Abstract
Since the external regions of the envelopes of rapidly rotating early-type stars are unstable to convection, a coupling may exist between the convection and the internal rotation. We explore what can be learned from spectroscopic and interferometric observations about the properties of the rotation law in the external layers of these objects. Using simple relations between the entropy and specific rotational quantities, some of which are found to be efficient at accounting for the solar differential rotation in the convective region, we derived analytical solutions that represent possible differential rotations in the envelope of early-type stars. A surface latitudinal differential rotation may not only be an external imprint of the inner rotation, but induces changes in the stellar geometry, the gravitational darkening, the aspect of spectral line profiles, and the emitted spectral energy distribution. By studying the equation of the surface of stars with non-conservative rotation laws, we conclude that objects undergo geometrical deformations that are a function of the latitudinal differential rotation able to be scrutinized both spectroscopically and by interferometry. The combination of Fourier analysis of spectral lines with model atmospheres provides independent estimates of the surface latitudinal differential rotation and the inclination angle. Models of stars at different evolutionary stages rotating with internal conservative rotation laws were calculated to show that the Roche approximation can be safely used to account for the gravitational potential. The surface temperature gradient in rapid rotators induce an acceleration to the surface angular velocity. A non-zero differential rotation parameter may indicate that the rotation is neither rigid nor shellular underneath the stellar surface.
Highlights
Many attempts have been made to obtain information on the internal rotation from detailed studies of: a) the position of stars in the HR diagram; b) the evolution of the Vsin i parameter during the main sequence (MS) phase; c) the shape of absorption lines, whose characteristics can depend upon the rotational law in layers close to the stellar surface; d) the global stellar geometry described with interferometric data
T/tMS is the fractional age of the star, where tMS is the time that a non-rotating star of mass M spends on the main sequence, Ωcr is the critical angular velocity, Ω/Ωcr represents the angular velocity for which the model was calculated, ρc is the core density of the rotating object, Re/R is the equatorial radius of the model star in solar units and Re/Rp the equatorial-to-polar radii ratio, Veq is the equatorial linear velocity in km s−1, J/M is the total specific angular momentum, η = Ω2R3e/GM is the ratio of centrifugal to the gravitational acceleration in the equator, and K/|W| is the ratio of the kinetic rotational energy (K) to the absolute value of the gravitational potential energy (W)
Recent theoretical works suggest that rapidly rotating early-type stars should have rather deep convective layers in the envelope. This situation may favor some coupling between the convective region and the differential rotation, as happens in the Sun
Summary
One of the most enduring unknowns in stellar physics has been the inner distribution of the angular momentum in a star. Many attempts have been made to obtain information on the internal rotation from detailed studies of: a) the position of stars in the HR diagram; b) the evolution of the Vsin i parameter during the main sequence (MS) phase; c) the shape of absorption lines, whose characteristics can depend upon the rotational law in layers close to the stellar surface; d) the global stellar geometry described with interferometric data. Since in many cases the observed ratios RLC were found to be situated in-between the two extreme theoretical predictions, it was suggested that stars should be differential rotators These studies could not provide any information about the characteristics of the internal rotational law (Sandage 1955; Danziger & Faber 1972; Zorec et al 1987; Zorec 2004).
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