THE SCALING RELATIONS AND STAR FORMATION LAWS OF MINI-STARBURST COMPLEXES

The ¹²CO 1–0 data from the CO all-sky survey is used to catalog and characterize MCCs in the Galaxy. This survey was conducted with the CfA 1.2 m telescopes in both northern and southern hemispheres. The observations are sub-Nyquist sampled with an effective angular resolution of 88. The spectral cube contains data obtained from various observations starting in 1986 (Dame et al. 1986; Bronfman et al. 1989) and ending in 2001 (Dame et al. 2001), stored in the CO survey archive.¹⁴ The combined data cube is provided in main beam antenna temperature and has a sensitivity of ∼1.5 K per 0.65 $\mathrm{km}\,{{\rm{s}}}^{-1}$, but individual surveys have sensitivity ∼0.12–1 K per 0.65 $\mathrm{km}\,{{\rm{s}}}^{-1}$ channel. We use the "whole galaxy cube" covering the entire 360° longitude range and −40° to 40° latitude range. The integrated map of the Galactic plane, where MCCs reside, is shown in Figure 1.

Zoom In Zoom Out Reset image size

To derive the SFRs of MCCs, we use the 21 cm radio continuum data from the VLA Galactic Plane Survey (VGPS, Stil et al. 2006), the Canadian Galactic Plane Survey (CGPS, Taylor et al. 2003), and the Southern Galactic Plane Survey (SGPS, Haverkorn et al. 2006). VGPS has a full width at half maximum (FWHM) beam of 1' and covers the Galactic longitude 18°–67°, CGPS has an FWHM beam of 1' and covers the Galactic longitude 63°–175°, and SGPS has an FWHM of 22 covering the Galactic longitude 253°–358°.

We complement our data with literature data to increase the dynamical ranges of all parameters. For the Larson relations (Section 5), we add cloud properties data from the following sources:

• 1.  
  cores/clumps/GMCs: Maruta et al. (2010), Onishi et al. (2002), Shimajiri et al. (2015), Heyer et al. (2009), Roman-Duval et al. (2010), Evans et al. (2014).
• 2.  
  MCCs: García et al. (2014), Murray (2011), Donovan Meyer et al. (2013), Rosolowsky (2007), Miura et al. (2012, 2014), Wei et al. (2012).
• 3.  
  Galaxies: Leroy et al. (2013), Tacconi et al. (2013), Genzel et al. (2010).

For the Schmidt–Kennicutt scaling relation (Section 6), we add SFR data from the following sources:

• 1.  
  clumps/GMCs: Heiderman et al. (2010), Lada et al. (2010), Evans et al. (2014).
• 2.  
  MCCs: García et al. (2014), Bolatto et al. (2008), Murray (2011), Donovan Meyer et al. (2013), Rosolowsky (2007), Miura et al. (2012, 2014), Wei et al. (2012).
• 3.  
  Galaxies: Leroy et al. (2013), Tacconi et al. (2013), Genzel et al. (2010).

For the galaxies sample, we calculate the velocity dispersion assuming that the entire galaxy is a dynamical system. Thus, we do not distinguish between elliptical and disk galaxies or starburst and normal galaxies. We convert the orbital time from the Leroy et al. (2013) sample or the dynamical time from Genzel et al. (2010) and Tacconi et al. (2013) samples to a rotation velocity as ${V}_{\mathrm{rot}}=\tfrac{2\pi R}{{t}_{\mathrm{dyn},\mathrm{orb}}}$. Then, the velocity dispersion is written as $\sigma =\tfrac{2{\pi }^{2}{GR}\sigma }{1.5{V}_{\mathrm{rot}}}$. The radii used here are either the optical B-band 25th magnitude isophote (Leroy et al. 2013) or the half-light radii (Genzel et al. 2010; Tacconi et al. 2013). The local galaxies in Leroy et al. (2013) have velocity dispersions smaller than their rotation velocity, but the high-z galaxies in Genzel et al. (2010) and Tacconi et al. (2013) have velocity dispersion larger than the rotation velocity. This is probably because the high-z galaxies are more turbulent than local galaxies.

We also add data from individual mini-starburst regions in a galaxy at z = 1.987 (Zanella et al. 2015), in the SDP81 galaxy at z = 3.042 (Hatsukade et al. 2015), and in Arp 220 (Scoville 2015).

Systematic errors caused by different measurements are unavoidable but we assume that these errors have little impact on our analysis. For example, errors caused by: mass determinations from different tracers such as ¹²CO, ¹³CO, or dust emission; the variation of the abundances for  CO/H₂ conversion as a function of position in the galaxy; or the exclusion of the CO-dark gas, are not taken into account. So, the mass estimates are the lower limits of true values. We are aware of these error sources, but due to our large statistical sample, and assuming an uncertainty of typically 50%, we are still able to derive robust scaling relations. Actually, these errors can be safely ignored because they typically vary around about 30%–100% of the measured values and all relations that we will discuss are presented in log–log space.

