Geometry of the Huygens-Roche figure

Abstract

Theory of the figures of celestial bodies, which are in the state of hydrostatic equilibrium under gravitational, centrifugal, and pressure forces, took the form of a rigorous mathematical theory in the second part of the XX century. Fundamental physical laws served as its basis. The Huygens-Roche figure (total mass is concentrated in the centre, while the rotating atmosphere takes the equilibrium form) plays an important role in the theory. Properties of the figure are carefully examined. In particular, it is known that each isobar (surface of equal pressure) represents one of the three-parameter family of the Huygens- Roche surfaces. Meanwhile, convexity (or its absence) was not dicussed in the literature as far as we know. It is worth noting that there are non-convex figures between equilibrium ones. In the present paper, we find the curvature of a meridional section of an arbitrary Huygens-Roche figure (in the closed form as well as in the form of a series in powers of the Clairaut parameter, main in the theory of equilibrium figures). We succeed to prove that the curvature is positive and is separated from zero. Hence, each surface from the Huygens-Roche figures family is convex, and has no points of flattening. Moreover, no curve on its surface has points of straightening.