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Constructing multi-scale gravitational energy spectra from molecular cloud surface density PDF -- Interplay between turbulence and gravity

(Abridged) We derive an analytical formula which provides estimates on multiscale gravitational energy distribution using the observed surface density PDF. Our analytical formalism also enables one to convert the observed column density PDF into an estimated volume density PDF, and to obtain average radial density profile $ρ(r)$. For a region with $N_{\rm col} \sim N^{-γ_{\rm N}}$, the gravitational energy spectra is $E_{\rm p}(k)\sim k^{-4(1 - 1/γ_{\rm N})}$. We apply the formula to observations of molecular clouds, and find that a scaling index of $-2$ of the surface density PDF implies that $ρ\sim r^{-2}$ and $E_{\rm p}(k) \sim k^{-2}$. The results are valid from the cloud scale (a few parsec) to around $\sim 0.1 \;\rm pc$. Because of the resemblance the scaling index of the gravitational energy spectrum and the that of the kinetic energy power spectrum of the Burgers turbulence (where $E\sim k^{-2}$), our result indicates that gravity can act effectively against turbulence over a multitude of physical scales. This is the critical scaling index which divides molecular clouds into two categories: clouds like Orion and Ophiuchus have shallower power laws, and the amount of gravitational energy is too large for turbulence to be effective inside the cloud. Because gravity dominates, we call this type of cloud g-type clouds. On the other hand, clouds like the California molecular cloud and the Pipe nebula have steeper power laws, and turbulence can overcome gravity if it can cascade effectively from the large scale. We call this type of cloud t-type clouds. The analytical formula can be used to determine if gravity is dominating cloud evolution when the column density probability distribution function (PDF) can be reliably determined.