We use the CO 1–0 emission as a proxy to estimate the total gas mass content of molecular cloud structure. For this purpose, using CO 1–0 alone is subject to two major problems. First, CO may not trace all molecular hydrogen in molecular clouds and it misses the 'CO-dark' gas (Langer et al. 2014). Second, the CO 1–0 emission becomes optically thick quickly in the dense parts of the molecular clouds. Moreover, Barnes et al. (2015) proposed that the linear conversion from CO integrated intensity to H₂ gas column density might underestimate the true column density. However, for a first-order global mass estimate this is acceptable. The X factor $X=2\times {10}^{20}\,{\mathrm{cm}}^{-2}$ ${{\rm{K}}}^{-1}\,{(\mathrm{km}{{\rm{s}}}^{-1})}^{-1}$ is used to convert the CO integrated intensity to H₂ gas column density as recommended by Bolatto et al. (2013), which was established after an exhaustive investigation of all possible measurements. The mean molecular mass ${m}_{{{\rm{H}}}_{2}+\mathrm{He}}=\mu {m}_{{\rm{H}}}=2.8{m}_{{\rm{H}}}$, also accounting for helium contained in the gas, is used to convert from number to mass column density. We note that additional intrinsic uncertainties on the X factor can also contribute to the error of the estimated column density and mass (see, e.g., Shetty et al. 2011).

Eventually, we derived the following physical parameters:

• 1.  
  Velocity-integrated intensity ${W}_{{\rm{CO}}}$ (K $\mathrm{km}\,{{\rm{s}}}^{-1}$) by integrating the channel maps within the velocity range derived from the Gaussian fit of the integrated spectrum.
• 2.  
  Equivalent radius $R=\sqrt{A/\pi }$ (pc) from the surface area A (pc²) measured in the integrated map.
• 3.  
  Velocity dispersion $\sigma =\tfrac{{\rm{\Delta }}{v}_{,\mathrm{FWHM}}}{\sqrt{8\mathrm{ln}2}}$ ($\mathrm{km}\,{{\rm{s}}}^{-1}$) from the FWHM linewidth ${\rm{\Delta }}{v}_{,\mathrm{FWHM}}$ resulting from Gaussian fitting of the integrated spectra.
• 4.  
  CO luminosity ${L}_{\mathrm{CO}}={{AW}}_{{\rm{CO}}}$ $(\mathrm{K\; km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2})$.
• 5.  
  Total gas mass $M={L}_{\mathrm{CO}}{X}_{\mathrm{CO}}{m}_{{\rm{H2}}}({\text{}}{M}_{\odot })$.
• 6.  
  Gas surface density ${{\rm{\Sigma }}}_{\mathrm{gas}}=\tfrac{M}{A}$ $({\text{}}{M}_{\odot }{\mathrm{pc}}^{-2})$.
• 7.  
  Virial parameter ${\alpha }_{\mathrm{vir}}=5{\sigma }_{1{\rm{D}}}^{2}R/{GM}$ with the gravitational constant G.

Our focus is on the 44 MCCs that are more massive than ${10}^{6}\,{\text{}}{M}_{\odot }$ (Table 1). All of them lie in the Galactic plane within longitude ranging from 0° to 90° or 310° to 355° and the latitude ranging from −1° to +1°. Thus, they lie mainly in the first and fourth quadrants (see Figure 1). They coincide spatially with all massive GMCs having mass larger than ${10}^{6}\,{\text{}}{M}_{\odot }$ in other CO surveys (Heyer et al. 2009; Roman-Duval et al. 2010; García et al. 2014) or MCCs hosting massive star clusters (Murray 2011). We also cover well-known MCCs such as W43 (Nguyen Luong et al. 2011a), Cygnus X (Schneider et al. 2006), W49 (Galván-Madrid et al. 2013), and W51 (Ginsburg et al. 2015). Therefore, our MCC catalog is a robust catalog of nearby MCCs. Table 1 lists for each MCC the location (l, b, ${V}_{\mathrm{VLSR}}$), extent (A, R, σ), CO luminosity (${L}_{\mathrm{CO}}$) along with its associated mass (${M}_{\mathrm{gas}}$), gas surface density (${{\rm{\Sigma }}}_{\mathrm{gas}}$), and viral parameter (${\alpha }_{\mathrm{vir}}$). Their typical sizes and masses range from 40 pc to 100 pc and from ${10}^{6}\,{\text{}}{M}_{\odot }$ to $5\times {10}^{7}\,{\text{}}{M}_{\odot }$. One exception is the Central Molecular Zone, having a mass of $1.3\times {10}^{8}\,{\text{}}{M}_{\odot }$ over an area with equivalent radius of 406 pc.

