A Unified Magnetohydrodynamic Scaling Relation for the Multiphase Interstellar Medium

Abstract

The interplay of magnetic fields, turbulence, and gravity governs the structural evolution of the interstellar medium (ISM) and the initial conditions of star formation, yet observational gaps have long enforced a broken power-law description of the magnetic field--density relation. Here we assemble a unified dataset spanning ten orders of magnitude in density ($10^{-26}$--$10^{-16}\,\mathrm{g\,cm^{-3}}$) by combining pulsar and Zeeman observations. The unified data are consistent with a continuous magnetic-field evolution organised by the Alfv\'en Mach number $\mathcal{M}_{\rm A}=\sqrt{E_K/E_B}$. Within this interpretation, the low-density gas is magnetically dominated ($\mathcal{M}_{\rm A}<1$), whereas the high-density gas becomes kinetically dominated ($\mathcal{M}_{\rm A}>1$) as gravity increasingly contributes to the kinetic-energy budget, with magnetic tension continuing to influence the collapse geometry. Within the Gradual Transition interpretation, the empirical break density traces the vicinity of the trans-Alfv\'enic equipartition point, $\mathcal{M}_{\rm A}=1$. This Gradual Transition model makes three predictions tested here. Its exponent and background field are fixed in advance by turbulent physics and recovered by the fits ($\beta\approx0.15$--$0.21$ against a predicted $0.147$; $B_c\approx2.0\,\mu$G). A dense-gas fit, extrapolated blindly across four decades, passes through the diffuse pulsar data. And, under the adopted scale mappings, the implied magnetic-energy spectrum approaches a $k^{-5/3}$-like scaling on large scales and departs from this extrapolation on small scales, where gravitational compression amplifies the field. The broken power law can therefore be viewed as a piecewise approximation to the continuous magnetic equation of state, with its fitted transition density potentially retaining a physical connection to the onset of gravity-driven motions.