Issue 
A&A
Volume 636, April 2020



Article Number  A100  
Number of page(s)  7  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201937398  
Published online  27 April 2020 
Period spacings of gravity modes in rapidly rotating magnetic stars
II. The case of an oblique dipolar fossil magnetic field
^{1}
AIM, CEA, CNRS, Université ParisSaclay, Université Paris Diderot, Sorbonne Paris Cité, 91191 GifsurYvette, France
email: vincent.prat@cea.fr
^{2}
LESIA, Paris Observatory, PSL University, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France
^{3}
Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
^{4}
Dept. of Astrophysics, IMAPP, Radboud University Nijmegen, 6500 GL Nijmegen, The Netherlands
^{5}
Max Planck Institute for Astronomy, Koenigstuhl 17, 69117 Heidelberg, Germany
Received:
23
December
2019
Accepted:
12
March
2020
Context. Stellar internal magnetic fields have recently been shown to leave a detectable signature on period spacing patterns of gravity modes.
Aims. We aim to investigate the effect of the obliquity of a mixed (poloidal and toroidal) dipolar internal fossil magnetic field with respect to the rotation axis on the frequency of gravity modes in rapidly rotating stars.
Methods. We used the traditional approximation of rotation to compute nonmagnetic modes, and a perturbative treatment of the magnetic field to compute the corresponding frequency shifts. We applied the new formalism to HD 43317, a magnetic, rapidly rotating, slowly pulsating Btype star, whose field has an obliquity angle of about 80°.
Results. We find that frequency shifts induced by the magnetic field on highradialorder gravity modes are larger with increasing obliquity angle, when the magnetic axis is closer to the equatorial region, where these modes are trapped. The maximum value is reached for an obliquity angle of 90°. This trend is observed for all mode geometries.
Conclusions. Our results predict that the signature of an internal oblique dipolar magnetic field is detectable using asteroseismology of gravity modes.
Key words: asteroseismology / waves / stars: magnetic field / stars: oscillations / stars: rotation
© V. Prat et al. 2020
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.
1. Introduction
Thanks to highprecision space photometry with missions such as CoRoT (Baglin et al. 2006), Kepler (Borucki et al. 2010), K2 (Howell et al. 2014), and now TESS (Ricker et al. 2015), the study of stellar pulsations has rapidly developed over the last 15years. It has provided important new information on the internal structure of stars through seismic modelling (see Aerts et al. 2019, for a recent review). In parallel, the observational study of magnetism in stars, for example, through spectropolarimetry from groundbased facilities, has provided major results on surface magnetic fields and rotation (e.g. Donati & Landstreet 2009; Neiner et al. 2015; Wade et al. 2016). Studying both physical processes (pulsations and magnetism) at the same time offers further opportunities. However, this new combined technique, called magnetoasteroseismology, requires the identification of diagnostic tools.
As a first step, in Prat et al. (2019; hereafter Paper I), we investigated the effect of an axisymmetric mixed (with both poloidal and toroidal components) magnetic field on gravitoinertial modes computed in the traditional approximation of rotation (TAR; e.g. Lee & Saio 1997; Townsend 2003; Bouabid et al. 2013). Theoretical and numerical studies have shown that the stability of fossil magnetic fields requires such a mixed configuration (e.g. Tayler 1980; Braithwaite & Spruit 2004; Braithwaite 2008; Duez et al. 2010a). We found that a dipolar field with a nearcore strength of order 100 kG, corresponding to a few kG at the surface as typically observed in approximately 10% of hot stars (Shultz et al. 2019), shifts the frequencies of gravity modes, and modifies the period spacing patterns, which have been used to probe nearcore mixing and rotation (Van Reeth et al. 2015a,b; Ouazzani et al. 2017; Christophe et al. 2018), in a way that should be detectable in current observations.
However, spectropolarimetric surveys show that most stars with a fossil magnetic field have a tilted magnetic axis with respect to the rotation axis (e.g. Grunhut et al. 2015). This obliquity may affect the frequency shifts induced by the magnetic field. Therefore, in the present paper, we address the effect of an oblique mixed magnetic field on the frequency of gravitoinertial modes.
2. Frequency shifts
In the present work, we consider a magnetic field that is weak enough that the effect of the unperturbed Lorentz force on the hydrostatic equilibrium state is negligible (e.g. Duez et al. 2010b). We used the perturbation theory for rotating stars presented in Paper I to compute the frequency shifts induced by the perturbed Lorentz force:
where μ_{0} is the vacuum permeability, B is the largescale magnetic field, and δB are the fluctuations of the magnetic field due to the oscillation displacement. Those are given in the adiabatic case by the induction equation
where ξ_{0} is the unperturbed displacement of the mode. The frequency shifts induced by the magnetic field read
where ρ is the density of the background model, ω_{0} is the unperturbed angular frequency of the mode in the corotating frame, Ω is the rotation vector, the scalar product is defined by
and the asterisk (^{*}) denotes the complex conjugate.
In the TAR, the horizontal component of the rotation vector is neglected (Ω ≃ Ω cos θe_{r}), and unperturbed eigenmodes for gravitoinertial waves are given in spherical coordinates (r, θ, φ) by
where m is the azimuthal order, e_{r}, e_{θ}, and e_{φ} are radial, latitudinal, and azimuthal unit vectors, and H_{r}, H_{θ}, and H_{φ} are radial, latitudinal, and azimuthal Hough functions, respectively (Hough 1898; Lee & Saio 1997; Townsend 2003). Their precise definitions are given in Paper I.
In the present study, we consider the magnetic field as a small pertubation of a rapidly rotating system. Thus, the oscillation axis can be approximated as the rotation axis. In the magnetic frame, which is inclined by an angle β with respect to the rotation axis as illustrated in Fig. 1, the magnetic field reads:
Fig. 1. Sketch of the field geometry. The magnetic field lines (decomposed into poloidal and toroidal components) are drawn in red. An example of a critical latitude for gravitoinertial waves is drawn in blue. 

