# Rotating stars

3.1 Introduction

Consider a single star that rotates about a fixed direction in space, with some assigned angular velocity. As we know, the star then assumes the shape of an oblate figure. However, we are at once faced with the following questions. What is the geometrical shape of the free boundary? What is the form of the surfaces upon which the physical variables (such as pressure, density, ... ) remain a constant? To sum up, what is the actual stratification of a rotating star, and how does it depend on the angular velocity distribution? For rotating stars, we have no a priori knowledge of this stratification, which is itself an unknown that must be derived from the basic equations of the problem. This is in sharp contrast to the case of a nonrotating star, for which a spherical stratification can be assumed ab initio.*

In principle, by making use of the equations derived in Section 2.2, one should be able to calculate at every instant the angular momentum distribution and the stratification in a rotating star. Obviously, this is an impossible task at the present level of knowledge of the subject, even were the initial conditions known. Until very recently, the standard procedure was to calculate in an approximate manner an equilibrium structure that corresponds to some prescribed rotation law, ruling out those configurations that are dynamically or thermally unstable with respect to axisymmetric disturbances (see Sections 3.4.2 and 3.5). Unfortunately, the results presented in Section 3.3 indicate that, no matter whether radiation or convection is providing the energy transport, the large-scale motion in a star is always the combination of a pure rotation and a circulation in meridian planes passing through the rotation axis. Moreover, as we shall see in Section 3.4.3, no dynamically stable model can possibly exist when nonaxisymmetric disturbances are taken into account. These barotropic and baroclinic instabilities, which have their roots in the geophysical literature, are mild ones in the sense that they continuously generate small-scale, eddy motions that interact with the large-scale flow. Lacking any better description of these transient motions, we shall further assume that the eddy flux

* Following the publication of Newton's (1687) Principia, the effects of rotation upon the internal structure of a self-gravitating body were investigated mainly with a view to their possible applications to geodesy and planetary physics. Many a classical result derived during the period 1740-1940 still retains its usefulness today when applied to centrally condensed stars. For a brief historical account of these and related matters, see J. L. Tassoul, Theory of Rotating Stars, Section 1.3, Princeton: Princeton University Press, 1978.

momentum can be represented parametrically by means of suitable coefficients of eddy viscosity. This is known as the eddy-mean flow interaction, which is presented in Section 3.6. This approach has been familiar to geophysicists since the late 1940s. It is particularly convenient because it resolves in a very simple manner the many contradictions and inconsistencies that have beset the theory of rotating stars.

3.2 Basic concepts

Because molecular viscosity is negligible for large-scale motions in a star, the momentum equation (2.7) reduces to

Dv 1

Dt p where v is the velocity in an inertial frame of reference, V is the gravitational potential, p is the density, and p is the pressure. The gravitational potential and the density are related by the Poisson equation

where G is the constant of gravitation. Mass conservation implies that

Finally, neglecting the dissipation function and thermal conductivity, we may recast Eq. (2.12) in the form

where eNuc is the rate of energy released by the thermonuclear reactions per unit mass and unit time, and F is the radiative flux vector. If we except the outermost surface layers of a star, this vector is given by

where

4ac T3