Stellar Rotation Rate Research Articles

Characterizing the features of the low-amplitude peaks in delta Scuti stars with TESS

Abstract The presence of low-amplitude peaks over the noise in the power spectra of δ Scuti stars is frequently disregarded. These seemingly insignificant peaks, collectively referred to as grass, might contain valuable information about the origin of these stars and the reasons behind the occurrence or absence of a plateau. It is crucial to systematically parameterize the grass phenomenon throughout a comprehensive sample that covers the entire δ Scuti star parameter range. Thus, we conduct a quantitative study of long-duration, high-duty-cycle TESS light curves, leading to improved detection methods for plateaus and a deeper understanding of their nature. This approach minimizes the impact of unresolved peaks caused by mode variations over time. Additionally, we present appropriate analysis techniques to mitigate window effects and identify and eliminate spurious peaks. We demonstrate here that the grass can be effectively parameterized based on peak density. With such parameterization two distinct regimes are found: the sparse grass regime, characterized by low peak density and the absence of a plateau in the power spectra, and the dense grass regime, characterized by high peak densities and the presence of an observable plateau. Our study is the first rigorous quantification of the emergence of such a plateau in the power spectra of δ Scuti stars. Since the grass might be related with fractality, mode variability, and stellar rotation rate, its parameterization opens a new way to analyze these stars.

The Role of Meridional Flow in the Generation of Solar/Stellar Magnetic Fields and Cycles

Meridional flow is crucial in generating the solar poloidal magnetic field by facilitating poleward transport of the field from decayed bipolar magnetic regions (BMRs). As the meridional circulation changes with the stellar rotation rate, the properties of stellar magnetic cycles are expected to be influenced by this flow. In this study, we explore the role of meridional flow in generating magnetic fields in the Sun and Sun-like stars using the STABLE (surface flux transport and Babcock–Leighton) dynamo model. We find that a moderate meridional flow increases the polar field by efficiently driving the trailing polarity flux toward the pole, while a strong flow tends to transport both polarities of BMRs poleward, potentially reducing the polar field. Our findings are in perfect agreement with what one can expect from the surface flux transport model. Similarly, the toroidal field initially increases with moderate flow speeds and then decreases beyond a certain value. This trend is due to the competitive effects of shearing and diffusion. Furthermore, our study highlights the impact of meridional flow on the strength and duration of stellar cycles. By including the meridional flow from a mean-field hydrodynamics model in STABLE, we show that the magnetic field strength initially increases with the stellar rotation rate and then declines in rapidly rotating stars, offering an explanation of the observed variation of stellar magnetic field with rotation rate.

A study of the electrostatic properties of the interiors of low-mass stars: Possible implications for the observed rotational properties

Context. In the partially ionized material of stellar interiors, the strongest forces acting on electrons and ions are the Coulomb interactions between charges. The dynamics of the plasma as a whole depend on the magnitudes of the average electrostatic interactions and the average kinetic energies of the particles that constitute the stellar material. An important question is how these interactions of real gases are related to the observable stellar properties. Specifically, the relationships between rotation, magnetic activity, and the thermodynamic properties of stellar interiors are still not well understood. These connections are crucial for understanding and interpreting the abundant observational data provided by space-based missions, such as Kepler/K2 and TESS, and the future data from the PLATO mission. Aims. In this study, we investigate the electrostatic effects within the interiors of low-mass main sequence (MS) stars. Specifically, we introduce a global quantity, a global plasma parameter, which allows us to compare the importance of electrostatic interactions across a range of low-mass theoretical models (0.7 − 1.4 M⊙) with varying ages and metallicities. We then correlate the electrostatic properties of the theoretical models with the observable rotational trends on the MS. Methods. We use the open-source 1D stellar evolution code MESA to compute a grid of main-sequence stellar models. Our models span the log g − Teff space of a set of 66 Kepler main-sequence stars. Results. We identify a correlation between the prominence of electrostatic effects in stellar interiors and stellar rotation rates. The variations in the magnitude of electrostatic interactions with age and metallicity further suggest that understanding the underlying physics of the collective effects of plasma can clarify key observational trends related to the rotation of low-mass stars on the MS. These results may also advance our understanding of the physics behind the observed weakened magnetic braking in stars.