To assess the detection completeness, we compare the number of MCCs in our catalog (44) with the total expected number of MCCs in the Milky Way from power-law mass distribution, $\tfrac{{dN}}{{dm}}\propto {m}_{\mathrm{GMC}}^{\gamma }$ with $\gamma =-1.5$ (e.g., Simon et al. 2001). Assuming that the Galactic molecular cloud mass ranges from minimum mass ${M}_{{\rm{L}}}={10}^{2}\,{\text{}}{M}_{\odot }$ to maximum mass ${M}_{{\rm{U}}}={10}^{7}\,{\text{}}{M}_{\odot }$, we calculate the number of GMCs above a certain mass m by integrating the mass distribution function over the mass range $[{M}_{{\rm{L}}}\mbox{--}m]$. This yields:

$$\begin{eqnarray}&&N(\gt m)=\displaystyle \frac{2-\gamma }{\gamma -1}\times \left(\displaystyle \frac{{\left(\tfrac{{M}_{{\rm{U}}}}{m}\right)}^{\gamma -1}-1}{1-{\left(\tfrac{{M}_{{\rm{L}}}}{{M}_{{\rm{U}}}}\right)}^{2-\gamma }}\right)\times \displaystyle \frac{{M}_{\mathrm{Tot}}}{{M}_{{\rm{U}}}}\,,\end{eqnarray} \tag{ 1 }$$

where ${M}_{\mathrm{Tot}}={10}^{9}\,{\text{}}{M}_{\odot }$ is the total gas mass in the Milky Way (Dame et al. 2001). A total number of ∼300 MCCs with masses larger than ${10}^{6}\,{\text{}}{M}_{\odot }$ is expected in the entire Galaxy, about six times the number of detected MCCs. Our detection scheme mostly focuses on the near side of the Milky Way, covering only ∼1/5 of the Galactic plane, the part in the first and fourth quadrants (see Figure 2). Therefore, the total number of 44 MCCs is complete within this Galactic region.

Zoom In Zoom Out Reset image size

MCCs scatter along the Galactic plane and closely follow the spiral arms structure (see Figures 1 and 2). 70% of the MCCs lie along the Scutum-Centaurus or Sagittarius arms, the two most prominent spiral arms in the Milky Way (Dame et al. 2001), and a few others lie along the Norma arm, Perseus arms, and interarm regions. We note that YMCs, the potential descendants of mini-starburst clouds, also lie along the spiral arms (Portegies Zwart et al. 2010) (see also Figure 2). This is in line with extragalactic observations and numerical simulations, which find that the most massive, most turbulent, and most actively star-forming regions are located in the spiral arms rather than in the interarm zones (Dobbs et al. 2006; Koda et al. 2012; Fujimoto et al. 2014). MCCs are concentrated in the midplane, and distribute within the latitude range $\pm 1^\circ$ or approximately ∼200 pc from the Galactic plane equator (see Figure 1).

To calculate the immediate past (last $5\times {10}^{6}$ year, or the timescale of one OB star generation) SFR of MCCs, we follow the same approach as in Nguyen et al. (2015). Following Mezger & Henderson (1967), we use the 21 cm continuum emission to calculate the total Lyα continuum photons emitted by young massive stars that drive H ii regions as

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{\mathrm{Ly}\alpha }}{8.9\times {10}^{46}\,{{\rm{s}}}^{-1}}=\displaystyle \frac{{S}_{\nu }}{\mathrm{Jy}}{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{0.1}{\left(\displaystyle \frac{{T}_{e}}{{10}^{4}{\rm{K}}}\right)}^{-0.45}{\left(\displaystyle \frac{d}{\mathrm{kpc}}\right)}^{2}.\end{eqnarray} \tag{ 2 }$$

where ${T}_{e}=8000$ K (Wilson et al. 2012) is the electron temperature, $\nu =1.42\,\mathrm{GHz}$ is the observing frequency, and d is the distance to the region.

Assuming that an O7 star (M > 25 ${\text{}}{M}_{\odot }$) emits on average ${N}_{\mathrm{Ly}\alpha }=5\times {10}^{48}\,{{\rm{s}}}^{-1}$ (Martins et al. 2005), we calculate the SFR of the MCCs based on SFR calibration for a full typical mass spectrum derived by Murray & Rahman (2010) as

$$\begin{eqnarray}&&\displaystyle \frac{\mathrm{SFR}}{{\text{}}{M}_{\odot }\ {\mathrm{yr}}^{-1}}=4.1\times {10}^{-54}\displaystyle \frac{{N}_{\mathrm{Ly}\alpha }}{{{\rm{s}}}^{-1}}.\end{eqnarray} \tag{ 3 }$$

The SFR is then calculated for all MCCs using the area derived from CO emission (see Table 1). The SFR densities of our MCCs range from 1 to 10 $\,{\text{}}{M}_{\odot }{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$, which are in the high range of the Gould Belt dense cores (Heiderman et al. 2010), although the sizes of MCCs are hundreds of times as large. SFR densities of MCCs are comparable with the SFR of super giant H ii regions in M33 (Miura et al. 2014). The fact that these high SFR densities fill up the missing part in the SFR–mass diagram derived by Lada et al. (2010) suggests that MCCs are good candidates to link the SFRs from the local GMC scale to the global galaxy scale (see Figure 7).

