Quantifying the interplay between gravity and magnetic field in molecular clouds - a possible multi-scale energy equipartition in NGC6334

The interplay between gravity, turbulence and the magnetic field determines the evolution of the molecular ISM and the formation of the stars. In spite of growing interests, there remains a lack of understanding of the importance of magnetic field over multiple scales. We derive the magnetic energy spectrum -- a measure that constraints the multi-scale distribution of the magnetic energy, and compare it with the gravitational energy spectrum derived in Li \& Burkert (2016). In our formalism, the gravitational energy spectrum is purely determined by the surface density PDF, and the magnetic energy spectrum is determined by both the surface density PDF and the magnetic-field-density relation. If regions have density PDFs close to $P(\Sigma)\sim \Sigma^{-2}$ and a universal magnetic field-density relation $B\sim \rho^{1/2}$, we expect a multi-scale near equipartition between gravity and the magnetic fields. This equipartition is found to be true in NGC6334 where estimates of magnetic fields over multiple scales (from 0.1 pc to a few parsec) are available. However, the current observations are still limited in sample size. In the future, it is necessary to obtain multi-scale measurements of magnetic fields from different clouds with different surface density PDFs and apply our formalism to further study the gravity-magnetic field interplay.