), where the [C ii] acts as a gas coolant (e.g., Wolfire et al. 2010; Cormier et al. 2015). In the case of the three XMPs that possess upper limit CO determinations, these provide [C ii]/CO ratios that can be several hundred thousand (I Zw 18) to less than several thousand (SBS 0335-052; Cormier et al. 2014, 2015). It is to be noted, however, that the [C ii] may also be tracing the H i gas, which is known to be significant in XMPs (Filho et al. 2013).

The existence of such large amounts of gas in XMPs would imply the suppression of star formation. Four potentially interdependent factors are proposed to have a relevant impact on the observed low star formation efficiency in XMPs, compared to the main disks of spiral galaxies:

• 1.  
  The entire H i reservoir may participate in the star formation process (Figure 3(c)), but its metal-poor nature (Filho et al. 2013) may inevitably lead to a quashing of the global SFR with respect to the KS law, such as the type of effect modeled by Krumholz (2013).
• 2.  
  Most of the H i reservoir is inert, and only a small fraction participates in the star formation process when forced by an external triggering event. This is consistent with the large range in H i mass observed for a fixed SFR (Figure 3(d)) and with the detection of metallicity drops associated with starbursts in XMPs (Sánchez Almeida et al. 2013, 2015). As is observed in many BCDs (e.g., Adamo et al. 2011), most of the star formation activity occurs in massive clumps. As such, it has been suggested that the star formation in these regions is fed by the accretion of metal-poor H i gas via cosmological cold gas accretion (e.g., Östlin et al. 2001; Ekta & Chengalur 2010; Filho et al. 2013, 2015; Sánchez Almeida et al. 2013, 2014, 2015). Cold gas accretion is predicted to occur by numerical simulations and is considered a relevant driver of star formation, even at low redshift (e.g., Birnboim & Dekel 2003; Kereš et al. 2005; Dekel & Birnboim 2006; Brooks et al. 2009). Star formation often occurs before the infalling gas is mixed with the preexisting gas, and starbursts ensue in metal-poor environments (Ceverino et al. 2016).
• 3.  
  The time lag to trigger star formation upon the arrival of fresh H i gas may also play a significant role; XMPs with extremely low star formation efficiencies (Figure 1(b)) may be sources that have just received a fresh supply of H i gas but have not had time to appropriately adjust their SFR (e.g., Cormier et al. 2014).
• 4.  
  Galaxy winds are expected to be important in dwarf galaxies (e.g., De Young & Gallagher 1990; Silich & Tenorio-Tagle 2001; Sánchez Almeida et al. 2014). For typical mass-loading factors, more than 80% of the gas may be returned back to the circumgalactic medium (Figure 5), contributing to the lowering of the star formation efficiency.

Therefore, in XMPs, the global H i mass is likely not a good tracer of the H₂ mass, nor of the SFR (as given by the KS law).

It is becoming increasingly clear that star formation triggering, timescales, gas accretion, mass ratios, high gas mass and density, and mass loading may have a significant impact on the (local and global) efficiency of the star formation process in low-metallicity environments, perhaps more so than the metallicity itself (e.g., Hunt et al. 2014; Roychowdhury et al. 2015; Rubio et al. 2015).

Whether, in XMPs, star formation is locally efficient (e.g., Sánchez Almeida et al. 2015) or not (e.g., Shi et al. 2014; Rubio et al. 2015), the net global effect is that XMPs are H i-dominated, high-specific-star-forming (sSFR ≳ 10⁻¹⁰ yr⁻¹), low-star-formation efficiency (SFE_gas ≲ 10⁻⁹ yr⁻¹) systems.

Traditional H₂ mass tracers, such as CO and dust, and, more recently, the use of the [C ii] line, appear to provide, in metal-poor environments, discrepant estimations of the H₂ mass (Section 1). This stems from the unique conditions for star formation that are available in such environments. In order to probe these unique environments, we have explored the low surface density and SFR regime of the KS diagram, comparing with empirical and theoretical star formation laws, as well as with observational data.

