Geometry of the Huygens—Roche Figure

Аннотация

The theory of figures of celestial bodies, which are in the state of hydrostatic equilibrium under the action of pressure, gravitational, and centrifugal forces, took the form of a rigorous mathematical theory in the second part of the 20th century. Fundamental physical laws served as its basis. The Huygens–Roche figure (its total mass is concentrated in the center, 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) itself is one of the three-parameter family of Huygens—Roche surfaces. However, as far as we know, convexity (or its absence) has not been discussed in the literature. Meanwhile, there are non-convex figures between equilibrium ones. In the present paper, we find the curvature of the meridional section of an arbitrary Huygens—Roche figure both in closed form and in the form of a series in powers of the Clairaut parameter, which is basic in the theory of equilibrium figures. We succeeded in proving that the curvature is positive and is bounded away from zero. Hence, every surface of the family of Huygens—Roche figures is convex and has no points of flattening. Moreover, none of the curves on its surface has points of straightening.