When using the radio continuum flux to estimate the SFR of MCCs, a few assumptions have to be made, such as the mass spectrum over which the total stellar mass is calculated, the maximum cut-off mass of the mass spectrum, the non-dispersal property of the gas clouds, or the independence of the mass spectrum on the total gas mass. These assumption might add up to uncertainties in our SFR measurements.

For the line width–size relation (Figure 3), we obtain the following results:

$$\begin{eqnarray}&&\mathrm{GMC}:\,\,\sigma ={10}^{-0.2}{R}^{0.4},{r}_{p}=0.9\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\mathrm{MCC}:\,\,\sigma ={10}^{-0.6}{R}^{0.7},{r}_{p}=0.6\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\mathrm{All}:\,\,\sigma ={10}^{-0.2}{R}^{0.5},{r}_{p}=0.9.\end{eqnarray} \tag{ 6 }$$

The slope 0.4 of the power law fitted to GMCs is close to $\beta \sim 0.38$ measured by Larson (1981). This is in between the slopes of the pure Kolmogorov turbulent structure ($\sigma \propto {R}^{0.33}$) and the Burger shock-generated one ($\sigma \propto {R}^{0.5}$). Simulations of cloud formation through turbulent shocks, for examples those of Bonnell et al. (2006) or Dobbs & Bonnell (2007), also produce a $\sigma \propto {R}^{0.4}$ relation. On the other hand, the slope 0.7 of the fit to the MCC population is much higher than the GMC one, even higher than the slope of Burger shock-generated fractals.

As for the Pearson coefficient, while the GMC and entire populations produce a good correlation between σ and R (${r}_{{\rm{p}}}=0.9$ and ${r}_{{\rm{p}}}=0.9$, respectively), the MCC population has a weaker correlation (${r}_{{\rm{p}}}=0.6$) and the galaxy population has no correlation (${r}_{{\rm{p}}}=-0.2$). Similarly, no relation between line-width and size is found for MCCs in nearby galaxies (Hughes et al. 2013; Rebolledo et al. 2015; Utomo et al. 2015). Together with our MCC data, we suggest that there is a break at the MCC scale in addition to an apparent universal $\sigma \mbox{--}R$ relation. However, this break may be artificial due to different measurements, such as tracers, methods, resolution, and thresholds for the mass determination used, or due to the intrinsic different properties of the samples. A more homogeneous investigation is needed to confirm this break.

The second Larson relation describes the tendency of molecular clouds to reach the virial equilibrium state. We check this hypothesis by examing the virial parameter $\sigma =5{\sigma }_{1{\rm{D}}}^{2}R/{GM}$. A cloud structure is dominated by gravitational energy if $\sigma \leqslant 1$, otherwise external pressure takes control. In Figure 4, we plot a histogram of virial parameters for different populations. The mean virial parameter of the GMCs is 1.0, of MCCs is 1.8, and of the entire sample is 1.2. Since σ of GMCs is closer to 1, the internal gravitational energy is stronger than kinetic energy, as opposed to MCCs which are dominated by kinetic energy as their σ is larger than 1. In other words, GMCs are closer to being gravitationally bound than super virial and unbound MCCs. Similar results were found for MCCs in other galaxies (Hughes et al. 2013). This picture is also seen in simulations where most of the large structures are unbound and the small structures are bound (Dobbs et al. 2011; Fujimoto et al. 2014). The large velocity dispersion may be caused by the fact that MCCs are more dynamic systems that can form a compressive environment. Susequently, this results in a higher specific SFR or star formation density as we discuss later.

