Traditional Approximation Of Rotation Research Articles

Non-linear three-mode coupling of gravity modes in rotating slowly pulsating B stars

Context. Slowly pulsating B (SPB) stars display multi-periodic variability in the gravito-inertial mode regime with indications of non-linear resonances between modes. Several have undergone asteroseismic modeling in the past few years to infer their internal properties, but only in a linear setting. These stars rotate fast, so that rotation is typically included in the modeling by means of the traditional approximation of rotation (TAR). Aims. We aim to extend the set of tools available for asteroseismology, by describing time-independent (stationary) resonant non-linear coupling among three gravito-inertial modes within the TAR. Such coupling offers the opportunity to use mode amplitude ratios in the asteroseismic modeling process, instead of only relying on frequencies of linear eigenmodes, as has been done so far. Methods. Following observational detections, we derive expressions for the resonant stationary non-linear coupling between three gravito-inertial modes in rotating stars. We assess selection rules and stability domains for stationary solutions. We also predict non-linear frequencies and amplitude ratio observables that can be compared with their observed counterparts. Results. The non-linear frequency shifts of stationary couplings are negligible compared to typical frequency errors derived from observations. The theoretically predicted amplitude ratios of combination frequencies match with some of their observational counterparts in the SPB targets. Other, unexplained observed ratios could be linked to other saturation mechanisms, to interactions between different modes, or to different opacity gradients in the driving zone. Conclusions. For the purpose of asteroseismic modeling, our non-linear mode coupling formalism can explain some of the stationary amplitude ratios of observed resonant mode couplings in single SPB stars monitored during 4 years by the Kepler space telescope.

Perturbative analysis of the effect of a magnetic field on gravito-inertial modes

Context. Magnetic fields have been measured recently in the cores of red giant stars thanks to their effects on stellar oscillation frequencies. The search for magnetic signatures in pulsating stars, such as γ Doradus (γ Dor) or slowly pulsating B stars, requires us to adapt the formalism developed for slowly rotating red giants to rapidly rotating stars. Aims. We perform a theoretical analysis of the effects of an arbitrary magnetic field on high radial order gravity and Rossby modes in a rapidly rotating star. Methods. The magnetic effects were treated as a perturbation. For high radial order modes, the contribution of the radial component of the magnetic field is likely to dominate over the azimuthal and latitudinal components. The rotation is taken into account through the traditional approximation of rotation. Results. General expressions of the frequency shift induced by an arbitrary radial magnetic field are derived. Approximate analytical forms are obtained in the high-order, high-spin-parameter limits for the modes most frequently observed in γ Dor stars. We propose simple methods to detect seismic magnetic signatures and measure possible magnetic fields in such stars. Conclusions. These methods offer new possibilities to look for internal magnetic fields in future observations, such as those of the PLATO mission, or of revisiting existing Kepler or TESS data.

Internal rotation and buoyancy travel time of 60 γ Doradus stars from uninterrupted TESS light curves spanning 352 days

Context. Gamma Doradus (hereafter γ Dor) stars are gravity-mode pulsators whose periods carry information about their internal structure. These periods are especially sensitive to the internal rotation and chemical mixing, two processes that are currently not well constrained in the theory of stellar evolution. Aims. We aim to identify the pulsation modes and deduce the internal rotation and buoyancy travel time for 106 γ Dor stars observed by the Transiting Exoplanet Survey Satellite (TESS) mission in its southern continuous viewing zone (hereafter S-CVZ). We rely on 140 previously detected period-spacing patterns, that is, series of (near-)consecutive pulsation mode periods. Methods. We used the asymptotic expression to compute gravity-mode frequencies for ranges of the rotation rate and buoyancy travel time that cover the physical range in γ Dor stars. Those frequencies were fitted to the observed period-spacing patterns by minimising a custom cost function. The effects of rotation were evaluated using the traditional approximation of rotation, using the stellar pulsation code GYRE. Results. We obtained the pulsation mode identification, internal rotation, and buoyancy travel time for 60 TESS γ Dor stars. For the remaining 46 targets, the detected patterns were either too short or contained too many missing modes for unambiguous mode identification, and longer light curves are required. For the successfully analysed stars, we found that period-spacing patterns from 1-yr-long TESS light curves can constrain the internal rotation and buoyancy travel time to a precision of 0.03 d-1 and 400 s, respectively, which is about half as precise as literature results based on 4-yr-long Kepler light curves of γ Dor stars.

