tokens: could regulate SFRs to some extent (Shi et al. 2011 , 2018 ). 4.2 Theoretical models to explain the relation between σ<sub>gas</sub> and SFR There are two dominant models for the origin of the turbulence: star formation feedback or gravitational instability of the gas (Krumholz Burkhart 2016 ). Both models predict that the velocity dispersions will correlate with their SFRs. The work of Krumholz Burkhart ( 2016 ) gives the details about these two models. In Fig. 8 , we overlaid the gravitational-instability model (left-hand panel) and star-formation-feedback model (right-hand panel). The gravitational-instability model gives SFR as following (Krumholz Burkhart 2016 ): where v c is the rotational velocity, f g is the gas fraction, ε ff = 0.01 is the SFR per free-fall time, φ P = 3 is a factor that accounts for the presence of stars in the disc, and a Coulomb logarithm-like term ln ( r 1 / r 0 ), which measures the radial extent of the star-forming disc ( r 1 = 10, r 0 = 0.1). From equation ( 3 ) we find that a gravitational-instability model has a strong dependence on the gas fraction. The star formation feedback model gives the SFR as following (Krumholz Burkhart 2016 ): where Q g is the Toomre parameters of the gas, φ ≈ 1 and are constants of order unity that parametrize various uncertainties, P * / m * = 3000 km s − 1 is the momentum per unit mass (Faucher-Giguère, Quataert Hopkins 2013 ). We found that the star formation feedback origin for the turbulence predicts that a velocity dispersion rises more steeply with the SFR, and it does not depend on the gas fraction. For the comparisons with our observations, we give the plausible range of f g and Q g following with the works of Krumholz Burkhart ( 2016 ) or Johnson et al. ( 2018 ). We also used the rotation velocity v c = 90–190 , which spans the plausible range for local and high z galaxies. From Fig. 8 (right panel), we find that a star-formation-feedback model provides a rather poor match to the observation, while the gravitational-instability model shows better agreement with the observations. Johnson et al. ( 2018 ) also test two analytic models but found that both provide an adequate description of the data, and need further observations to rule out either model. In the work of Krumholz Burkhart ( 2016 ), they also compared with observations and found that gravity is the ultimate source of ISM turbulence, at least in rapidly star-forming, high-velocity dispersion galaxies for which our test is most effective. Figure 8. The relationship between SFR and gas velocity dispersion. In the left-hand panel, lines show the predictions of the gravitational-instability model (equation 6) for f g = 0.2, 0.5, and 1.0, as indicated in the legend. Lines in the right panel show the prediction of the star formation feedback model (equation 8) for Q g = 0.5, 1.0, and 2.5. The solid lines are for a circular velocity v c = 140 km s −1 (the median rotation velocity of local and high z galaxies), and the
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