The third Larson relation implies that mass and size obey a power relation ${M}_{\mathrm{gas}}\propto {R}^{\alpha }$ with $\alpha \sim 2$, or all molecular cloud structures have similar mass surface density, which we examine in Figure 5. The best fits to the GMCs, MCCs, and entire samples yield slopes of 1.9, 2.2, and 2.0, respectively:

$$\begin{eqnarray}&&\mathrm{GMC}:\,\,{M}_{\mathrm{gas}}={10}^{2.6}{R}^{1.9},{r}_{p}=1.0\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\mathrm{MCC}:\,\,{M}_{\mathrm{gas}}={10}^{2.1}{R}^{2.2},{r}_{p}=0.7\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\mathrm{All}:\,\,{M}_{\mathrm{gas}}={10}^{2.6}{R}^{2.0},{r}_{p}=1.0.\end{eqnarray} \tag{ 9 }$$

All three slopes are very close to ${M}_{\mathrm{gas}}\propto {R}^{2}$ although that of MCCs slightly deviates from 2. They are in between the value of $\sim 1.6$ for substructures within individual clouds (Lombardi et al. 2010; Kauffmann et al. 2010) and the value of ∼2.4 derived for GMCs in the Galactic plane GRS survey (Roman-Duval et al. 2010). The good correlation between mass and size is also present in the large Pearson coefficients: 1.0 for GMCs, 0.7 for MCCs, and 1.0 for the entire population.

In summary, there is an apparent universal relation between velocity dispersion and size, and mass and size of cloud structures across eight orders of magnitude in size and 13 orders of magnitude in mass. However, there is a break at the MCC scale in the $\sigma \mbox{--}R$ relation and the slopes of individual populations are slightly different. The virial parameters of GMCs and galaxies are close to 1 while those of MCCs are larger than 1, signifying the importance of the kinetic energy contribution in regulating MCC structures.

The ${{\rm{\Sigma }}}_{\mathrm{SFR}}\mbox{--}{{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}$ relation was first constructed by Kennicutt (1998) for the integrated SFR density ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ and the total gas mass density ${{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}$ of normal star-forming and starburst galaxies. Since then, it has been shown to be one of the most applicable tools to explain the universal role of gravity in forming stars. Recently, the relation has been extended to the GMC scale (Heiderman et al. 2010; Krumholz et al. 2012) in an endeavour to establish a universal star formation law that connects local to global scales. We reinvestigate this relation for our combined data set by constructing the SFR density and gas mass density relation in Figure 6 and deriving their best linear fits to the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–σ relations in the log–log space as

$$\begin{eqnarray}&&\mathrm{GMC}:\,\,{{\rm{\Sigma }}}_{\mathrm{SFR}}={10}^{-4.9}{{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}^{2.4},{r}_{p}=0.8\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\mathrm{MCC}:\,\,{{\rm{\Sigma }}}_{\mathrm{SFR}}={10}^{-3.0}{{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}^{1.3},{r}_{p}=0.5\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\mathrm{GALAXY}:\,\,{{\rm{\Sigma }}}_{\mathrm{SFR}}={10}^{-3.8}{{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}^{1.4},{r}_{p}=1.0\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\mathrm{ALL}:\,\,{{\rm{\Sigma }}}_{\mathrm{SFR}}={10}^{-3.4}{{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}^{1.5},{r}_{p}=0.7.\end{eqnarray} \tag{ 13 }$$

Zoom In Zoom Out Reset image size

The slopes of the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–${{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}$ relations are different among different populations. While GMCs have the steepest slope of 2.4, MCCs have the most shallow slope of 1.3, and galaxies have a slope that is closest to the original value of 1.4 derived by Kennicutt (1998). The steeper slope of the GMC population is consistent with that of the star-forming clump population (Heiderman et al. 2010) and of the resolved individual MCCs (Willis et al. 2015). From a global view of our data, we derive a universal Schmidt–Kennicutt scaling relation with a slope of 1.5 and large scatters in the GMC and MCC populations. The Pearson coefficients show that ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ and ${{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}$ correlate strongly (r_p = 0.9) in the galaxy population while they correlate least in the MCC population (r_p = 0.5).

Instead of comparing the surface density quantities of SFR and mass, we examine the integrated SFR and the integrated gas mass relation in Figure 7. Our literature and new data of MCCs fill the $\mathrm{SFR}\mbox{--}{M}_{\mathrm{gas}}$ plane and connect the smallest cloud scale to galaxy scale. We plot in Figure 7 the relation between SFR and ${M}_{\mathrm{gas}}$ in the log–log space and fit linear functions to obtain the following results:

$$\begin{eqnarray}&&\mathrm{GMC}:\,\,\mathrm{SFR}={10}^{-7.4}{{M}_{\mathrm{gas}}}^{0.8},{r}_{p}=0.7\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\mathrm{MCC}:\,\,\mathrm{SFR}={10}^{-6.2}{{M}_{\mathrm{gas}}}^{0.6},{r}_{p}=0.4\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\mathrm{GALAXY}:\,\,\mathrm{SFR}={10}^{-11.7}{{M}_{\mathrm{gas}}}^{1.3},{r}_{p}=0.9\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\mathrm{ALL}:\,\,\mathrm{SFR}={10}^{-7.6}{{M}_{\mathrm{gas}}}^{0.9},{r}_{p}=1.0.\end{eqnarray} \tag{ 17 }$$