Forward modelling and the quest for mode identification in rapidly rotating stars

Asteroseismology has opened a window on the internal physics of thousands of stars, by relating oscillation spectra properties to the internal physics of stars. Mode identification, namely the process of associating a measured oscillation frequency to the corresponding mode geometry and properties, is the cornerstone of this analysis of seismic spectra. In rapidly rotating stars this identification is a challenging task that remains incomplete, as modes assume complex geometries and regular patterns in frequencies get scrambled under the influence of the Coriolis force and centrifugal flattening. In this article, I will first discuss the various classes of mode geometries that emerge in rapidly rotating stars and the related frequency and period patterns, as predicted by ray dynamics, complete (non-)adiabatic calculations, or using the traditional approximation of rotation. These patterns scale with structural quantities and help us derive crucial constraints on the structure and evolution of these stars. I will summarize the amazing progress accomplished over the last few years for the deciphering of gravity-mode pulsator oscillation spectra, and recent developments based on machine-learning classification techniques to distinguish oscillation modes and pattern analysis strategies that let us access the underlying physics of pressure-mode pulsators. These approaches pave the way to ensemble asteroseismology of classical pulsators. Finally, I will highlight how these recent progress can be combined to improve forward seismic modelling. I will focus on the example of Rasalhague, a well-known rapid rotator, to illustrate the process and the needed advances to obtain à-la-carte modelling of such stars.

Detecting deep axisymmetric toroidal magnetic fields in stars

Context. Asteroseismology has revealed small core-to-surface rotation contrasts in stars in the whole Hertzsprung–Russell diagram. This is the signature of strong transport of angular momentum (AM) in stellar interiors. One of the plausible candidates to efficiently carry AM is magnetic fields with various topologies that could be present in stellar radiative zones. Among them, strong axisymmetric azimuthal (toroidal) magnetic fields have received a lot of interest. Indeed, if they are subject to the so-called Tayler instability, the accompanying triggered Maxwell stresses can transport AM efficiently. In addition, the electromotive force induced by the fluctuations of magnetic and velocity fields could potentially sustain a dynamo action that leads to the regeneration of the initial strong axisymmetric azimuthal magnetic field. Aims. The key question we aim to answer is whether we can detect signatures of these deep strong azimuthal magnetic fields. The only way to answer this question is asteroseismology, and the best laboratories of study are intermediate-mass and massive stars with external radiative envelopes. Most of these are rapid rotators during their main sequence. Therefore, we have to study stellar pulsations propagating in stably stratified, rotating, and potentially strongly magnetised radiative zones, namely magneto-gravito-inertial (MGI) waves. Methods. We generalise the traditional approximation of rotation (TAR) by simultaneously taking general axisymmetric differential rotation and azimuthal magnetic fields into account. Both the Coriolis acceleration and the Lorentz force are therefore treated in a non-perturbative way. Using this new formalism, we derive the asymptotic properties of MGI waves and their period spacings. Results. We find that toroidal magnetic fields induce a shift in the period spacings of gravity (g) and Rossby (r) modes. An equatorial azimuthal magnetic field with an amplitude of the order of 105 G leads to signatures that are detectable in period spacings for high-radial-order g and r modes in γ Doradus (γ Dor) and slowly pulsating B (SPB) stars. More complex hemispheric configurations are more difficult to observe, particularly when they are localised out of the propagation region of MGI modes, which can be localised in an equatorial belt. Conclusions. The magnetic TAR, which takes into account toroidal magnetic fields in a non-perturbative way, is derived. This new formalism allows us to assess the effects of the magnetic field in γ Dor and SPB stars on g and r modes. We find that these effects should be detectable for equatorial fields thanks to modern space photometry using observations from Kepler, TESS CVZ, and PLATO.

