Jupiter and Saturn Multi-layer Models Rotating Differentially

Abstract

In a past work, models for Jupiter were constructed in base to a set of concentric distorted spheroids (“spheroidals”) rotating differentially—whose semi-axes are independent of one another—a task that was achieved with a law of rotation deduced from a generalization of Bernoulli’s theorem, and which holds exclusively for axial-symmetric masses. The shape of the mass is that of a spheroid whose surface equation contains an extra term, d/z4, where d is a parameter which measures the degree of distortion. Each layer rotates with its own profile of angular velocity. The rotation law has a simple dependence on the derivative of the gravitational potential. No magnetic fields or equations of state were involved. The multi- structures were demanded, firstly, to reproduce the gravitational moments of the planets, as surveyed by space missions; and, secondly, to be equilibrium figures. For the calculation of the gravitational moments, a minimization procedure was employed. Paying attention on the outermost laye—the relevant one in the present context—of the formerly reported models for Jupiter, we became aware that they all share an angular velocity profile that decreases from the pole towards the equator, an event that, so far, has not been verified observationally. Since figures with profiles of the opposite tendency turned out to be also possible, they should be included as candidates for our purpose, as effectively they are herein as a complement of that work. The same procedure is here entailed to Saturn, for which figures to show one or the other tendencies are as well obtained. The dual behavior of the rotation profiles may be explained by arguments involving the centripetal force. According to this standpoint, the double behavior is a consequence of the algebraic sign assigned to d: if positive, so that the surface is more bloated than that of a spheroid, the decreasing tendency results; whereas if negative, so that the surface is more depressed, the increasing tendency shows up. This, in turn, is because for d negative the radial force increases more rapidly from pole to the equator than for d positive. We point out that the rotation profiles of the current figures are determined from their equilibrium, rather than being imposed ad initio.