Zoom In Zoom Out Reset image size

The slopes of our fits are super-linear for the galaxy population and sublinear for all other cases. The slope of the fit to the combined data is 0.87, which does not agree with the unity slopes derived by Wu et al. (2005) or Lada et al. (2010). To first order, this universal scaling relation is useful to establish a common relation between SFR and the total gas mass for all star-forming objects. However, at the individual scales, this relation has strong scatter and the linear fits to the data vary strongly. The strong correlation between SFR and ${M}_{\mathrm{gas}}$ of the entire data set can also be seen in the large Pearson coefficient (r_p = 0.95). Similar to the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–${{\rm{\Sigma }}}_{\mathrm{gas}}$ relation, MCCs have very low Pearson coefficient (r_p = 0.37), indicating the low correlation between SFR and ${M}_{\mathrm{gas}}$. For example, the total SFRs of MCCs vary almost over four orders of magnitude. They are consistent with the recent numerical simulation of star formation activity in MCCs; this simulation shows large SFR spread in the SFR–${M}_{\mathrm{gas}}$ diagram (Howard et al. 2016). Lada et al. (2012) suggested that the spread in the SFR–M diagram is the effect of the dense gas fraction which we could not address with the current data. While we expect that the slopes of the SFR–M relations are superlinear, similar to those of ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–${{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}$ relations, they are sublinear. The reason is that the gas surface density also scales with area as $A\sim {{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}^{-q}$. Therefore, if $\mathrm{SFR}\propto {M}_{\mathrm{gas}}^{p}$, we should get ${{\rm{\Sigma }}}_{\mathrm{SFR}}\propto {A}^{-1}{{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}^{p}={A}^{p-1}{{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}^{p}$. The dependence of gas surface density on area has been investigated by Burkert & Hartmann (2013), who showed that ${{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}$ scales with area as $A\sim {{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}^{-3}$ for low-mass star-forming regions and $A\sim {{\rm{\Sigma }}}_{{M}_{\mathrm{gas}}}^{-1}$ for massive cores.

Ideally, if all of the scaling relations in Sections 5.1, 5.3, and 6.2 hold together, we can deduce a relation between ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–σ from the other relations. Because $\sigma \propto {R}^{0.3-0.5}$ and $\ {M}_{\mathrm{gas}}\propto {R}^{2}$, we have the relation $\ {M}_{\mathrm{gas}}\propto {\sigma }^{4-7}$. As a consequence of the relation $\mathrm{SFR}\propto {{M}_{\mathrm{gas}}}^{(0.8-1.3)}$, we should have a relation $\mathrm{SFR}\propto {\sigma }^{(3.1-8.8)}$. But as shown in Figure 8, we obtain different slopes from fitting power laws to the individual populations and also the combined data set

$$\begin{eqnarray}&&\mathrm{MCC}:\,\,\mathrm{SFR}={10}^{-2.9}\ {\sigma }^{0.9},{r}_{p}=0.3\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\mathrm{GALAXY}:\,\,\mathrm{SFR}={10}^{-1.8}\ {\sigma }^{1.9},{r}_{p}=0.8\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\mathrm{ALL}:\,\,\mathrm{SFR}={10}^{-3.8}\ {\sigma }^{2.7},{r}_{p}=0.8.\end{eqnarray} \tag{ 20 }$$

Zoom In Zoom Out Reset image size

Empirical fits show that SFR increases with velocity dispersion but the slopes are much shallower than the theoretical prediction from other scaling laws. A global fit with a slope of 2.7 fits well to the entire data set. But fits to GMC and MCC populations are uncertain because their velocity dispersion ranges are less than an order of magnitude.

Krumholz & Burkhart (2016) created feedback-driven models and gravity-driven models to explain the SFR–$\sigma$ relations in the galaxy interstellar media and found that the former model produces a steeper relation SFR$-{\sigma }^{2}$ while the later model produces a shallower and dense-gas-mass-fraction dependence relation SFR$-{f}_{\mathrm{DG}}\sigma$. These slopes are close to the slope of our galaxy population data. However, they are not applicable to MCCs and GMCs, which seem to be fitted with steeper slopes. Similar to previous cases, MCCs have the smallest Pearson coefficient (r_p = 0.3) or their SFR and σ are not correlated well.