The traditional approximation of rotation for rapidly rotating stars and planets

Context. We examine the dynamics of low-frequency gravito-inertial waves (GIWs) in differentially rotating deformed radiation zones in stars and planets by generalising the traditional approximation of rotation (TAR). The TAR treatment was built on the assumptions that the star is spherical (i.e. its centrifugal deformation is neglected) and uniformly rotating. However, it has been generalised in our previous work by including the effects of the centrifugal deformation using a non-perturbative approach. In the meantime, TAR has been generalised in spherical geometry to take the differential rotation into account. Aims. We aim to carry out a new generalisation of the TAR treatment to account for the differential rotation and the strong centrifugal deformation simultaneously. Methods. We generalise our previous work by taking into account the differential rotation in the derivation of our complete analytical formalism that allows the study of the dynamics of GIWs in differentially and rapidly rotating stars. Results. We derived the complete set of equations that generalises the TAR, simultaneously taking the full centrifugal acceleration and the differential rotation into account. Within the validity domain of the TAR, we derived a generalised Laplace tidal equation for the horizontal eigenfunctions and asymptotic wave periods of the GIWs, which can be used to probe the structure and dynamics of differentially rotating deformed stars with asteroseismology. Conclusions. A new generalisation of the TAR, which simultaneously takes into account the differential rotation and the centrifugal acceleration in a non-perturbative way, was derived. This generalisation allowed us to study the detectability and the signature of the differential rotation on GIWs in rapidly rotating deformed stars and planets. We found that the effects of the differential rotation in early-type deformed stars on GIWs is theoretically largely detectable in modern space photometry using observations from Kepler and TESS.

The traditional approximation of rotation for rapidly rotating stars and planets

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.

The effect of tides on near-core rotation: analysis of 35 Kepler γ Doradus stars in eclipsing and spectroscopic binaries

ABSTRACT We systematically searched for gravity- and Rossby-mode period spacing patterns in Kepler eclipsing binaries with γ Doradus pulsators. These stars provide an excellent opportunity to test the theory of tidal synchronization and angular momentum transport in F- and A-type stars. We discovered 35 systems that show clear patterns, including the spectroscopic binary KIC 10080943. Combined with 45 non-eclipsing binaries with γ Dor components that have been found using pulsation timing, we measured their near-core rotation rates and asymptotic period spacings. We find that many stars are tidally locked if the orbital periods are shorter than 10 d, in which the near-core rotation periods given by the traditional approximation of rotation are consistent with the orbital period. Compared to the single stars, γ Dor stars in binaries tend to have slower near-core rotation rates, likely a consequence of tidal spin-down. We also find three stars that have extremely slow near-core rotation rates. To explain these, we hypothesize that unstable tidally excited oscillations can transfer angular momentum from the star to the orbit, and slow the star below synchronism, a process we refer to as ‘inverse tides’.

Detecting axisymmetric magnetic fields using gravity modes in intermediate-mass stars

Context.Angular momentum (AM) transport models of stellar interiors require improvements to explain the strong extraction of AM from stellar cores that is observed with asteroseismology. One of the frequently invoked mediators of AM transport are internal magnetic fields, even though their properties, observational signatures, and influence on stellar evolution are largely unknown.Aims.We study how a fossil, axisymmetric internal magnetic field affects period spacing patterns of dipolar gravity mode oscillations in main sequence stars with masses of 1.3, 2.0, and 3.0 M⊙. We assess the influence of fundamental stellar parameters on the magnitude of pulsation mode frequency shifts.Methods.We computed dipolar gravity mode frequency shifts due to a fossil, axisymmetric poloidal–toroidal internal magnetic field for a grid of stellar evolution models, varying stellar fundamental parameters. Rigid rotation was taken into account using the traditional approximation of rotation, and the influence of the magnetic field was computed using a perturbative approach.Results.We find magnetic signatures for dipolar gravity mode oscillations in terminal-age main sequence stars that are measurable for a near-core field strength larger than 105G. The predicted signatures differ appreciably from those due to rotation.Conclusions.Our formalism demonstrates the potential for the future detection and characterization of strong fossil, axisymmetric internal magnetic fields in gravity-mode pulsators near the end of core-hydrogen burning fromKeplerphotometry, if such fields exist.