Open with DEXTER 
where B_{0} is the strength of the magnetic field, b_{r}, b_{θ}, and b_{φ} are radial functions that model realistic stable fossil configurations (Paper I, Duez et al. 2010a), and the spherical coordinates in the magnetic frame (e_{r}, e_{θ}′, e_{φ}′) are (r, θ′,φ′). In the corotating frame, this leads to
where
is axisymmetric (with the axis aligned with the polar axis), and
is fully oblique (with the axis in the equatorial plane).
Because of the dependence of B_{eq} on φ, when computing ⟨ξ_{0}, δF_{L}/ρ⟩, the integral over φ cancels out mixed terms in B_{pol} and B_{eq}, thus leading to
where and are the perturbed Lorentz forces based on B_{pol} and B_{eq} only, respectively. For highradialorder modes, the eigenfunctions are rapidly oscillating in the radial direction, and the dominant terms in the full expansion given in Appendix A yield
where R is the stellar radius,
with x = r/R,
and ν = 2Ω/ω_{0} is the spin factor. Equation (11) shows that the magnetic frequency shifts scale with the square of the field strength, as in the axisymmetric case. In addition, Eq. (13) implies that the dependence on β of the frequency shifts induced by the magnetic field is of the form
3. Application to HD 43317
Similarly to Paper I, we now compute the new frequency shifts for a representative stellar model of HD 43317, which is a rapidly rotating, magnetic, slowly pulsating Btype star exhibiting g modes (Buysschaert et al. 2018). The adopted model has a mass of 5.8 M_{⊙}, a radius of 3.39 R_{⊙}, an effective temperature of 17 822 K, a solarlike metallicity, and a central hydrogen mass fraction of 0.54, which corresponds to an age of 28.4 Myr. The identified gmode frequencies of HD 43317range from 0.69162 d^{−1} to 5.00466 d^{−1}. The rotation period of the star is P_{rot} = 0.897673(4)d (Pápics et al. 2012), which corresponds to 33% of the Keplerian angular velocity. Its mostly dipolar surface magnetic field has a strength of 1312 ± 332 G and an obliquity angle of 81 ± 6° (Buysschaert et al. 2018). However, we investigate here how the frequency shifts vary with the obliquity. Therefore, we consider the latter as a variable of the problem hereafter. For nonaxisymmetric modes, it is important to distinguish between the frequency in the corotating frame ω and the frequency in the inertial frame ω_{i}, where
For nonaxisymmetric magnetic fields, the observed frequencies are different from the intrinsic frequencies in the inertial frame, due to the fact that the rotation axis is not exactly the same as the oscillation axis. However, since we consider only the perturbative effect of the magnetic field while rotation is treated nonperturbatively, we choose to neglect this effect and impose that the oscillation axis is the same as the rotation axis.
In this section, we investigate the effect of an oblique magnetic field on zonal (Sect. 3.1), prograde (Sect. 3.2), and retrograde (Sect. 3.3) dipole modes. The impact of rotation is discussed in Sect. 3.4.
3.1. Zonal modes
Figure 2 represents for ℓ = 1 and m = 0 the period spacing ΔP between modes of consecutive radial orders as a function of the period P = 2π/ω_{i}. The first result is that the period spacing patterns for an oblique field are very similar to those obtained in the axisymmetric case. Indeed, the dips related to nearcore chemical gradients are transformed at long periods into a sawtoothlike pattern. In addition, it shows that oblique fields induce stronger signatures in the period spacing patterns than axisymmetric fields, with a maximum for an obliquity angle of 90°. This can be explained by the fact that longperiod gravity modes, which are most affected by the magnetic field, are trapped near the equatorial plane, as illustrated in Fig. 1. These modes are mainly sensitive to the radial component of the magnetic field, as shown by Eq. (12), which is maximal on the magnetic axis. Therefore, the expected signature is stronger when the magnetic axis is closer to the equatorial plane, with a maximum for an obliquity angle of 90°. For the n = −69 mode, the dependence of the frequency shift induced by the magnetic field on β, as predicted by Eq. (14), is shown in Fig. 3.
Fig. 2. Period spacings of g modes with radial orders from −1 (left) to −74 (right) as a function of the period in the inertial frame for ℓ = 1, m = 0, B_{0} = 10^{5} G, and different obliquity angles. The vertical black bar on the left represents a typical observational error bar of 250 s (Van Reeth et al. 2015b). 

