On the Use of the Self-consistent-Field Method in the Construction of Models for Rapidly Rotating Main-Sequence Stars

A new formulation of the self-consistent-field (SCF) method for computing models of rapidly, differentially rotating stars is described. The angular velocity is assumed to depend only on the distance from the axis of rotation. In the modified SCF iterative scheme, normalized distributions of two thermodynamic variables—pressure and temperature—are used as trial functions, while the central values of the pressure and temperature are adjusted by a Newton-Raphson iteration. A two-dimensional trial density distribution, which is needed to compute the gravitational potential, is readily obtained from the pressure and temperature through the equation of state in conjunction with a third trial function specifying the two-dimensional shape of the constant-density surfaces. Rotating models of chemically homogeneous main-sequence stars have been computed as necessary in order to illustrate the algorithm and to make comparisons with existing models. Unlike previous implementations of the SCF method, the method described here is not limited to the upper main sequence: it converges for all main-sequence masses, including those well below 9 M⊙. Moreover, the method converges for values of the parameter t = T/|W| (the ratio of rotational kinetic energy to gravitational potential energy) that are at least as high as those obtained by Clement's relaxation technique. The method is also capable of producing models with deep concavities about the poles as well as models with extreme oblateness (far greater than that possible in uniformly rotating stars). For cases with moderate degrees of differential rotation (say for Ω0/Ωe < 10, where Ω0 and Ωe denote the angular velocity at the center and at the equator, respectively), the method has been found to be remarkably robust. For higher degrees of differential rotation, models are restricted to a portion of parameter space away from two regions of nonconvergence, inside which some of the models evidently develop toroidal level surfaces.