Leading elliptic relationship for supercritical fluids in the Widom region

It is well-known that the behaviour of supercritical fluids near the critical point changes considerably in the region around the Widom lines. We have found, for pure fluids, that the relationship between the thermal expansion coefficient and the isothermal compressibility is not only highly nonlinear in this region but in a certain temperature range can also be approximated well at constant pressure by a quadratic relationship, i.e., an ellipse. In this paper, we show this analytically, and demonstrate it for the van der Waals, Berthelot and IAPWS-95 water models numerically. Furthermore, the parameters of the ellipse are discovered to obey power and linear laws. The parameters of these laws are determined for the three aforementioned models.