Shear-gravity transition determines the steep velocity dispersion-size relation in molecular clouds: confronting analytical formula with observations

The velocity dispersion-size relation ($\sigma_{\rm v}\sim R^{\beta}$) is a crucial indicator of the dynamic properties of interstellar gas, where the slope is considered as $\beta \sim 0.5$. Recent observations reveal a steep velocity dispersion-size relation with the slope $\beta> 0.6$, which cannot be explained by a single mechanism with only gravity ($\beta\sim0.5$) or shear ($\beta \sim 1$). We present a two-component model $\sigma_{\rm v_{total}} = \lambda_1 \sigma_{\rm v_{g}} + \lambda_2 \sigma_{\rm v_{shear}} = A[(GM/R)^{\frac{1}{2}} + f(R/t_{\rm shear})]$ to explain the steep velocity dispersion-size relation for clouds larger than several parsecs in observations from e.g. Miville-Desch\^{e}nes et al. (2017), Zhou et al. (2022) and Sun et al. (2024). We find that, above several parsecs, the velocity dispersion of small clouds is mainly caused by self-gravity, while large clouds are primarily affected by shear, and these two regimes are linked by a gradual transition with a transition scale $\sim100$ pc -- the scale height of the Galactic molecular gas disk. The variation of cloud velocity dispersion-size relation with the Galactocentric distance results from the variation of both cloud internal density structure and Galactic shear rate. Our two-component model captures how the dynamics of the molecular gas can be affected by both internal and external factors, and we expect it to be applied to data from galaxies with different physical conditions to reveal the physics.