An alternative method to construct a consistent second-order theory on the equilibrium figures of rotating celestial bodies

Abstract

The main objective of this work is to construct a new method to develop a consistent second-order amplitudes theory to evaluate the potential of a rotating deformable celestial body when the hydrostatic system equilibrium has been achieved. In this case, we have: , , where is the pressure, is the density, is the total potential, is Laplace operator, is the gravitational constant and is the angular velocity of the system. To integrate these equations in a general case of mass distribution a state equation relating pressure and density is needed.

To assess the full potential, , it is necessary to calculate the self-gravitational potential, , and the centrifugal potential, . The equilibrium configuration involves the hydrostatic equilibrium, it is, the rigid rotation of the system corresponding to the minimum potential and, according to Kopal, this state involves the identification of equipotential, isobaric, isothermal and isopycnic surfaces.

To study the structure of the body we define a coordinate system where is the center of mass of the component, is an axis fixed in an arbitrary point of the body equator, an axis parallel to angular velocity and defining a direct trihedron. For an arbitrary point in the rotating body the Clairaut coordinates are given by where is the radius of the sphere that contains the same mass that the equipotential surface that contains and are the angular spherical coordinates of .

This problem has been solved in the first order in following two techniques: the first one is based on the asymptotic properties of the numerical quadrature formulae. The second is similar to the one used by Laplace to develop the inverse of the distance between two planets. The second-order theory based on the first method has been developed by the authors in a recent paper. In this work we develop a consistent second-order theory about the equilibrium figures of rotating celestial bodies based on the second method.

Finally, to show the performance of the method it is interesting to study a numerical example based on a convective star.