Open with DEXTER 
Fig. 3. Relative frequency shift induced by the magnetic field of the mode ℓ = 1, m = 0, n = −69 as a function of the obliquity angle for B_{0} = 10^{5} G. This mode has an unperturbed period of about 2.12 d. 

Open with DEXTER 
As illustrated in Fig. 4, the enhancement of magnetic frequency shifts obtained when increasing the obliquity angle is similar to the effect of a stronger magnetic field. However, it is not strictly equivalent. Indeed, the scaling of the frequency shifts induced by the magnetic field with the period for a larger obliquity angle is different from the scaling found for a stronger field. In theory, it is thus possible to distinguish between the two effects if enough modes are observed (and correctly identified).
Fig. 4. Same as Fig. 2 (m = 0), but with different values of the field strength and the obliquity angle. 

Open with DEXTER 
3.2. Prograde modes
Figure 5 represents the period spacings for m = 1 between modes of consecutive radial orders as a function of the oscillation period. We observe that oblique magnetic fields have a stronger impact on prograde modes than axisymmetric magnetic fields, similarly to the case of zonal modes. However, their period spacings go to zero more rapidly for high radial orders than for those of zonal modes, which makes magnetic signatures potentially more difficult to detect. This difficulty might be counterbalanced by the fact that prograde sectoral dipole (ℓ = m = 1) g modes are predominantly observed in rapidly rotating intermediatemass stars (see e.g. Van Reeth et al. 2016; Moravveji et al. 2016; Pápics et al. 2017; Li et al. 2019a,b, 2020).
Fig. 5. Same as Fig. 2, but for m = 1. 

Open with DEXTER 
3.3. Retrograde modes
Figure 6 represents the period spacings for m = −1 between modes of consecutive radial orders as a function of the oscillation period. Because of the Doppler shift, retrograde modes are split into two branches. For m = −1, this split occurs at ω = Ω (in the corotating frame). The impact of the magnetic field is negligible on the loworder branch, while it is clearly visible on the highorder branch. Again, oblique fields have stronger signatures than axisymmetric ones. Most of the modes identified in the spectrum of HD 43317 are retrograde modes, but they are all on the loworder branch, and only weak signatures are thus expected. This is consistent with the absence of a detected observational signature of a magnetic field.
Fig. 6. Same as Fig. 2, but for m = −1. 