Differential Rotation of CoRoT Stars and a Kepler Binary Star from Starspot Transit Mapping

Abstract While direct magnetic field measurements are rare for slowly rotating stars, spots observed during planetary transits provide a potential indicator of magnetic activity on stellar surfaces. Moreover, the rotation of the stellar surface can be probed by monitoring the spots’ position with time in subsequent transits. This study investigates the dynamic interplay of rotational shear and stellar rotation rate in six stars of spectral types F to M, all hosting exoplanets observed by the CoRoT space mission, except for one binary star from Kepler. The analysis, facilitated by the ECLIPSE code, unveils the physical properties of stellar spots, including radius, intensity, temperature, and position. The five CoRoT stars exhibit spot characteristics consistent with those observed in solar type stars. The determination of a spot longitude during different transits allows for the inference of the star’s differential rotation profile, revealing a decreasing trend of rotational shear with the mean stellar rotation period, given by Δ Ω ∝ P ¯ − 0.9 . This implies that, at least for slow rotators (mean rotation period >5 days), as stars age, their differential rotation decreases. Additionally, the six stars analyzed here seem to fall into two categories, according to their spot flux deficit: those with ΔF spot > 0.005 and those with ΔF spot < 0.002.

Exploring the Effects of Stellar Magnetism on the Potential Habitability of Exoplanets

Considerable interest has centered on Earth-like planets orbiting in the circumstellar habitable zone (CHZ) of its star. However, the potential habitability of an exoplanet depends upon a number of additional factors, including the presence and strength of any planetary magnetic field and the interaction of this field with that of the host star. Not only must the exoplanet have a strong enough magnetic field to shield against stellar activity, but it must also orbit far enough from the star to avoid direct magnetic connectivity. We characterize stellar activity by the star’s Rossby number, Ro, the ratio of stellar rotation rate to convective turnover time. We employ a scaled model of the solar magnetic field to determine the star’s Alfvén radius, the distance at which the stellar wind becomes super-Alfvénic. Planets residing within the Alfvén surface may have a direct magnetic connection to the star and therefore not be the most viable candidates for habitability. Here, we determine the Rossby number of a sample of 1053 exoplanet-hosting stars for which the rotation rates have been observed and for which a convective turnover time can be calculated. We find that 84 exoplanets in our sample have orbits which lie inside the CHZ and that also lie outside the star’s Alfvén surface: 34 of these have been classified as terran (11) or superterran (23) planets. Applying the Alfvén surface habitability criterion yields a subset of the confirmed exoplanets that may be optimal targets for future observations in the search for signatures of life.

Rotational Evolution of Classical T Tauri Stars: Models and Observations

We developed a grid of stellar rotation models for low-mass and solar-type classical T Tauri stars (CTTS; 0.3M ⊙ < M * < 1.2M ⊙). These models incorporate the star–disk interaction and magnetospheric ejections to investigate the evolution of the stellar rotation rate as a function of the mass of the star M *, the magnetic field (B *), and stellar wind ( Ṁwind ). We compiled and determined stellar parameters for 208 CTTS, such as projected rotational velocity vsin(i) , mass accretion rate Ṁacc , stellar mass M *, ages, and estimated rotational periods using Transiting Exoplanet Survey Satellite (TESS) data. We also estimated a representative value of the mass-loss rate for our sample using the [O i] λ 6300 spectral line. Our results confirm that vsin(i) measurements in CTTS agree with the rotation rates provided by our spin models in the accretion-powered stellar winds picture. In addition, we used the approximate Bayesian computation technique to explore the connection between the model parameters and the observational properties of CTTS. We find that the evolution of vsin(i) with age might be regulated by variations in (1) the intensity of B * and (2) the fraction of the accretion flow ejected in magnetic winds, removing angular momentum from these systems. The youngest stars in our sample (∼1 Myr) show a median branching ratio Ṁwind/Ṁacc∼0.16 and median B * ∼ 2000 G, in contrast to ∼0.01 and 1000 G, respectively, for stars with ages ≳3 Myr.

