Supersonic compressible turbulence is ubiquitous in star-forming regions. However, predicting measurable statistical properties of the density fluctuations remains a major challenge in physics. In 1951, Chandrasekhar derived a mass invariant under the assumptions of statistical homogeneity and isotropy of the turbulent density field, together with stationarity of the background density. This invariant depends on both the variance and the correlation length of the density field.

In this work, we perform high-resolution (2048^3) numerical simulations of homogeneous and isotropic compressible turbulence in order to investigate the validity of this invariant in media subject either to decaying turbulence or to self-gravity.

We then explore several applications of this invariant to improve our understanding of the statistical properties of compressible turbulent flows. We show that it allows one to predict the evolution of the slope of the density power spectrum as a function of the Mach number in supersonic turbulence. These predictions are validated through comparison with numerical simulations.

Finally, we present a model based on this invariant that may explain the observed universality of the characteristic mass of prestellar cores in Milky Way-like star-forming regions, as well as its systematic variations in more extreme environments, such as high-mass protoclusters and massive early-type galaxies.