Gravity-mode period spacings and near-core rotation rates of 611 γ Doradus stars with Kepler

Abstract We report our survey of γ Dor stars from the 4-yr Kepler mission. These stars pulsate mainly in g modes and r modes, showing period-spacing patterns in the amplitude spectra. The period-spacing patterns are sensitive to the chemical composition gradients and the near-core rotation, hence they are essential for understanding the stellar interior. We identified period-spacing patterns in 611 γ Dor stars. Almost every star pulsates in dipole g modes, while about 30% of stars also show clear patterns for quadrupole g modes and 16% of stars present r mode patterns. We measure periods, period spacings, and the gradient of the period spacings. These three observables guide the mode identifications and can be used to estimate the near-core rotation rate. We find many stars are hotter and show longer period-spacing patterns than theory. Using the Traditional Approximation of Rotation (TAR), we inferred the asymptotic spacings, the near-core rotation rates, and the radial orders of the g and r modes. Most stars have a near-core rotation rate around 1 d−1and an asymptotic spacing around 4000 s. We also find that many stars rotate more slowly than predicted by theory for unclear reasons. 11 stars show rotational splittings with fast rotation rates. We compared the observed slope–rotation relation with the theory and find a large spread. We detected rotational modulations in 58 stars and used them to derive the core-to-surface rotation ratios. The interiors rotate faster than the cores in most stars, but by no more than 5%.

The traditional approximation of rotation, including the centrifugal acceleration for slightly deformed stars

Context. The traditional approximation of rotation (TAR) is a treatment of the dynamical equations of rotating and stably stratified fluids in which the action of the Coriolis acceleration along the direction of the entropy (and chemicals) stratification is neglected, while assuming that the fluid motions are mostly horizontal because of their inhibition in the vertical direction by the buoyancy 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) that become separable, such as in the non-rotating case, while they are not separable in the case in which the full Coriolis acceleration is taken into account. This approximation, first introduced in geophysical fluid dynamics for thin atmospheres and oceans, has been broadly applied in stellar (and planetary) astrophysics to study low-frequency GIWs that have short vertical wavelengths. The appoximation is now being tested thanks to direct 2D oscillation codes, which constrain its domain of validity. The mathematical flexibility of this treatment allows us to explore broad parameter spaces and to perform detailed seismic modelling of stars. Aims. The TAR treatment is built on the assumptions that the star is spherical (i.e. its centrifugal deformation is neglected) and uniformly rotating while an adiabatic treatment of the dynamics of the waves is adopted. In addition, their induced gravitational potential fluctuations is neglected. However, it has been recently generalised with including the effects of a differential rotation. We aim to carry out a new generalisation that takes into account the centrifugal acceleration in the case of deformed stars that are moderately and uniformly rotating. Methods. We construct an analytical expansion of the equations for the dynamics of GIWs in a spheroidal coordinates system by assuming the hierarchies of frequencies and amplitudes of the velocity components adopted within TAR in the spherical case. Results. We derive the complete set of equations that generalises TAR by taking the centrifugal acceleration into account. As in the case of a differentially rotating spherical star, the problem becomes 2D but can be treated analytically if we assume the anelastic and JWKB approximations, which are relevant for low-frequency GIWs. This allows us to derive a generalised Laplace tidal equation for the horizontal eigenfunctions and asymptotic wave periods, which can be used to probe the structure and dynamics of rotating deformed stars thanks to asteroseismology. A first numerical exploration of its eigenvalues and horizontal eigenfunctions shows their variation as a function of the pseudo-radius for different rotation rates and frequencies and the development of avoided crossings.

