Tidal interactions and principle of corresponding states: from micro to macro cosmos. A century after van der Waals' Nobel Prize

Abstract

The current attempt is aimed to honor the first centennial of Johannes Diderik van der Waals (VDW) awarding Nobel Prize in Physics. The VDW theory of ordinary fluids is reviewed in the first part of the paper, where special effort is devoted to the equation of state and the law of corresponding states. In addition, a few mathematical features involving properties of cubic equations are discussed, for appreciating the intrinsic beauty of the VDW theory. A theory of astrophysical fluids is shortly reviewed in the second part of the paper, grounding on the tensor virial theorem for two-component systems, and an equation of state is formulated with a convenient choice of reduced variables. Additional effort is devoted to particular choices of density profiles, namely a simple guidance case and two cases of astrophysical interest. The related macroisothermal curves are found to be qualitatively similar to VDW isothermal curves below the critical threshold and, for sufficiently steep density profiles, a critical macroisothermal curve exists, with a single horizontal inflexion point. Under the working hypothesis of a phase transition (assumed to be gas-stars) for astrophysical fluids, similar to the vapour-liquid phase transition in ordinary fluids, the location of gas clouds, stellar systems, galaxies, cluster of galaxies, on the plane scanned by reduced variables, is tentatively assigned. A brief discussion shows how van der Waals' two great discoveries, namely a gas equation of state where tidal interactions between molecules are taken into account, and the law of corresponding states, related to microcosmos, find a counterpart with regard to macrocosmos. In conclusion, after more than a century since the awarding of the Nobel Prize in Physics, van der Waals' ideas are still valid and helpful to day for a full understanding of the universe.