Differential rotation on rapidly rotating stars

Theories of meridional circulation and differential rotation in stellar convective zones predict trends in surface flow patterns on main-sequence stars that are amenable to direct observational testing. Here I summarise progress made in the last few years in determining surface differential rotation patterns on rapidly-rotating young main-sequence stars of spectral types F, G, K and M. Differential rotation increases strongly with increasing effective temperature along the main sequence. The shear rate appears to increase with depth in the sub-photospheric layers. Tidal locking in close binaries appears to suppress differential rotation, but better statistics are needed before this conclusion can be trusted. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differential rotation of cool active stars

The surface differential rotation of active solar‐type stars can be investigated by means of Doppler and Zeeman‐Doppler Imaging, both techniques enabling one to estimate the short‐term temporal evolution of photospheric structures (cools spots or magnetic regions). After describing the main modeling tools recently developed to guarantee a precise analysis of differential rotation in this framework, we detail the main results obtained for a small number of active G and K fast rotating stars. We evoke in particular some preliminary trends that can be derived from this sample, bearing the promise that major advances in this field will be achieved with the new generation of spectropolarimeters (ESPaDOnS/CFHT, NARVAL/TBL). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling the differential rotation of F stars

We model stellar differential rotation based on the mean-field theory of fluid dynamics. DR is mainly driven by Reynolds stress, which is anisotropic and has a non-diffusive component because the Coriolis force affects the convection pattern. Likewise, the convective heat transport is not strictly radial but slightly tilted towards the rotation axis, causing the polar caps to be slightly warmer than the equator. This drives a flow opposite to that caused by differential rotation and so allows the system to avoid the Taylor-Proudman state. Our model reproduces the rotation pattern in the solar convection zone and allows predictions for other stars with outer convection zones. The surface shear turns out to depend mainly on the spectral type and only weakly on the rotation rate. We present results for stars of spectral type F which show signs of very strong differential rotation in some cases. Stars just below the mass limit for outer convection zones have shallow convection zones with short convective turnover times. We find solar-type rotation and meridional flow patterns at much shorter rotation periods and horizontal shear much larger than on the solar surface, in agreement with recent observations. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differential rotation of cool active stars: the case of intermediate rotators

In this paper, we present a new method for measuring the surface differential rotation of cool stars with rotation periods of a few days, for which the sparse phase coverage achievable from single-site observations generally prevents the use of more conventional techniques. The basic idea underlying this new analysis is to obtain the surface differential rotation pattern that minimizes the information content of the reconstructed Doppler image through a simultaneous fit of all available data.
Simulations demonstrate that the performance of this new method in the case of cool stars is satisfactory for a variety of observing strategies. Differential rotation parameters can be recovered reliably as long as the total data set spans at least 4 per cent of the time for the equator to lap the pole by approximately one complete cycle. We find in particular that these results hold for potentially complex spot distributions (as long as they include a mixture of low- and high-latitude features), and for various stellar inclination angles and rotation velocities. Such measurements can be obtained from either unpolarized or polarized data sets, provided their signal-to-noise ratio is larger than approximately 500 and 5000 per 2 km s⁻¹ spectral bin, respectively.
This method should therefore be very useful for investigating differential rotation in a much larger sample of objects than what has been possible up to now, and should hence give us the opportunity of studying how differential rotation reacts to various phenomena operating in stellar convective zones, such as tidal effects or dynamo magnetic field generation.     

Differential rotation of relativistic superfluid in neutron stars

It is shown how to set up a mathematically elegant and fully relativistic superfluid model that can provide a realistic approximation (neglecting small anisotropies due to crust solidity, magnetic fields, etc., but allowing for the regions with vortex pinning) of the global structure of a rotating neutron star, in terms of just two independently moving constituents. One of these represents the differentially rotating neutron superfluid, while the other part represents the combination of all the other ingredients, including the degenerate electrons, the superfluid protons in the core, and the ions in the crust, the electromagnetic interactions of which will tend to keep them locked together in a state of approximately rigid rotation. Order of magnitude estimates are provided for relevant parameters such as the resistive drag coefficient.     