Period spacings of gravity modes in rapidly rotating magnetic stars

Context.Stellar magnetic fields are often invoked to explain the missing transport of angular momentum observed in models of stellar interiors. However, the properties of an internal magnetic field and the consequences of its presence on stellar evolution are largely unknown.Aims.We study the effect of an axisymmetric internal magnetic field on the frequency of gravity modes in rapidly rotating stars to check whether gravity modes can be used to detect and probe such a field.Methods.Rotation is taken into account using the traditional approximation of rotation and the effect of the magnetic field is computed using a perturbative approach. As a proof of concept, we compute frequency shifts due to a mixed (i.e. with both poloidal and toroidal components) fossil magnetic field for a representative model of a known magnetic, rapidly rotating, slowly pulsating B-type star: HD 43317.Results.We find that frequency shifts induced by the magnetic field scale with the square of its amplitude. A magnetic field with a near-core strength of the order of 150 kG (which is consistent with the observed surface field strength of the order of 1 kG) leads to signatures that are detectable in period spacings for high-radial-order gravity modes.Conclusions.The predicted frequency shifts can be used to constrain internal magnetic fields and offer the potential for a significant step forward in our interpretation of the observed structure of gravity-mode period spacing patterns in rapidly rotating stars.

Period spacings of γ Doradus pulsators in the Kepler field: Rossby and gravity modes in 82 stars

Rossby modes are the oscillations in a rotating fluid, whose restoring force is the Coriolis force. They provide an additional diagnostic to understand the rotation of stars, which complicates asteroseismic modelling. We report 82 $\gamma$\,Doradus stars for which clear period spacing patterns of both gravity and Rossby modes have been detected. The period spacings of both show a quasi-linear relation with the pulsation period but the slope is negative for the gravity modes and positive for the Rossby modes. Most Rossby modes have $k=-2, m=-1$. For only one star a series of $k=-1,m=-1$ modes is seen. For each pattern, the mean pulsation period, the mean period spacing, and the slope are measured. We find that the slope correlates with the mean period for Rossby mode patterns. The traditional approximation of rotation is used to measure the near-core rotation rate, assuming the star rotates rigidly. We report the near-core rotation rates, the asymptotic period spacings, and the radial orders of excited modes of these 82 main-sequence stars. The near-core rotation rates lie between $0.6\,\mathrm{d^{-1}}$ and $2.3\,\mathrm{d^{-1}}$. Six stars show surface rotation modulations, among which only KIC\,3341457 shows differential rotation while the other five stars have uniform rotations. The radial orders of excited modes show different distributions for the dipole and quadrupole gravity modes versus the Rossby modes.

Deciphering the oscillation spectrum of γ Doradus and SPB stars

Context. The space-based Kepler mission provided four years of highly precise and almost uninterrupted photometry for hundreds of γ Doradus stars and tens of slowly pulsating B-type (SPB) stars, finally allowing us to apply asteroseismology to these gravity mode pulsators. Without rotation, gravity modes are equally spaced in period. This simple structure does not hold in rotating stars for which rotation needs to be taken into account to accurately interpret the oscillation spectrum. Aims. We aim to develop a stellar-model-independent method to analyse and interpret the oscillation spectrum of γ Dor and SPB stars. Methods. Within the traditional approximation of rotation, we highlight the possibility of recovering the equidistance of period spacings by stretching the pulsation periods. The stretching function depends on the degree and azimuthal order of gravity modes and the rotation rate of the star. In this new stretched space, the pulsation modes are regularly spaced by the stellar buoyancy radius. Results. On the basis of this property, we implemented a method to search for these new regularities and simultaneously infer the rotation frequency and buoyancy radius. Tests on synthetic spectra computed with a non-perturbative approach show that we can retrieve these two parameters with reasonable accuracy along with the mode identification. In uniformly rotating models of a typical γ Dor star, and for the most observed prograde dipole modes, we show that the accuracy on the derived parameters is better than 5% on both the internal rotation rate and the buoyancy radius. Finally, we apply the method to two stars of the Kepler field, a γ Dor and an SPB, and compare our results with those of other existing methods. Conclusions. We provide a stellar-model-independent method to obtain the near-core rotation rate, the buoyancy radius, and the mode identification from gravity-mode spectra of γ Dor and SPB stars.