We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow ...very different laws as the Ekman number
$E$
becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to
$E^{1/6}$
,
$E^{1/4}$
and
$E^{1/3}$
control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter
$E^{1/12}$
in these oscillatory flows. With a simplified model of the viscous Poincaré equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to
$\sin (\unicodeSTIX{x03C0}/4)$
and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.
Context. Interpretation of interferometric observations of rapidly rotating stars requires a good model of their surface effective temperature. Until now, laws of the form ...\hbox{$T_\mathrm{eff}\propto g_\mathrm{eff}^\beta$}Teff∝geffβ have been used, but they are only valid for slowly rotating stars. Aims. We propose a simple model that can describe the latitudinal variations in the flux of rotating stars at any rotation rate. Methods. This model assumes that the energy flux is a divergence-free vector that is antiparallel to the effective gravity. Results. When mass distribution can be described by a Roche model, the latitudinal variations in the effective temperature only depend on a single parameter, namely the ratio of the equatorial velocity to the Keplerian velocity. We validate this model by comparing its predictions to those of the most realistic two-dimensional models of rotating stars issued from the ESTER code. The agreement is very good, as it is with the observations of two rapidly rotating stars, α Aql and α Leo. Conclusions. We suggest that as long as a gray atmosphere can be accepted, the inversion of data on flux distribution coming from interferometric observations of rotating stars uses such a model, which has just one free parameter.
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Context.
Fast rotation is responsible for important changes in the structure and evolution of stars and the way we see them. Optical long baseline interferometry now allows for the study of its ...effects on the stellar surface, mainly gravity darkening and flattening.
Aims.
We aim to determine the fundamental parameters of the fast-rotating star Altair, in particular its evolutionary stage (represented here by the core hydrogen mass fraction
X
c
), mass, and differential rotation, using state-of-the-art stellar interior and atmosphere models together with interferometric (ESO-VLTI), spectroscopic, and asteroseismic observations.
Methods.
We use ESTER two-dimensional stellar models to produce the relevant surface parameters needed to create intensity maps from atmosphere models. Interferometric and spectroscopic observables are computed from these intensity maps and several stellar parameters are then adjusted using the publicly available MCMC algorithm Emcee.
Results.
We determined Altair’s equatorial radius to be
R
eq
= 2.008 ± 0.006
R
⊙
, the position angle PA = 301.1 ± 0.3°, the inclination
i
= 50.7 ± 1.2°, and the equatorial angular velocity Ω = 0.74 ± 0.01 times the Keplerian angular velocity at equator. This angular velocity leads to a flattening of
ε
= 0.220 ± 0.003. We also deduce from the spectroscopically derived
v
sin
i
≃ 243 km s
−1
, a true equatorial velocity of ∼314 km s
−1
corresponding to a rotation period of 7h46m (∼3 cycles/day). The data also impose a strong correlation between mass, metallicity, hydrogen abundance, and core evolution. Thanks to asteroseismic data, and provided our frequencies identification is correct, we constrain the mass of Altair to 1.86 ± 0.03
M
⊙
and further deduce its metallicity
Z
= 0.019 and its core hydrogen mass fraction
X
c
= 0.71, assuming an initial solar hydrogen mass fraction
X
= 0.739. These values suggest that Altair is a young star ∼100 Myr old. Finally, the 2D ESTER model also gives the internal differential rotation of Altair, showing that its core rotates approximately 50% faster than the envelope, while the surface differential rotation does not exceed 6%.
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Context. The observations of rapidly rotating stars are increasingly detailed and precise thanks to interferometry and asteroseismology; two-dimensional models taking into account the hydrodynamics ...of these stars are very much needed. Aims. A model to study the dynamics of baroclinic stellar envelopes is presented. Methods. This model treats the stellar fluid with the Boussinesq approximation and assumes that it is contained in a rigid spherical domain. The temperature field and the rotation of the system generate the baroclinic flow. Results. We give an analytical solution to the asymptotic problem at small Ekman and Prandtl numbers. We show that, provided the Brunt-Vaeisaelae frequency profile is smooth enough, differential rotation of a stably stratified envelope takes the form a fast rotating pole and a slow equator while it is the opposite in a convective envelope. We also show that at low Prandtl numbers and without D*m-barriers, the jump in viscosity at the core-envelope boundary generates a shear layer staying along the tangential cylinder of the core. Its role in mixing processes is discussed. Conclusions. Such a model provides an interesting tool to investigate the fluid dynamics of rotating stars in particular for the study of the various instabilities affecting baroclinic flows or a dynamo effect.
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Aims. This work aims at presenting the first two-dimensional models of an isolated rapidly rotating star that include the derivation of the differential rotation and meridional circulation in a ...self-consistent way. Methods. We use spectral methods in multidomains, together with a Newton algorithm to determine the steady state solutions including differential rotation and meridional circulation for an isolated non-magnetic, rapidly rotating early-type star. In particular we devise an asymptotic method for small Ekman numbers (small viscosities) that removes the Ekman boundary layer and lifts the degeneracy of the inviscid baroclinic solutions. Results. For the first time, realistic two-dimensional models of fast-rotating stars are computed with the actual baroclinic flows that predict the differential rotation and the meridional circulation for intermediate-mass and massive stars. These models nicely compare with available data of some nearby fast-rotating early-type stars like Ras Alhague (α Oph), Regulus (α Leo), and Vega (α Lyr). It is shown that baroclinicity drives a differential rotation with a slow pole, a fast equator, a fast core, and a slow envelope. The differential rotation is found to increase with mass, with evolution (here measured by the hydrogen mass fraction in the core), and with metallicity. The core-envelope interface is found to be a place of strong shear where mixing will be efficient. Conclusions. Two-dimensional models offer a new view of fast-rotating stars, especially of their differential rotation, which turns out to be strong at the core-envelope interface. They also offer more accurate models for interpreting the interferometric and spectroscopic data of early-type stars.