Differential rotation of giant stars

From a set of high-resolution spectral observations of late type giant stars we used Doppler imaging to derive time-series temperature maps of the stellar surfaces. Using these temperature maps, it is possible to track the temporal changes of the spot features and derive estimates of the strength and sign of the differential surface rotation of these stars. Looking into the latitudinal changes of the surface maps, it is also possible to derive meridional flows on these stars. But due to the lower accuracy of the latitudes of the reconstructed spot features, the data requirements are higher than for the detection of differential rotation. Nevertheless, a correlation between the differential rotation and meridional flow estimates is suggested. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rapid differential rotation in intermediate mass stars

ZAMS models with internal differential rotation are computed for 6.5 M_⊙ and 10 M_⊙ models with the 2D evolution code ROTORC. Two applications of these models, their pulsational frequencies and deduced locations in the HR diagram, are discussed in order to show how the internal differential rotation could be investigated. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differential rotation and meridional flow in the solar convection zone and beneath

The influence of the basic rotation on anisotropic and inhomogeneous turbulence is discussed in the context of differential rotation theory. An improved representation for the original turbulence leads to a Λ‐effect which complies with the results of 3D numerical simulations. The resulting rotation law and meridional flow agree well with both the surface observations (∂Ω/∂r < 0 and meridional flow towards the poles) and with the findings of helioseismology. The computed equatorward flow at the bottom of convection zone has an amplitude of about 10 m/s and may be significant for the solar dynamo. The depth of the meridional flow penetration into the radiative zone is proportional to ν^0.5_core, where ν_core is the viscosity beneath the convection zone. The penetration is very small if the tachocline is laminar. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differential rotation of some HK-Project stars and the butterfly diagrams

We analyze the long-term variability of the chromospheric radiation of 20 stars monitored in the course of the HK-Project at the Mount Wilson Observatory. We apply the modified wavelet algorithm for this set of gapped time series. Besides the mean rotational periods for all these stars, we find reliable changes of the rotational periods from year to year for a few stars. Epochs of slower rotation occur when the activity level of the star is high, and the relationship repeats again during the next maximum of an activity cycle. Such an effect is traced in two stars with activity cycles that are not perfectly regular (but labeled “Good” under the classification in [Baliunas, S.L., Donahue, R.A., Soon, W.H., Horne, J.H., Frazer, J., Woodard-Eklund, L., Bradford, M., Rao, L.M., Wilson, O.C., Zhang, Q. et al., 1995. ApJ 438, 269.]) but the two stars have mean activity levels exceed that of the Sun. The averaged rotational period of HD 115404 is 18.5 days but sometimes the period increases up to 21.5 days. The sign of the differential rotation is the same as the Sun’s, and the value ΔΩ/Ω=-0.14. For the star HD 149661, this ratio is −0.074. Characteristic changes of rotational periods occur over around three years when the amplitude of the rotational modulation is large. These changes can be transformed into latitude-time butterfly diagrams with minimal a priori assumptions. We compare these results with those for the Sun as a star and conclude that epochs when surface inhomogeneities rotate slower are synchronous with the reversal of the global magnetic dipole.     

Differential rotation and meridional flow of Arcturus

The spectroscopic variability of Arcturus hints at cyclic activity cycle and differential rotation. This could provide a test of current theoretical models of solar and stellar dynamos. To examine the applicability of current models of the flux transport dynamo to Arcturus, we compute a mean‐field model for its internal rotation, meridional flow, and convective heat transport in the convective envelope. We then compare the conditions for dynamo action with those on the Sun. We find solar‐type surface rotation with about 1/10th of the shear found on the solar surface. The rotation rate increases monotonically with depth at all latitudes throughout the whole convection zone. In the lower part of the convection zone the horizontal shear vanishes and there is a strong radial gradient. The surface meridional flow has maximum speed of 170 m/s and is directed towards the equator at high and towards the poles at low latitudes. Turbulent magnetic diffusivity is of the order 10¹⁵–10¹⁶ cm²/s. The conditions on Arcturus are not favorable for a circulation‐dominated dynamo (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the time dependence of differential rotation in young late-type stars