Open with DEXTER 
3.4. Influence of rotation
We now investigate the effect of the rotation rate on the frequency shifts induced by oblique fields. To do so, we computed modes and the associated frequency shifts induced by the magnetic field with 50% and 150% of Ω_{⋆}, the measured rotation rate of HD 43317. This corresponds to 17% and 50% of the Keplerian angular velocity, respectively. For ℓ = 1 and m = 0, we obtain the period spacings plotted in Fig. 7. Similarly to the axisymmetric case, the frequency shift induced by the magnetic field for a given mode (at a given radial order) is larger at the lower rotation rate, due to the fact that the mode has a larger period and is thus more sensitive to the magnetic field, according to Eq. (11). In contrast, however, for modes at different rotation rates but close to the same given period, the frequency shift induced by the magnetic field is larger for the larger rotation rate when the obliquity is large. This comes from the more efficient equatorial trapping of largeperiod modes for larger rotation rates.
Fig. 7. Same as Fig. 2 (m = 0), for two different rotation rates: Ω = 0.5Ω_{⋆} (Ω/Ω_{K} = 0.17) and Ω = 1.5Ω_{⋆} (Ω/Ω_{K} = 0.50), where Ω_{K} is the Keplerian angular velocity. 

Open with DEXTER 
Since oblique fields can generate much larger frequency shifts, they require a smaller amplitude to generate negative period spacing values (visible in Fig. 7 for the lower rotation rate). Such negative values mean that the perturbative treatment of the magnetic field is no longer valid. Such frequency shifts would indeed lead to avoided crossings between consecutive modes, and the description of such interactions between modes requires a nonperturbative treatment (e.g. Morsink & Rezania 2002; Mathis & de Brye 2011) of the magnetic field (see also the discussion in Van Beeck et al. 2020).
4. Discussion and conclusions
The analysis presented here shows that the results of Paper I for an axisymmetric, mixed, internal largescale magnetic field are still valid for the more realistic case of an oblique field. The magnetic effect on the oscillation frequencies of gravitoinertial modes in the traditional approximation of rotation indeed modifies the period spacing patterns. The major new result of this study is that the signatures of the presence of a magnetic field in the period spacing patterns are stronger for an oblique field compared to an axisymmetric field, typically up to a factor ten when the field is fully inclined. This is because the magnetic axis is closer to the equatorial region, where lowfrequency modes are trapped. As a consequence, weaker fields can be detected if they are inclined. This is true for all mode geometries, including prograde sectoral modes, which are predominantly observed in rapidly rotating stars, and retrograde modes, which are predominantly observed in the particular case of HD 43317 (Buysschaert et al. 2018). The change in the patterns due to the strength of the field and to its obliquity can be distinguished from each other if enough modes are identified in the appropriate frequency regime, especially if several period spacing patterns for different mode geometries are observed. In addition, these signatures are more easily observed if the star rotates slowly. As a consequence, the search for patterns like those shown here would allow us to indirectly discover internal magnetic fields from observations of gravity modes in hot stars.
The influence of various stellar parameters, such as the mass, the age, and the metallicity on the expected magnetic signatures is investigated in an extensive parallel paper (Van Beeck et al. 2020). The main result of this study is that magnetic signatures are detectable in Kepler data and should be more easily detectable for stars near the terminalage main sequence. A dedicated search for magnetic signatures such as those predicted in this work has not yet been undertaken in Kepler data for intermediatemass gmode pulsators. Additionally, many hot stars have recently been observed with the TESS mission (Bowman et al. 2019; Pedersen et al. 2019). The identification and detailed seismic modelling of pulsating magnetic hot stars will provide observational constraints on the internal properties of their magnetic field. In addition, including magnetic effects on oscillation frequencies will also provide much improved constraints on stellar interiors for earlytype stars. This would be an important new step into the recent domain of magnetoasteroseismology, but such observational detections have not been made yet. The new TESS samples of pulsating OB stars are promising in this respect (DavidUraz et al. 2019; Neiner et al., in prep.).
To progress even further on magnetoseismic diagnostics, a nonperturbative treatment of the magnetic field would be needed to be able to consider (i) a magnetic field that is not confined inside the star, and (ii) stronger fields such as those of Ap stars. Moreover, the formalism presented here could be applied to Rossby modes, which are observed in many intermediatemass stars (Van Reeth et al. 2016; Saio et al. 2018; Li et al. 2019b), to provide additional constraints on internal magnetic fields of stars.
Acknowledgments
The research leading to these results received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreements No. 647383: SPIRE with PI S.M. and No. 670519: MAMSIE with PI C.A.). V.P. and S.M. acknowledge support from the CNES PLATO grant at CEA/DAp. The authors thank the anonymous referee for useful comments.