The observational data employed in the analysis consist of auxiliary data from the literature for dwarf and LSB galaxies (Section 2.2), plus data for a 43-source sample of XMPs (Section 2.1). For the latter data set, the observational parameters have been compiled from published H i data, SDSS or DSS-II images, and SFR data from the MPA-JHU. The relevant quantities have been derived from these parameters without performing inclination corrections (owing to the irregular optical morphology of the XMPs) and assuming a very simplified model, based on an exponential H i profile, to estimate the H i mass within the optical radius and the H i gas surface density (Section 2.1). H₂ masses (Section 4) and total gas surface densities (Section 2.1) for the 43 sample XMPs have been derived assuming several empirical molecular mass scaling relations calibrated for samples of dwarf and spiral galaxies.

The XMP H i data (Figure 1(b)) approximately follow the relation found for FIGGS galaxies (yellow solid line; Figure 1(b)) and are roughly consistent with the position of many of the DGS sources (brown open triangles; Figure 1(a)) on the KS plot. However, the H i data for many XMPs already fall close to, or below, the KS law, suggesting that the addition of an H₂ component will veer the XMPs toward extremely low (net) star formation efficiencies (Figures 6(b) and (c)). Because no direct H₂ measurements are available for the XMPs (except for three upper limits), the application of different molecular mass scaling relations, using the metallicity, H i mass, and SFR, provides a wide range of possible H₂ masses, anywhere from 0.01 to 5 times the H i mass (Figure 6(a)). These H₂ masses place the XMPs in the low star formation efficiency regime (SFE_gas ≃ 10⁻⁹–10⁻¹² yr⁻¹; Figures 6(b) and (c)). For the lower range of H₂ masses (green crosses; Figures 6(a) and (b)), the data may still be consistent with the KMT+ model (black solid line; Figure 6(b)) in the H i-dominated regime, but for elevated H₂ abundances (red and orange crosses; Figures 6(a)–(c)), this model does not provide a satisfactory interpretation. Further investigation shows that XMPs generally possess large sSFRs (sSFR ≃ 10⁻¹⁰–10⁻⁷ yr⁻¹), similar to high-redshift star-forming galaxies, but low H i star formation efficiencies (SFE_{H i} ≡ 1/τ_{H i} ≃ 10⁻¹¹–10⁻⁸ yr⁻¹; Figure 4(b)). The H i gas depletion timescale is found to be correlated with the H i mass; the larger the H i mass, the longer the H i depletion timescale (Figure 3(c)). In addition, there is a large variation of the total H i mass present on large scales (M_{H i}/M_⊙ ≃ 10⁷–10¹⁰) for a fixed SFR (Figure 3(d)). These findings suggest that the global H i content is likely not an appropriate tracer of the H₂ mass or of the SFR (as given by the KS law).

The results could be interpreted as the metal-poor H i reservoir (Filho et al. 2013) unavoidably guiding the system toward a lower global star formation efficiency, as in the model developed by Krumholz (2013). Alternatively, the large-scale HI reservoir may not be directly involved in the star formation process. In XMPs, the star formation is clearly dominated by compact star-forming regions that are fed by metal-poor H i clumps, likely accreted via cold cosmological accretion flows (e.g., Filho et al. 2013, 2015; Sánchez Almeida et al. 2013, 2014, 2015). Star formation triggering timescales (relative to the gas depletion and star formation timescales; Figure 4(b)) and galaxy winds (Figure 5), which are known to be relevant in low-mass galaxies such as XMPs, likely also make a significant contribution to the low star formation efficiency in these metal-poor environments. While on small scales, the individual star-forming regions may (e.g., Sánchez Almeida et al. 2015) or may not (e.g., Shi et al. 2014; Rubio et al. 2015) be efficient at producing stars, the net effect, on large scales, is the appearance of H i-dominated, high-sSFR (sSFR ≳ 10⁻¹⁰ yr⁻¹), inefficient star-forming (SFE_gas ≲ 10⁻⁹ yr⁻¹) systems.

This research has been supported by the Estallidos IV project (AYA2010-21887-C04-04) and Estallidos V project (AYA2013-47742-C4-2-P), funded by the Spanish Ministerio de Economia e Competitividad (MINECO).

We would like to thank D. Hunter for help regarding WLM data and M. Krumholz for a useful discussion on the KS relation in low-metallicity environments.

We thank the anonymous referee for their comments and suggestions, which have helped to improve this manuscript.

This research has made use of the SDSS DR7, the DSS-II, the NED, and the MPA-JHU DR7 release of spectrum measurements.

A.1. Dicussion on WLM

A.2. Discussion on the XMPs in the DGS Data Set

A.3. Discussion on the XMPs Sextans A and ESO 146-G14