A model for the angular momentum transfer within the convection zone of a rapidly rotating star is introduced and applied to the analysis of recent observations of temporal fluctuations of the differential rotation on the young late-type stars AB Doradus (AB Dor) and LQ Hydrae (LQ Hya). Under the hypothesis that the mean magnetic field produced by the stellar dynamo rules the angular momentum exchanges and that the angular velocity depends only on the distance s from the rotation axis and the time, the minimum azimuthal Maxwell stress  | B_sB _φ|  , averaged over the convection zone, is found to range from ∼0.04 to  ∼0.14 T²  . If the poloidal mean magnetic field   B _s   is of the order of 0.01 T, as indicated by the Zeeman–Doppler imaging maps of those stars, then the azimuthal mean field   B _φ  can reach an intensity of several teslas, which significantly exceeds equipartition with the turbulent kinetic energy. Such strong fields can account also for the orbital period modulation observed in cataclysmic variables and RS Canum Venaticorum systems with a main-sequence secondary component. Moreover, the model allows us to compute the kinetic energy dissipation rate during the maintenance of the differential rotation. Only in the case of the largest surface shear observed on LQ Hya may the dissipated power exceed the stellar luminosity, but the lack of a sufficient statistic on the occurrence of such episodes of large shear does not allow us to estimate their impact on the energy budget of the convection zone.     

Extrasolar planets and the rotation and axisymmetric mass-loss of evolved stars

I examine the implications of the recently found extrasolar planets on the planet-induced axisymmetric mass-loss model for the formation of elliptical planetary nebulae (PNe). This model attributes the low departure from spherical mass-loss of upper asymptotic giant branch (AGB) stars to envelope rotation which results from deposition of orbital angular momentum of the planets. Since about half of all PNe are elliptical, i.e., have low equatorial to polar density contrast, it was predicted that about 50 per cent of all Sun-like stars have Jupiter-like planets around them, i.e., a mass about equal to that of Jupiter, M _J, or more massive. In the light of the new findings that only 5 per cent of Sun-like stars have such planets, and a newly proposed mechanism for axisymmetric mass-loss, the cool magnetic spots model, I revise this prediction. I predict that indeed ∼50 per cent of PN progenitors do have close planets around them, but the planets can have much lower masses, as low as ∼0.01 M _J, in order to spin-up the envelopes of AGB stars efficiently. To support this claim, I follow the angular momentum evolution of single stars with main-sequence mass in the range of 1.3–2.4 M_⊙ , as they evolve to the post-AGB phase. I find that single stars rotate much too slowly to possess any significant non-spherical mass-loss as they reach the upper AGB. It seems, therefore, that planets, in some cases even Earth-like planets, are sufficient to spin-up the envelope of these AGB stars for them to form elliptical PNe. The prediction that on average several such planets orbit each star, as in the Solar system, still holds.     

Physical properties of active stars and stellar systems

We study spectra of 18 active stars and stellar systems, drawn from lists of known active binaries or newly discovered extreme-ultraviolet and soft X-ray sources. Using spectral synthesis and template modelling, we measure (often for the first time) a variety of physical properties of the stars, including spectral types, luminosities and v  sin  i values. We report direct observation of previously undetected secondary components in three systems, and refine the orbital elements for two of these (HR 7275 and HD 217344).     

Dark matter and rotation curves of stars in galaxies

The dark matter accretion theory (around a central body) of the author on the basis of his 5‐dimensional Projective Unified Field Theory (PUFT) is applied to the orbital motion of stars around the center of the Galaxy. The departure of the motion from Newtonian mechanics leads to approximately flat rotation curves being in rough accordance with the empirical facts. The spirality of the motion is investigated.     