References
 Aerts, C., Mathis, S., & Rogers, T. M. 2019, ARA&A, 57, 35 [NASA ADS] [CrossRef] [Google Scholar]
 Baglin, A., Auvergne, M., Barge, P., et al. 2006, in The CoRoT Mission PreLaunch Status  Stellar Seismology and Planet Finding, eds. M. Fridlund, A. Baglin, J. Lochard, L. Conroy, ESA Spec. Pub., 1306, 33 [NASA ADS] [Google Scholar]
 Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Bouabid, M.P., Dupret, M.A., Salmon, S., et al. 2013, MNRAS, 429, 2500 [NASA ADS] [CrossRef] [Google Scholar]
 Bowman, D. M., Burssens, S., Pedersen, M. G., et al. 2019, Nat. Astron., 3, 760 [NASA ADS] [CrossRef] [Google Scholar]
 Braithwaite, J. 2008, MNRAS, 386, 1947 [NASA ADS] [CrossRef] [Google Scholar]
 Braithwaite, J., & Spruit, H. C. 2004, Nature, 431, 819 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Buysschaert, B., Aerts, C., Bowman, D. M., et al. 2018, A&A, 616, A148 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Christophe, S., Ballot, J., Ouazzani, R.M., Antoci, V., & Salmon, S. J. A. J. 2018, A&A, 618, A47 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 DavidUraz, A., Neiner, C., Sikora, J., et al. 2019, MNRAS, 487, 304 [NASA ADS] [CrossRef] [Google Scholar]
 Donati, J.F., & Landstreet, J. D. 2009, ARA&A, 47, 333 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Duez, V., Braithwaite, J., & Mathis, S. 2010a, ApJ, 724, L34 [Google Scholar]
 Duez, V., Mathis, S., & TurckChièze, S. 2010b, MNRAS, 402, 271 [NASA ADS] [CrossRef] [Google Scholar]
 Grunhut, J. H., & Neiner, C. 2015, in Polarimetry, eds. K. N. Nagendra, S. Bagnulo, R. Centeno, & M. Jesús Martínez González, IAU Symp., 305, 53 [Google Scholar]
 Hough, S. S. 1898, Philos. T. Roy. Soc. Lon. A, 191, 139 [NASA ADS] [CrossRef] [Google Scholar]
 Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398 [NASA ADS] [CrossRef] [Google Scholar]
 Lee, U., & Saio, H. 1997, ApJ, 491, 839 [NASA ADS] [CrossRef] [Google Scholar]
 Li, G., Bedding, T. R., Murphy, S. J., et al. 2019a, MNRAS, 482, 1757 [NASA ADS] [CrossRef] [Google Scholar]
 Li, G., Van Reeth, T., Bedding, T. R., Murphy, S. J., & Antoci, V. 2019b, MNRAS, 487, 782 [NASA ADS] [CrossRef] [Google Scholar]
 Li, G., Van Reeth, T., Bedding, T. R., et al. 2020, MNRAS, 491, 3586 [NASA ADS] [CrossRef] [Google Scholar]
 Mathis, S., & de Brye, N. 2011, A&A, 526, A65 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Moravveji, E., Townsend, R. H. D., Aerts, C., & Mathis, S. 2016, ApJ, 823, 130 [NASA ADS] [CrossRef] [Google Scholar]
 Morsink, S. M., & Rezania, V. 2002, ApJ, 574, 908 [NASA ADS] [CrossRef] [Google Scholar]
 Neiner, C., Mathis, S., Alecian, E., et al. 2015, in Polarimetry, eds. K. N. Nagendra, S. Bagnulo, R. Centeno & M. Jesús Martínez González, IAU Symp., 305, 61 [NASA ADS] [Google Scholar]
 Ouazzani, R.M., Salmon, S. J. A. J., Antoci, V., et al. 2017, MNRAS, 465, 2294 [NASA ADS] [CrossRef] [Google Scholar]
 Pápics, P. I., Briquet, M., Baglin, A., et al. 2012, A&A, 542, A55 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Pápics, P. I., Tkachenko, A., Van Reeth, T., et al. 2017, A&A, 598, A74 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Pedersen, M. G., Chowdhury, S., Johnston, C., et al. 2019, ApJ, 872, L9 [NASA ADS] [CrossRef] [Google Scholar]
 Prat, V., Mathis, S., Buysschaert, B., et al. 2019, A&A, 627, A64 (Paper I) [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, J. Astron. Telesc. Instrum. Syst., 1, 014003 [NASA ADS] [CrossRef] [Google Scholar]
 Saio, H., Kurtz, D. W., Murphy, S. J., Antoci, V. L., & Lee, U. 2018, MNRAS, 474, 2774 [NASA ADS] [CrossRef] [Google Scholar]
 Shultz, M. E., Wade, G. A., Rivinius, T., et al. 2019, MNRAS, 490, 274 [NASA ADS] [CrossRef] [Google Scholar]
 Tayler, R. J. 1980, MNRAS, 191, 151 [NASA ADS] [Google Scholar]
 Townsend, R. H. D. 2003, MNRAS, 340, 1020 [NASA ADS] [CrossRef] [Google Scholar]
 Van Beeck, J.,Prat, V., Van Reeth, T., et al. 2020, A&A, submitted [Google Scholar]
 Van Reeth, T., Tkachenko, A., Aerts, C., et al. 2015a, A&A, 574, A17 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Van Reeth, T., Tkachenko, A., Aerts, C., et al. 2015b, ApJS, 218, 27 [NASA ADS] [CrossRef] [Google Scholar]
 Van Reeth, T., Tkachenko, A., & Aerts, C. 2016, A&A, 593, A120 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Wade, G. A., Neiner, C., Alecian, E., et al. 2016, MNRAS, 456, 2 [NASA ADS] [CrossRef] [Google Scholar]
Appendix A: Nonzeroaverage terms of the Lorentz work
In this section, we list all nonzeroaverage terms of the work of the Lorentz force induced by a totally oblique dipolar field (for convenience, we take here the product with μ_{0}). Interestingly, all these terms involve either only poloidal components of the magnetic field, or only the toroidal component. Therefore, we group them accordingly. All terms have a purely radial part multiplied by a purely latitudinal part, and the prime symbol (′) denotes a total derivative, either radial or latitudinal depending on the context.
A.1. Poloidal terms
Noting A = [(rb_{θ})′ + b_{r}], the terms involving poloidal components are
A.2. Toroidal terms
The terms that involve the toroidal component are
All Figures
Fig. 1. Sketch of the field geometry. The magnetic field lines (decomposed into poloidal and toroidal components) are drawn in red. An example of a critical latitude for gravitoinertial waves is drawn in blue. 

