From Maclaurin Spheroids to the Roche Models

The stability of rotating configurations has been studied since long (see review in [315]), for example with Maclaurin spheroids, where the density q is supposed constant or with the Roche model, which assumes an infinite central condensation. The complex reality lies between these two extreme cases.

In the case of the Maclaurin spheroids, the equilibrium configurations flatten for high rotation. For extremely high angular momentum, it tends toward an infinitely thin circular disk. The maximum value of the angular velocity Q (supposed to be constant in the body) is Qm^ = 0.4494kGq. In reality, some instabilities would occur before this limit is reached.

In the case of the Roche model with constant Q (this is not a necessary assumption), the equilibrium figure also flattens to reach a ratio of 2/3 between the polar

A. Maeder, Physics, Formation and Evolution of Rotating Stars, Astronomy and 19

Astrophysics Library, DOI 10.1007/978-3-540-76949-1-2, © Springer-Verlag Berlin Heidelberg 2009

and the equatorial radii, with a maximum angular velocity Q^^ = 0.7215 nOq, where q is the mean density (see Sect. 4.4.2). Interestingly enough, for all stellar masses the rotational energy of the Roche model amounts to at most about 1% of the absolute value of the potential energy of the models considered with their real density distributions. Except for the academic case of stars with constant density or nearly constant density, the Roche approximation better corresponds to the stellar reality. Recent results from long-baseline interferometry [94, 465] support the application of the Roche model in the cases of Altair and Achernar, which both rotate very fast close to their break-up velocities (see Sect. 4.2.3). These new possibilities of observations open interesting perspectives.

Here, we consider models of real stars, with no a priori given density distributions and obeying a general equation of state. The properties of rotating stars depend on the distribution Q(r) in the stellar interiors. The first models were applied to solid body rotation, i.e., Q = const. throughout the stellar interior. More elaborate models consider differential rotation, in particular the case of the so-called shellular rotation [632], i.e., with a rotation law Q(r) constant on isobaric shells and depending on the first order of the distance to the stellar center (see Sect. 2.2). The reason for such a rotation law rests on the strong horizontal turbulence in differentially rotating stars, which imposes a constancy of Q on isobars [632]. In the vertical direction, the turbulence is weak due to the stable density stratification.

Interestingly enough, recent models with rotation and magnetic fields give rotation laws Q(r) rather close to solid body rotation (Sect. 13.6), nevertheless with some significant deviations from constant Q. Thus, whether or not magnetic fields play a role, it is necessary to account for rotation laws which are not constant in stellar interiors during evolution.

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