On the relation between activity and rotation in main-sequence stars

Es wird ein modifizierter Index für das chromosphärische Aktivitätsniveau bei Sternen, A'_Call, vorgeschlagen. Dieses wird auf Grund der in den H und K Ca II-Linienkernen gemessenen Strahlungsflußdichte berechnet, ist aber im Unterschied zu anderen bekannten Indizes frei vom Einfluß der Sternfarbe. Die Anwendung des neuen Indexes führt zu einer modifizierten Aktivität-Rotation-Beziehung bei Sternen im unteren Teil der Hauptreihe. Es ist nachgewiesen, daß im Koordinatensystem log A'_Call, log Ro (Ro ist die Rossby Zahl) F-M-Sterne einer allgemeinen Beziehung mit dem Parameter α = 1.6 gehorchen. Die Kurve weist keine Zeichen von Aktivitätssättigung bei Sternen mit niedrigen Rossby-Zahlen auf, aber das Aktivitätsniveau strebt gegen einen konstanten Wert bei langsam rotierenden Sternen mit dünnen Konvektionszonen.     

Toroidal versus poloidal magnetic fields in Sun-like stars: a rotation threshold

From a set of stellar spectropolarimetric observations, we report the detection of surface magnetic fields in a sample of four solar-type stars, namely HD 73350, HD 76151, HD 146233 (18 Sco) and HD 190771. Assuming that the observed variability of polarimetric signal is controlled by stellar rotation, we establish the rotation periods of our targets, with values ranging from 8.8 d (for HD 190771) to 22.7 d (for HD 146233). Apart from rotation, fundamental parameters of the selected objects are very close to the Sun's, making this sample a practical basis to investigate the specific impact of rotation on magnetic properties of Sun-like stars.
We reconstruct the large-scale magnetic geometry of the targets as a low-order  (ℓ < 10)  spherical harmonic expansion of the surface magnetic field. From the set of magnetic maps, we draw two main conclusions. (i) The magnetic energy of the large-scale field increases with rotation rate. The increase in chromospheric emission with the mean magnetic field is flatter than observed in the Sun. Since the chromospheric flux is also sensitive to magnetic elements smaller than those contributing to the polarimetric signal, this observation suggests that a larger fraction of the surface magnetic energy is stored in large scales as rotation increases. (ii) Whereas the magnetic field is mostly poloidal for low rotation rates, more rapid rotators host a large-scale toroidal component in their surface field. From our observations, we infer that a rotation period lower than ≈12 d is necessary for the toroidal magnetic energy to dominate over the poloidal component.     

A new look at the relationship between activity, dynamo number and Rossby number in late-type stars

The correlation between stellar activity, as measured by the indicator Δ R _HK, and the Rossby number Ro in late-type stars is revisited in light of recent developments in solar dynamo theory. Different stellar interior models, based on both mixing-length theory and the full spectrum of turbulence, are used in order to see to what extent the correlation of activity with Rossby number is model dependent, or otherwise can be considered universal. Although we find some modest model dependence, we find that the correlation of activity with Rossby number is significantly better than with rotation period alone for all the models we consider. Dynamo theory suggests that activity should scale with the dynamo number. A current model of the solar dynamo, the so-called interface dynamo, proposes that the amplification of the toroidal magnetic field by differential rotation (the ω -effect) and the production of the poloidal magnetic field from toroidal by helical turbulence (the α -effect) take place in different, adjacent layers near the base of the convection zone. A new scale analysis based on the interface dynamo shows that the appropriate dynamo number does not depend on the Rossby number alone, but also depends on an additional dimensionless factor related to the differential rotation. This leads to a new interpretation of the correlation between activity and Rossby number, which in turn leads to some conclusions about the magnitude of differential rotation in the dynamo layers of late-type main-sequence stars.