Asteroseismology with the space-borne missions CoRoT and Kepler provides a powerful mean of testing the modeling of transport processes in stars. Rotational splittings are currently measured for a large number of red giant stars and can provide strin
gent constraints on the rotation profiles. The aim of this paper is to obtain a theoretical framework for understanding the properties of the observed rotational splittings of red giant stars with slowly rotating cores. This allows us to establish appropriate seismic diagnostics for rotation of these evolved stars. Rotational splittings for stochastically excited dipolar modes are computed adopting a first-order perturbative approach for two $1.3 M_odot$ benchmark models assuming slowly rotating cores. For red giant stars with slowly rotating cores, we show that the variation of the rotational splittings of $ell=1$ modes with frequency depends only on the large frequency separation, the g-mode period spacing, and the ratio of the average envelope to core rotation rates (${cal R}$). This leds us to propose a way to infer directly ${cal R}$ from the observations. This method is validated using the Kepler red giant star KIC 5356201. Finally, we provide a theoretical support for the use of a Lorentzian profile to measure the observed splittings for red giant stars.
Rotational splittings are currently measured for several main sequence stars and a large number of red giants with the space mission Kepler. This will provide stringent constraints on rotation profiles. Our aim is to obtain seismic constraints on the
internal transport and surface loss of angular momentum of oscillating solar-like stars. To this end, we study the evolution of rotational splittings from the pre-main sequence to the red-giant branch for stochastically excited oscillation modes. We modified the evolutionary code CESAM2K to take rotationally induced transport in radiative zones into account. Linear rotational splittings were computed for a sequence of $1.3 M_{odot}$ models. Rotation profiles were derived from our evolutionary models and eigenfunctions from linear adiabatic oscillation calculations. We find that transport by meridional circulation and shear turbulence yields far too high a core rotation rate for red-giant models compared with recent seismic observations. We discuss several uncertainties in the physical description of stars that could have an impact on the rotation profiles. For instance, we find that the Goldreich-Schubert-Fricke instability does not extract enough angular momentum from the core to account for the discrepancy. In contrast, an increase of the horizontal turbulent viscosity by 2 orders of magnitude is able to significantly decrease the central rotation rate on the red-giant branch. Our results indicate that it is possible that the prescription for the horizontal turbulent viscosity largely underestimates its actual value or else a mechanism not included in current stellar models of low mass stars is needed to slow down the rotation in the radiative core of red-giant stars.
