Physics of rotation in stellar models

In these lecture notes, we present the equations presently used in stellar interior models in order to compute the effects of axial rotation. We discuss the hypotheses made. We suggest that the effects of rotation might play a key role at low metallicity. 1 Physics of rotation Axial rotation modifies the hydrostatic equilibrium configuration by adding a centrifugal acceleration term in the hydrostatic equation, induces many instabilities driving the transport of angular momentum and of chemical species in radiative zones and changes the mass loss rates. In the present lecture notes we shall consider the case of models without magnetic fields. 1.1 Hydrostatic effects of rotation The equations of stellar structure. In a rotating star, the equations of stellar structure need to be modified [23]. The usual spherical coordinates must be replaced by new coordinates characterizing the equipotentials. The classical method applies when the effective gravity can be derived from a potential Ψ = Φ− 12Ωr sin 2 θ, i.e. when the problem is conservative. There, Φ is the gravitational potential, Ω the angular velocity, r the radius at the colatitude θ. If the rotation law is shellular (i.e. such that Ω is constant on isobaric surfaces see below), the problem is non–conservative. Most existing models of rotating stars apply, rather inconsistently, the classical scheme by [23]. However, as shown by [44], the equations of stellar structure can still be written consistently, in term of a coordinate referring to the mass inside the isobaric surfaces . Thus, the problem of the stellar structure of a differentially rotating star in a shellular rotation state can be kept one–dimensional. The Roche model. In all the derivations, we shall use the Roche model, i.e. we approximate the gravitational potential by GMr/r where Mr is the mass inside the isobaric surface with a mean radius r. The radius r which labels each isobaric 1 For shellular rotation, the shape of the isobaric surfaces are given by the same expression as the one giving the shape of the equipotentials in conservative cases provided some changes of variables are performed.