2.5-MHD models of circumstellar discs around FS CMa post-mergers: II. Stationary accretion stage

ABSTRACT We study the star–disc interaction in the presence of the strong magnetic field (B⋆ = 6.2 kG) of a slowly rotating star. This situation describes a post-merger of the spectral type B and has not been previously investigated. We perform a set of resistive and viscosity 2.5D-magnetohydrodynamical simulations using the PLUTO code. Based on our previous work, we consider the initial gas disc density ρd0 = 10−13 g cm−3 since it describes the conditions around IRAS 17449+2320 well. We find that the fall of gas towards the star occurs in the mid-plane, and remarkably, intermittent backflow takes place in the mid-plane in all of our models for R ≥ 10R⋆. However, we do not rule out that the funnel effect may occur and cause the accretion closer to the poles. Also, when larger values of viscosity (αν = 1) and stellar rotation rate (δ⋆ = 0.2) are considered, we find that the disc exhibits a thickening which is characteristic of FS CMa-type stellar objects. Additionally, we find that the poloidal magnetic field lines twist over short periods of time, leading to magnetic reconnection causing coronal heating that could explain the presence of the Raman lines found observationally in several FS CMa stars. Lastly, we find the formation of several knots in the magnetic field lines near and in the mid-plane of the disc which produce perturbations in the density and velocity components, as well as the formation of shallow gaps whose position depends on the inflation of the magnetic field lines.

Rotational evolution of young-to-old stars with data-driven three-dimensional wind models

ABSTRACT Solar-type stars form with a wide range of rotation rates Ω. A wide Ω range persists until a stellar age of t ∼ 0.6 Gyr, after which solar-type stars exhibit Skumanich spin-down where Ω ∝ t−1/2. Rotational evolution models incorporating polytropic stellar winds struggle to simultaneously reproduce these two regimes, namely the initially wide Ω range and the Skumanich spin-down without imposing an a priori cap on the wind mass-loss rate. We show that a three-dimensional wind model driven by Alfvén waves and observational data yields wind torques that agree with the observed age distribution of Ω. In our models of the Sun and 27 open cluster stars aged from 0.04 to 0.6 Gyr that have observationally derived surface magnetic maps and rotation rates, we find evidence of exponential spin-down in young stars that are rapid rotators and Skumanich spin-down for slow rotators. The two spin-down regimes emerge naturally from our data-driven models. Our modelling suggests that the observed age distribution of stellar rotation rates Ω arises as a consequence of magnetic field strength saturation in rapid rotators.

Double Trouble: Two Transits of the Super-Earth GJ 1132 b Observed with JWST NIRSpec G395H

The search for rocky planet atmospheres with JWST has focused on planets transiting M dwarfs. Such planets have favorable planet-to-star size ratios, enhancing the amplitude of atmospheric features. Since the expected signal strength of atmospheric features is similar to the single-transit performance of JWST, multiple observations are required to confirm any detection. Here, we present two transit observations of the rocky planet GJ 1132 b with JWST NIRSpec G395H, covering 2.8–5.2 μm. Previous Hubble Space Telescope WFC3 observations of GJ 1132 b were inconclusive, with evidence reported for either an atmosphere or a featureless spectrum based on analyses of the same data set. Our JWST data exhibit substantial differences between the two visits. One transit is consistent with either an H2O-dominated atmosphere containing ∼1% CH4 and trace N2O () or stellar contamination from unocculted starspots (). However, the second transit is consistent with a featureless spectrum. Neither visit is consistent with a previous report of HCN. Atmospheric variability is unlikely to explain the scale of the observed differences between the visits. Similarly, our out-of-transit stellar spectra show no evidence of changing stellar inhomogeneity between the two visits—observed 8 days apart, only 6.5% of the stellar rotation rate. We further find no evidence of differing instrumental systematic effects between visits. The most plausible explanation is an unlucky random noise draw leading to two significantly discrepant transmission spectra. Our results highlight the importance of multivisit repeatability with JWST prior to claiming atmospheric detections for these small, enigmatic planets.