One application of the ${{\rm{\Sigma }}}_{\mathrm{SFR}}\mbox{--}{{\rm{\Sigma }}}_{\mathrm{gas}}$ diagram is to distinguish starburst from normal galaxies. Daddi et al. (2010), for example, argued that the two different regimes of star formation in galaxies are caused by the longer depletion time of normal galaxies than that of starburst galaxies. For the same gas surface density, starburst galaxies and mini-starburst MCCs have higher SFR density than normal star-forming galaxies and normal star-forming MCCs. Therefore, they lie at a higher location in the Schmidt–Kennicutt diagram. However, the scatter in ${{\rm{\Sigma }}}_{\mathrm{SFR}}\mbox{--}{{\rm{\Sigma }}}_{\mathrm{gas}}$ is getting larger as better observations are made, both in galactic and extragalactic scales, which makes the small deviations from the universal law noticeable. Using our combined data, we propose an alternative way to distinguish starburst from normal star-forming objects by applying the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ and σ thresholds to the Schmidt–Kennicutt diagram.

A ${{\rm{\Sigma }}}_{\mathrm{gas}}$ threshold of $\sim 100\mbox{--}120\,{\text{}}{M}_{\odot }{\mathrm{pc}}^{-2}$ was suggested as the borderline between star-forming and non star-forming clouds or between normal spiral galaxies and starburst galaxies (Heiderman et al. 2010; Lada et al. 2010). As we can already see in the Schmidt–Kennicutt diagram in Figure 6, some MCCs or galaxies, and even some GMCs have star formation at ${{\rm{\Sigma }}}_{\mathrm{gas}}\lt 100\,{\text{}}{M}_{\odot }{\mathrm{pc}}^{-2}$. However, this threshold can be safely used to divide different star formation modes: isolated versus clustered, normal versus starburst, with the latter always requiring more gas with ${{\rm{\Sigma }}}_{\mathrm{gas}}\gt 100\,-\,120\,{\text{}}{M}_{\odot }{\mathrm{pc}}^{-2}$ (Kennicutt 1998; Wu et al. 2005). There are exceptions, such as the Central Molecular Zone in the Milky Way, which has high gas mass surface density but low SFR density. Additionally, we use ${{\rm{\Sigma }}}_{\mathrm{SFR}}\sim 1\,{\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$ as the threshold between starburst and non-starburst objects. This SFR threshold was originally used to distinguish between starburst and normal-star-forming galaxies (Kennicutt 1998).

Putting these two thresholds into the Schmidt–Kennicutt diagram, as in Figure 6, the four quadrants are divided and named clockwise as the: low-density starburst quadrant, normal star-forming quadrant, inefficient-star-forming quadrant, and starburst quadrant. Structures in the starburst quadrant have high gas density and high star formation efficiency, such as starburst galaxies and ministarburst complexes. Structures in the inefficient star-formation quadrant also have high gas density, however their star formation is impeded. The Central Molecular Zone is a good example of this exception (Immer et al. 2012). The majority of structures in the normal-star formation quadrant are galaxies and they follow the Schmidt–Kennicutt fit with much less scatter than objects in the starburst quadrant. In the low-density starburst quadrant, the low gas density inhibits high SFR density higher than $1\,{\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$. Nevertheless, there are maybe a few candidates that have ${{\rm{\Sigma }}}_{\mathrm{SFR}}\gt 1\,{\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$, or they might be outliers or must be further examined.

The starburst phenomenon was first suggested by observations of the excess star formation activity in galactic nuclei (Arp & O'Connell 1975; Huchra 1977). More recently, Elbaz et al. (2011) proposed that a galaxy experiences a starburst phase if its "current SFR" is twice as high as its SFR average over time, and used this to define its "main sequence SFR." With a total gas mass of $1\times {10}^{9}\,{\text{}}{M}_{\odot }$ (Dame et al. 1993), the Milky Way becomes a starburst galaxy only if its SFR is as high as 20 times the current rate of $\sim 0.7\mbox{--}2\,{\text{}}{M}_{\odot }$ yr⁻¹ (Robitaille & Whitney 2010). The Milky Way as a whole is therefore far from experiencing a starburst phase. However, SFR is not uniformly distributed across the Milky Way but excess star formation activity exists in mini-starburst MCCs that form massive star clusters (Nguyen Luong et al. 2011a; Murray 2011).

We define the mini-starburst MCCs as objects having the following properties:

• 1.  
  Total gas mass larger than ${10}^{6}\,{\text{}}{M}_{\odot }$.
• 2.  
  Gravitationally unbound.
• 3.  
  Star formation rate density larger than 1 ${\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$ or its location is on the starburst quadrant in the Schmidt–Kennicutt diagram (see Figure 6).