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We investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation ...profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in
${E}^{1/ 3} $
with the Ekman number
$E$
, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with
${E}^{1/ 2} $
as Ekman boundary layers. The behaviour of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.
We investigate the linear properties of the steady and axisymmetric stress-driven spin-down flow of a viscous fluid inside a spherical shell, both within the incompressible and anelastic ...approximations, and in the asymptotic limit of small viscosities. From boundary layer analysis, we derive an analytical geostrophic solution for the three-dimensional incompressible steady flow, inside and outside the cylinder $\mathcal {C}$ that is tangent to the inner shell. The Stewartson layer that lies on $\mathcal {C}$ is composed of two nested shear layers of thickness $O(E^{2/7})$ and $O(E^{1/3})$ where E is the Ekman number. We derive the lowest-order solution for the $E^{2/7}$-layer. A simple analysis of the $E^{1/3}$-layer lying along the tangent cylinder, reveals it to be the site of an upwelling flow of amplitude $O(E^{1/3})$. Despite its narrowness, this shear layer concentrates most of the global meridional kinetic energy of the spin-down flow. Furthermore, a stable stratification does not perturb the spin-down flow provided the Prandtl number is small enough. If this is not the case, the Stewartson layer disappears and meridional circulation is confined within the thermal layers. The scalings for the amplitude of the anelastic secondary flow have been found to be the same as for the incompressible flow in all three regions, at the lowest order. However, because the velocity no longer conforms the Taylor–Proudman theorem, its shape differs outside the tangent cylinder $\mathcal {C}$, that is, where differential rotation takes place. Finally, we find the settling of the steady state to be reached on a viscous time for the weakly, strongly and thermally unstratified incompressible flows. Large density variations relevant to astro- and geophysical systems, tend to slightly shorten the transient.
Context. Star-planet tidal interactions may result in the excitation of inertial waves in the convective region of stars. In low-mass stars, their dissipation plays a prominent role in the long-term ...orbital evolution of short-period planets. Turbulent convection can sustain differential rotation in their envelopes with an equatorial acceleration (as in the Sun) or deceleration, which can modify the propagation properties of the waves. Aims. We explore in this first paper the general propagation properties of free linear inertial waves in a differentially rotating homogeneous fluid inside a spherical shell. We assume that the angular velocity background flow depends on the latitudinal coordinate alone, close to what is expected in the external convective envelope of low-mass stars. Methods. We use an analytical approach in the inviscid case to get the dispersion relation, from which we compute the characteristic trajectories along which energy propagates. This allows us to study the existence of attractor cycles and infer the different families of inertial modes. We also use high-resolution numerical calculations based on a spectral method for the viscous problem. Results. We find that modes that propagate in the whole shell (D modes) behave the same way as with solid-body rotation. However, another family of inertial modes exists (DT modes), which can only propagate in a restricted part of the convective zone. Our study shows that they are less common than D modes and that the characteristic rays and shear layers often focus towards a wedge - or point-like attractor. More importantly, we find that for non-axisymmetric oscillation modes, shear layers may cross a corotation resonance with a local accumulation of kinetic energy. Their damping rate scales very differently from the value we obtain for standard D modes, and we show an example where it is independent of viscosity (Ekman number) in the astrophysical regime in which it is small.
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We investigate the properties of forced inertial modes of a rotating fluid inside a spherical shell. Our forcing is tidal like, but its main property is that it is on the large scales. By numerically ...solving the linear equations of this problem, including viscosity, we first confirm some analytical results obtained on a two-dimensional model by Ogilvie (J. Fluid Mech., vol. 543, 2005, p. 19); some additional properties of this model are uncovered like the existence of narrow resonances associated with periodic orbits of characteristics. We also note that as the frequency of the forcing varies, the dissipation varies drastically if the Ekman number E is low (as is usually the case). We then investigate the three-dimensional case and compare the results to the foregoing model. The three-dimensional solutions show, like their two-dimensional counterpart, a spiky dissipation curve when the frequency of the forcing is varied; they also display small frequency intervals where the viscous dissipation is independent of viscosity. However, we show that the response of the fluid in these frequency intervals is crucially dominated by the shear layer that is emitted at the critical latitude on the inner sphere. The asymptotic regime, where the dissipation is independent of the viscosity, is reached when an attractor has been excited by this shear layer. This property is not shared by the two-dimensional model where shear layers around attractors are independent of those emitted at the critical latitude. Finally, resonances of the three-dimensional model correspond to some selected least damped eigenmodes. Unlike their two-dimensional counter parts these modes are not associated with simple attractors; instead, they show up in frequency intervals with weakly contracting webs of characteristics. Besides, we show that the inner core is negligible when its relative radius is less than the critical value 0.4E1/5. For these spherical shells, the full sphere solutions give a good approximation of the flows.
Stellar variability, at a variety of timescales, can strongly affect the ability to detect exoplanets, in particular when using radial velocity (RV) techniques. Accurately characterized solar ...variations are precious in this context to study the impact of stellar variations on planet detectability. Here we focus on the impact of small timescale variability. The objective of this paper is to model realistic RV time series due to granulation and super-granulation and to study in greater detail the impact of granulation and super granulation on RV times series in the solar case. We have simulated a collection of granules and super-granules evolving in time to reproduce solar photometric and RV time series. Synthetic time series are built over the full hemisphere over one solar cycle. To minimize the effect of granulation, the best strategy is to split the observing time during the night into several periods instead of observing over a consecutive duration. The results do not take the presence of pulsations into account.
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