Trends in torques acting on the star during a star-disk magnetospheric interaction

Aims. We assess the modification of angular momentum transport in various configurations of star-disk accreting systems based on numerical simulations with different parameters. In particular, we quantify the torques exerted on a star by the various components of the flow and field in our simulations of a star-disk magnetospheric interaction. Methods. In a suite of resistive and viscous numerical simulations, we obtained results using different stellar rotation rates, dipole magnetic field strengths, and resistivities. We probed a part of the parameter space with slowly rotating central objects, up to 20% of the Keplerian rotation rate at the equator. Different components of the flow in star-disk magnetospheric interaction were considered in the study: a magnetospheric wind (i.e., the “stellar wind”) ejected outwards from the stellar vicinity, matter infalling onto the star through the accretion column, and a magnetospheric ejection launched from the magnetosphere. We also took account of trends in the total torque in the system and in each component individually. Results. We find that for all the stellar magnetic field strengths, B⋆, the anchoring radius of the stellar magnetic field in the disk is extended with increasing disk resistivity. The torque exerted on the star is independent of the stellar rotation rate, Ω⋆, in all the cases without magnetospheric ejections. In cases where such ejections are present, there is a weak dependence of the anchoring radius on the stellar rotation rate, with both the total torque in the system and torque on the star from the ejection and infall from the disk onto the star proportional to Ω⋆B3. The torque from a magnetospheric ejection is proportional to Ω⋆4. Without the magnetospheric ejection, the spin-up of the star switches to spin-down in cases involving a larger stellar field and faster stellar rotation. The critical value for this switch is about 10% of the Keplerian rotation rate.

Magnetized rotating isothermal winds

ABSTRACT We consider the general problem of a Parker-type non-relativistic isothermal wind from a rotating and magnetic star. Using the magnetohydrodynamics code athena++, we construct an array of simulations in the stellar rotation rate Ω* and the isothermal sound speed cT, and calculate the mass, angular momentum, and energy loss rates across this parameter space. We also briefly consider the 3D case, with misaligned magnetic and rotation axes. We discuss applications of our results to the spin-down of normal stars, highly irradiated exoplanets, and to nascent highly magnetic and rapidly rotating neutron stars born in massive star core-collapse.

Non-LTE Monte Carlo radiative transfer – III. The thermal properties of tilted and warped Be star discs

ABSTRACT We use the three-dimensional Monte Carlo radiative transfer code hdust to model Be stars where the disc is tilted from the equatorial plane of the star. We compute 128 models across four spectral types, B0, B2, B5, and B8, tilting the disc by 0, 10○, 20○, and 40○, respectively, while varying disc density according to spectral type. We also compute every model for an average and high stellar rotation rate. We first discuss non-tilted disc temperatures and show its nonlinear dependence on stellar and disc parameters. We find that tilting the disc minimally affects the density-weighted average disc temperature, but tilting does create a temperature asymmetry in disc cross-sections, which is more pronounced for a faster rotation rate. We also investigate the effect tilting has on V-band magnitude, polarization, and the H$\rm \alpha$ line. Tilting the disc does affect these observables, but the changes are entirely dependent on the position of the observer relative to the direction of tilt. We find the observables that distinguish tilting from a change in density or geometry are the H$\rm \alpha$ line shapes, where it can transition between single-peaked and double-peaked, and the polarization position angle, whose value is dependent on the projected major elongation axis of the disc on the sky. We also present one early- and one late-type model with warped discs. We find their temperature structure varies a small amount from the uniformly tilted models, and the different observables correspond to different tilt angles, consistent with their expected volume of origin within the disc.