Open with DEXTER  
In the text 
Fig. 2. Period spacings of g modes with radial orders from −1 (left) to −74 (right) as a function of the period in the inertial frame for ℓ = 1, m = 0, B_{0} = 10^{5} G, and different obliquity angles. The vertical black bar on the left represents a typical observational error bar of 250 s (Van Reeth et al. 2015b). 

Open with DEXTER  
In the text 
Fig. 3. Relative frequency shift induced by the magnetic field of the mode ℓ = 1, m = 0, n = −69 as a function of the obliquity angle for B_{0} = 10^{5} G. This mode has an unperturbed period of about 2.12 d. 

Open with DEXTER  
In the text 
Fig. 4. Same as Fig. 2 (m = 0), but with different values of the field strength and the obliquity angle. 

Open with DEXTER  
In the text 
Fig. 5. Same as Fig. 2, but for m = 1. 

Open with DEXTER  
In the text 
Fig. 6. Same as Fig. 2, but for m = −1. 

Open with DEXTER  
In the text 
Fig. 7. Same as Fig. 2 (m = 0), for two different rotation rates: Ω = 0.5Ω_{⋆} (Ω/Ω_{K} = 0.17) and Ω = 1.5Ω_{⋆} (Ω/Ω_{K} = 0.50), where Ω_{K} is the Keplerian angular velocity. 

Open with DEXTER  
In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.