From the 44 massive MCCs detected in Section 3, we obtain 21 mini-starburst MCCs. Most of them are famous and are studied extensively in the literature, for example:

• 1.  
  RCW 106 is the second brightest MCC in our survey, resides in the Scutum-Centaurus arm, and surrounds the bright giant H ii region RCW 106 hosting a rich OB cluster (Rodgers et al. 1960; Nguyen et al. 2015).
• 2.  
  W43 lies at the meeting point of the Scutum-Centaurus (or Scutum-Crux) arm and the Bar. This mini-starburst is a prototypical example of a mini-starburst MCC (Nguyen Luong et al. 2011b; Carlhoff et al. 2013).
• 3.  
  W49 lies on the Perseus arm, hosts ongoing starburst events, and forms a very massive star with mass from 100–190 ${\text{}}{M}_{\odot }$(Galván-Madrid et al. 2013).
• 4.  
  Cygnus X is one of the most massive mini-starburst MCCs and is located in the Cygnus arm (Schneider et al. 2006).
• 5.  
  W51 is near the tangent point of the Sagittarius arm, has a high dense gas fraction, and hosts massive star formation events (Ginsburg et al. 2015).

We compare the Galactic mini-starbursts with the extragalactic mini-starbursts which are resolved to a comparable scale. First, we use data from Arp 220, a relatively nearby (d ∼ 75 Mpc) ultraluminous infrared galaxy (Soifer et al. 1984), containing two nuclei that are powered by extreme starburst activity. Sakamoto et al. (2008), Wilson et al. (2014), and Scoville et al. (2015) used ALMA to resolve Arp 220 down to a spatial scale of $\sim 100\,$ pc and obtained gas mass densities of $5.4\times {10}^{4}$ and $14\times {10}^{4}\,{\text{}}{M}_{\odot }\,{\mathrm{pc}}^{-2}$ for the Eastern and Western nucleus. Taking into account the SFR of $100-200\,{\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (Scoville et al. 2015), we obtain an SFR density ${10}^{4}-{10}^{4.5}\,{\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$, the highest SFR density at the MCC scale. Second, we use data of 14 starburst MCC clumps in the SDP.81 galaxy derived from ALMA observations. SDP.81 is one of the brightest galaxies at a redshift z = 3.042 (or a luminosity distance of $\sim 25\times {10}^{3}$ Mpc) and is gravitationally lensed by a foreground galaxy at z = 0.2999. Therefore, it allows us to resolve the gas properties down the scale of ∼200 pc by ALMA (Hatsukade et al. 2015). Finally, we also include in our comparison data from the first direct measurement of an MCC clump at z = 1.987 down to the scale of ∼500 pc (Zanella et al. 2015). Although larger than our mini-starbursts, we still compare them with our data, keeping in mind that their σ and ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ can be higher if we resolve them at a smaller scale.

The SFR and SFR density of the extragalactic mini-starbursts are ten to hundred of times higher than their Galactic counterparts, so that they are located in the upper part of the starburst quadrant and the upper part of the SFR$-{M}_{\mathrm{gas}}$ diagram. Highly compressed gas in these extragalactic mini-starbursts may be the origin of their star formation activity.

We advocate that the high ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ and high velocity dispersion σ of a mini-starburst MCC is caused by dynamical processes happening during MCC evolution. These processes are supported by externally induced pressure such as shocks, galactic disk gravitational instability, or colliding flow (Vazquez-Semadeni et al. 1996; Bonnell et al. 2013). Therefore, mini-starburst MCCs are often found in highly dynamic regions such as the overlapping regions in the Antennae galaxy (Herrera et al. 2012; Fukui et al. 2014) or at the end of the Galactic Bar (Nguyen Luong et al. 2011b). Simulation of molecular cloud evolution in a galaxy also agrees with this view by showing that MCCs are concentrated mostly in the Bar or spiral arms, and have high SFR (Fujimoto et al. 2014).

In all cases, continuous gas flows agglomerate clouds, compress material, and develop active star formation sites (e.g., Koyama & Inutsuka 2000; Bergin et al. 2004), which explains why mini-starburst MCCs have high SFR (see Section 7.2). This framework is also called cloud–cloud collision, advocated as the main formation mechanism of massive stars and stellar clusters (Inoue & Fukui 2013). For example, in W43, one of the most active massive star-forming regions, both large-scale and small-scale gas flows were observed as a means of forming dense gas and massive stars (Nguyen-Luong et al. 2013; Louvet et al. 2014; Motte et al. 2014).

In addition, a superlinear relation between $\mathrm{SFR}\mbox{--}\sigma$ (Figure 8) indicating that SFR increases with turbulence or compression degrees of the gas supports the dynamical view of MCC evolution. The steep slope of the $\mathrm{SFR}\mbox{--}\sigma$ relation, especially that of MCCs, disagrees with the slope produced by gravity-driven models or the feedback-driven model (Krumholz & Burkhart 2016). Therefore, cloud compression plays a stronger role in controlling the structure and star formation activity, beyond gravity and feedback. As a consequence, the majority of MCCs are gravitational unbound and form stars efficiently as shown in Figure 9, especially mini-starburst MCCs.

Zoom In Zoom Out Reset image size