The role of tidal interactions in the formation of slowly rotating early-type stars in young star clusters

ABSTRACT The split main sequences found in the colour–magnitude diagrams of star clusters younger than ∼600 Myr are suggested to be caused by the dichotomy of stellar rotation rates of upper main-sequence stars. Tidal interactions have been suggested as a possible explanation of the dichotomy of the stellar rotation rates. This hypothesis proposes that the slow rotation rates of stars along the split main sequences are caused by tidal interactions in binaries. To test this scenario, we measured the variations in the radial velocities of slowly rotating stars along the split main sequence of the young Galactic cluster NGC 2422 (∼90 Myr) using spectra obtained at multiple epochs with the Canada–France–Hawai’i Telescope. Our results show that most slowly rotating stars are not radial velocity variables. Using the theory of dynamical tides, we find that the binary separations necessary to fully or partially synchronize our spectroscopic targets, on time-scales shorter than the cluster age, predict much larger radial velocity variations across multiple-epoch observations, or a much larger radial velocity dispersion at a single epoch, than the observed values. This indicates that tidal interactions are not the dominant mechanism to form slowly rotating stars along the split main sequences. As the observations of the rotation velocity distribution among B- and A-type stars in binaries of larger separations hint at a much stronger effect of braking with age, we discuss the consequences of relaxing the constraints of the dynamical tides theory.

Living with a Red Dwarf: The Rotation–Age Relationships of M Dwarfs

Age is a fundamental stellar property, yet for many stars, it is difficult to reliably determine. For M dwarfs, it has been notoriously so. Due to their lower masses, core hydrogen fusion proceeds at a much slower rate in M dwarfs than it does in more massive stars like the Sun. As a consequence, more customary age determination methods (e.g., isochrones and asteroseismology) are unreliable for M dwarfs. As these methods are unavailable, many have searched for reliable alternatives. The M dwarfs comprise the overwhelming majority of the nearby stellar inventory, which makes the determination of their fundamental parameters even more important. Further, an ever-increasing number of exoplanets are being found to orbit M dwarfs, and recent studies have suggested that they may have a relatively higher number of low-mass planets than other spectral types. Determining the ages of M dwarfs then allows us to better study any hosted exoplanets as well. Fortunately, M dwarfs possess magnetic activity and stellar winds like other cool dwarf stars. This causes them to undergo the spin-down effect (rotating with longer periods) as they age. For this reason, stellar rotation rate has been considered a potentially powerful age determination parameter for over 50 yr. Calibrating reliable age–rotation relationships for M dwarfs has been a lengthy process, but here we present the age–rotation relationships for ∼M0–6.5 dwarfs, determined as part of the Living with a Red Dwarf program. These relationships should prove invaluable for a wide range of stellar astrophysics and exoplanetary science applications.

On the spatial distribution of electron energy loss due to gyro-cooling in hot star magnetospheres

ABSTRACT Hot magnetic stars often exhibit incoherent circularly polarized radio emission thought to arise from gyro-synchrotron emission by energetic electrons trapped in the circumstellar magnetosphere. Theoretical scalings for electron acceleration by magnetic reconnection driven by centrifugal breakout match well the empirical scalings for observed radio luminosity with both the magnetic field strength and the stellar rotation rate. This paper now examines how energetic electrons introduced near the top of closed magnetic loops are subsequently cooled by the energy loss associated with their gyro-synchrotron radio emission. For sample assumed distributions for energetic electron deposition about the loop apex, we derive the spatial distribution of the radiated energy from such ‘gyro-cooling’. For sub-relativistic electrons, we show explicitly that this is independent of the input energy, but also find that even extensions to the relativistic regime still yield a quite similar spatial distribution. However, cooling by Coulomb collisions with even a modest ambient density of thermal electrons can effectively quench the emission from sub-relativistic electrons, indicating that the observed radio emission likely stems from relativistic electrons that are less affected by such collisional cooling. The overall results form an initial basis for computing radio emission spectra in future models that account for such cooling and multimode excitation about the fundamental gyro-frequency. Though motivated in the context of hot stars, the basic results here could also be applied to gyro-emission in any dipole magnetospheres, including those of ultra-cool dwarfs and even (exo)-planets.

