Stars may be understood as self-gravitating masses of a compressible fluid whose radiative cooling is compensated by nuclear reactions or gravitational contraction. The understanding of their time evolution requires the use of detailed models that account for a complex microphysics including that of opacities, equation of state and nuclear reactions. The present stellar models are essentially one-dimensional, namely spherically symmetric. However, the interpretation of recent data like the surface abundances of elements or the distribution of internal rotation have reached the limits of validity of one-dimensional models because of their very simplified representation of large-scale fluid flows. In this article, we describe the ESTER code, which is the first code able to compute in a consistent way a two-dimensional model of a fast rotating star including its large-scale flows. Compared to classical 1D stellar evolution codes, many numerical innovations have been introduced to deal with this complex problem. First, the spectral discretization based on spherical harmonics and Chebyshev polynomials is used to represent the 2D axisymmetric fields. A nonlinear mapping maps the spheroidal star and allows a smooth spectral representation of the fields. The properties of Picard and Newton iterations for solving the nonlinear partial differential equations of the problem are discussed. It turns out that the Picard scheme is efficient on the computation of the simple polytropic stars, but Newton algorithm is unsurpassed when stellar models include complex microphysics. Finally, we discuss the numerical efficiency of our solver of Newton iterations. This linear solver combines the iterative Conjugate Gradient Squared algorithm together with an LU-factorization serving as a preconditionner of the Jacobian matrix.
Context. The understanding of the evolution of early-type stars is tightly related to that of the effects of rapid rotation. For massive stars, rapid rotation combines with their strong radiation-driven wind. Aims. The aim of this paper is to investigate two questions that are prerequisite to the study of the evolution of massive rapidly rotating stars: (i) What is the critical angular velocity of a star when radiative acceleration is significant in its atmosphere? (ii) How do mass and angular momentum loss depend on the rotation rate? Methods. To investigate fast rotation, which makes stars oblate, we used the 2D ESTER models and a simplified approach, the ω-model, which gives the latitudinal dependence of the radiative flux in a centrifugally flattened radiative envelope.Results. We find that radiative acceleration only mildly influences the critical angular velocity, at least for stars with masses lower than 40 M⊙. For instance, a 15 M⊙ star on the zero-age main sequence (ZAMS) would reach criticality at a rotation rate equal to 0.997 the Keplerian equatorial rotation rate. We explain this mild reduction of the critical angular velocity compared to the classical Keplerian angular velocity by the combined effects of gravity darkening and a reduced equatorial opacity that is due to the centrifugal acceleration. To answer the second question, we first devised a model of the local surface mass flux, which we calibrated with previously developed 1D models. The discontinuity (the so-called bi-stability jump) included in theṀ − T eff relation of 1D models means that the mass flux of a fast-rotating star is controlled by either a single wind or a two-wind regime. Mass and angular momentum losses are strong around the equator if the star is in the two-wind regime. We also show that the difficulty of selecting massive stars that are viewed pole-on makes detecting the discontinuity in the relation between mass loss and effective temperature also quite challenging.
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