Issue |
A&A
Volume 652, August 2021
|
|
---|---|---|
Article Number | A154 | |
Number of page(s) | 16 | |
Section | Stellar structure and evolution | |
DOI | https://doi.org/10.1051/0004-6361/202140615 | |
Published online | 31 August 2021 |
The traditional approximation of rotation for rapidly rotating stars and planets
I. The impact of strong deformation
1
Département d’Astrophysique-AIM, CEA/DRF/IRFU, CNRS/INSU, Université Paris-Saclay, Université Paris-Diderot, Universitéde Paris, 91191 Gif-sur-Yvette, France
e-mail: hachem.dhouib@cea.fr
2
Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
Received:
19
February
2021
Accepted:
17
May
2021
Context. The traditional approximation of rotation (TAR) is a treatment of the hydrodynamic equations of rotating and stably stratified fluids in which the action of the Coriolis acceleration along the direction of the entropy and chemical stratifications is neglected because it is weak in comparison with the buoyancy Archimedean force. This leads to the neglect of the horizontal projection of the rotation vector in the equations for the dynamics of gravito-inertial waves (GIWs). The dependent variables in those equations then become separable into radial and horizontal parts as in the non-rotating case. The TAR is built on the assumptions that the star is spherical (i.e., its centrifugal deformation is neglected) and uniformly rotating. However, it has recently been generalised to include the effects of a moderate centrifugal deformation using a perturbative approach.
Aims. We study the feasibility of carrying out a new generalisation to account for the centrifugal acceleration in the case of strongly deformed uniformly and rapidly rotating stars (and planets), and to identify the validity domain of this approximation.
Methods. We built a complete formalism analytically that allows the study of the dynamics of GIWs in spheroidal coordinates which take the flattening of uniformly and rapidly rotating stars into account by assuming the hierarchies of frequencies adopted within the TAR in the spherical case.
Results. Using 2D stellar models, we determine the validity domain of the generalised TAR as a function of the rotation rate of the star normalised by its critical angular velocity and its pseudo-radius. Assuming the anelastic and the two-dimensional Jeffreys-Wentzel-Kramers-Brillouin approximations, we derive a generalised Laplace tidal equation for the horizontal eigenfunctions of the GIWs and their asymptotic wave periods, which can be used to probe the structure and dynamics of rotating deformed stars with asteroseismology. The generalised TAR where the centrifugal deformation of a star (or planet) is taken into account non-perturbatively allows us to identify, within the framework of 2D Evolution STEllaire en Rotation models, the validity domain of this approximation which is reduced by increasing the rate of rotation. We can affirm with a level of confidence of 90% that the TAR remains applicable in all the space domain of deformed stars rotating at a rotation rate lower than 20% of the critical rotation rate.
Conclusions. A new generalisation of the TAR, which takes the centrifugal acceleration into account in a non-perturbative way, is derived. This generalisation allows us to study the detectability and the signature of the centrifugal effects on GIWs in rapidly rotating deformed stars (and planets). We found that the effects of the centrifugal acceleration in rapidly rotating early-type stars on GIWs are theoretically detectable in modern space photometry using observations from Kepler. We found also, by comparing the period spacing pattern computed with the standard and the generalised TAR, that the centrifugal acceleration affects the period spacing by increasing its values for low radial orders and by decreasing them slightly for high radial orders.
Key words: hydrodynamics / waves / stars: rotation / stars: oscillations / methods: analytical / methods: numerical
© H. Dhouib et al. 2021
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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