Discrepant Approaches to Modeling Stellar Tides and the Blurring of Pseudosynchronization

We examine the reasons for discrepancies between two alternative approaches to modeling small-amplitude tides in binary systems. The direct solution (DS) approach solves the governing differential equations and boundary conditions directly, while the modal decomposition (MD) approach relies on a normal-mode expansion. Applied to a model for the primary star in the heartbeat system KOI-54, the two approaches predict quite different behavior of the secular tidal torque. The MD approach exhibits the pseudosynchronization phenomenon, where the torque due to the equilibrium tide changes sign at a single, well-defined, and theoretically predicted stellar rotation rate. The DS approach instead shows “blurred” pseudosynchronization, where positive and negative torques intermingle over a range of rotation rates. We trace a major source of these differences to an incorrect damping coefficient in the profile functions describing the frequency dependence of the MD expansion coefficients. With this error corrected, some differences between the approaches remain; however, both are in agreement that pseudosynchronization is blurred in the KOI-54 system. Our findings generalize to any type of star for which the tidal damping depends explicitly or implicitly on the forcing frequency.

Accretion disc backflow in resistive MHD simulations

ABSTRACT We investigate accretion onto a central star, with the size, rotation rate, and magnetic dipole of a young stellar object, to study the flow pattern (velocity and density) of the fluid within and outside of the disc. We perform resistive magnetohydrodynamic (MHD) simulations of thin discs, varying the parameters such as the stellar rotation rate and (anomalous) coefficients of viscosity and resistivity in the disc. To provide a benchmark for the results and to compare them with known analytic results, we also perform purely hydrodynamic (HD) simulations for the same problem. Although obtained for different situations with differing inner boundary condition, the disc structure in the HD simulations closely follows the analytic solution of Kluzniak and Kita – in particular, a region of ‘mid-plane’ backflow exists in the right range of radii, depending on the viscosity parameter. In the MHD solutions, whenever the magnetic Prandtl number does not exceed a certain critical value, the mid-plane backflow exists throughout the accretion disc, extending all the way down to the foot point of the accretion funnel flow where the disc transitions to a magnetic funnel flow. For values of the magnetic Prandtl number close to the critical value the backflow and the inner disc undergo a quasi-periodic radial oscillation, otherwise the backflow is steady, as is the disc solution. From our results, supplemented by our reading of the literature, we suggest that mid-plane backflow is a real, physical, and not only numerical feature of at least some accretion discs.

How tidal waves interact with convective vortices in rapidly rotating planets and stars

Context. The dissipation of tidal inertial waves in planetary and stellar convective regions is one of the key mechanisms that drive the evolution of star–planet and planet–moon systems. This dissipation is particularly efficient for young low-mass stars and gaseous giant planets, which are rapid rotators. In this context, the interaction between tidal inertial waves and turbulent convective flows must be modelled in a realistic and robust way. In the state-of-the-art simulations, the friction applied by convection on tidal waves is commonly modeled as an effective eddy viscosity. This approach may be valid when the characteristic length scales of convective eddies are smaller than those of the tidal waves. However, it becomes highly questionable in the case where tidal waves interact with potentially stable large-scale vortices such as those observed at the poles of Jupiter and Saturn. The large-scale vortices are potentially triggered by convection in rapidly-rotating bodies in which the Coriolis acceleration forms the flow in columnar vortical structures along the direction of the rotation axis. Aims. We investigate the complex interactions between a tidal inertial wave and a columnar convective vortex. Methods. We used a quasi-geostrophic semi-analytical model of a convective columnar vortex, which is validated by numerical simulations. First, we carried out linear stability analysis using both numerical and asymptotic Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) methods. We then conducted linear numerical simulations of the interactions between a convective columnar vortex and an incoming tidal inertial wave. Results. The vortex we consider is found to be centrifugally stable in the range –Ωp ≤ Ω0 ≤ 3.62Ωp and unstable outside this range, where Ω0 is the local rotation rate of the vortex at its center and Ωp is the global planetary (stellar) rotation rate. From the linear stability analysis, we find that this vortex is prone to centrifugal instability with perturbations with azimuthal wavenumbers m = {0,1, 2}, which potentially correspond to eccentricity, obliquity, and asynchronous tides, respectively. The modes with m &gt; 2 are found to be neutral or stable. The WKBJ analysis provides analytic expressions of the dispersion relations for neutral and unstable modes when the axial (vertical) wavenumber is sufficiently large. We verify that in the unstable regime, an incoming tidal inertial wave triggers the growth of the most unstable mode of the vortex. This would lead to turbulent dissipation. For stable convective columns, the wave-vortex interaction leads to the mixing of momentum for tidal inertial waves while it creates a low-velocity region around the vortex core and a new wave-like perturbation in the form of a progressive wave radiating in the far field. The emission of this secondary wave is the strongest when the wavelength of the incoming wave is close to the characteristic size (radius) of the vortex. Incoming tidal waves can also experience complex angular momentum exchanges locally at critical layers of stable vortices. Conclusions. The interaction between tidal inertial waves and large-scale coherent convective vortices in rapidly-rotating planets (stars) leads to turbulent dissipation in the unstable regime and complex behaviors such as mixing of momentum and radiation of new waves in the far field or wave-vortex angular momentum exchanges in the stable regime. These phenomena cannot be modeled using a simple effective eddy viscosity.

Nonanalytic behavior of the relativistic r -modes in slowly rotating neutron stars

An inconsistency between the theoretical analysis and numerical calculations of the relativistic $r$-modes puzzles the neutron star community since the Kojima's finding of the continuous part in the $r$-mode oscillation spectrum in 1997. In this paper, after a brief review of the Newtonian $r$-mode theory and of the literature devoted to the continuous spectrum of $r$-modes, we apply our original approach to the study of relativistic oscillation equations. Working within the Cowling approximation, we derive the general equations, governing the dynamics of discrete relativistic $r$-modes for both barotropic (isentropic) and nonbarotropic stars. A detailed analysis of the obtained equations in the limit of extremely slow stellar rotation rate reveals that, because of the effect of inertial reference frame-dragging, the relativistic $r$-mode eigenfunctions and eigenfrequencies become {\it non-analytic} functions of the stellar angular velocity, $\Omega$. We also derive the explicit expressions for the $r$-mode eigenfunctions and eigenfrequencies for very small values of $\Omega$. These expressions explain the asymptotic behavior of the numerically calculated eigenfrequencies and eigenfunctions in the limit $\Omega\to 0$. All the obtained $r$-mode eigenfrequencies take discrete values in the frequency range, usually associated with the continuous part of the spectrum. No indications of the continuous spectrum, at least in the vicinity of the Newtonian $l=m=2$ $r$-mode frequency $\sigma=-4/3 \ \Omega$, are found.

On the Origin of the Dichotomy of Stellar Activity Cycles

The presence of possible correlations between stellar rotation rate Ω and the frequency of the activity cycle ω cyc is still much debated. We implement a new Bayesian classification algorithm based on a simultaneous regression analysis of multiple scaling laws and we demonstrate the existence of two different scalings in the – plane for an extended Mt. Wilson sample of 67 stars. Thanks to metallicity measurements obtained from both ESA Gaia and high-resolution spectroscopy, we argue that the origin of this dichotomy is likely related to the chemical composition: stars whose magnetic cycle frequency increases with rotation rate are less metallic than stars whose magnetic cycle frequency decreases with stellar rotation rates. On the contrary, no clear difference in chromospheric magnetic activity indicators characterizes the two branches.