Rotation Law Research Articles

Longitudinal distribution of major solar flares during 1975–2005

We have analyzed the coordinates of optical flares associated with the X-ray flares of class X observed by GOES satellite during the period from January 1975 through September 2005. The results can be summarized as follows. 1. The latitude distribution of X-ray flares of class X displays the Maunder butterfly diagram pattern. 2. The longitude distribution suggests that the place where X-ray flares of class X preferentially occur migrates in Carrington longitude. 3. In the new dynamic reference frame inferred from the differential rotation law on the Sun, the longitude distribution diagram shows that there are two persistent preferred longitudes of strong X-ray flare occurrence separated by about 180°. This suggests that strong X-ray flare active longitudes exist indeed for tens of years. 4. The strength of the two active longitudes alternates similarly to the “flip–flop” phenomenon. 5. The non-axisymmetry of the X-ray flare distribution is found to be highly significant 42.5% (43.0%) for Northern (Southern) hemisphere for the peak intensity and 55% (49%) for Northern (Southern) hemisphere for the flare number.

Advances in Secular Magnetohydrodynamics of Stellar Interiors Dedicated to Asteroseismic Spatial Missions

With the first light of COROT, the preparation of KEPLER and the future helioseismology spatial projects such as GOLF-NG, a coherent picture of the evolution of rotating stars from their birth to their death is needed. We describe here the modelling of the macroscopic transport of angular momentum and matter in stellar interiors that we have undertaken to reach this goal. First, we recall in detail the dynamical processes that are driving these mechanisms in rotating stars and the theoretical advances we have achieved. Then, we present our new results of numerical simulations which allow us to follow in 2D the secular hydrodynamics of rotating stars, assuming that anisotropic turbulence enforces a shellular rotation law. Finally, we show how this work is leading to a dynamical vision of the Hertzsprung-Russel diagram with the support of asteroseismology and helioseismology, seismic observables giving constraints on the modelling of the internal transport and mixing processes. In conclusion, we present the different processes that should be studied in the near future to improve our description of stellar radiation zones.

An asymmetry detected in the disk of κ Canis Majoris with AMBER/VLTI

Aims. We study the geometry and kinematics of the circumstellar environment of the Be star Kappa CMa in the Br gamma emission line and its nearby continuum. Methods. We use the VLTI/AMBER instrument operating in the K band which provides a spatial resolution of about 6 mas with a spectral resolution of 1500 to study the kinematics within the disk and to infer its rotation law. In order to obtain more kinematical constraints we also use an high spectral resolution Pa beta line profile obtain in December 2005 at the Observatorio do Pico do Dios, Brazil and we compile V/R line profile variations and spectral energy distribution data points from the literature. Results. Using differential visibilities and differential phases across the Br gamma line we detect an asymmetry in the disk. Moreover, we found that kappa CMa seems difficult to fit within the classical scenario for Be stars, illustrated recently by alpha Arae observations, i.e. a fast rotating B star close to its breakup velocity surrounded by a Keplerian circumstellar disk with an enhanced polar wind. Finally we discuss the possibility for kappa CMa to be a critical rotator with a Keplerian rotating disk and try to see if the detected asymmetry can be interpreted within the one-armed viscous disk framework.

Magnetorotational instability in electrically driven flow of liquid metal: Spectral analysis of global modes

The spectral magnetohydrodynamics stability of liquid metal differentially rotating in transverse magnetic field is studied numerically by solving the eigenvalue problem with rigid-wall boundary conditions. The equilibrium velocity profile used in calculations corresponds to the electrically driven flow in circular channel with the rotation law Ω(r)∝1∕r2. This type of flow profile is planned to be used in new experimental devices to test the magnetorotational instability (MRI) in the laboratory. Our analysis includes calculations of the eigenfrequency spectra for both axisymmetric (with azimuthal wavenumber m=0) and nonaxisymmetric (m≠0) modes. It is shown that for certain parameters the flow is unstable with respect to MRI with the fastest growth rate corresponding to the axisymmetric mode. For other parameters, the axisymmetric MRI modes can be suppressed and the instability develops only for modes with m≠0.

Doppler imaging of stellar surface structure

Aims. We present the first Doppler images of the bright RS CVn-type binary ζ And. The star is a magnetically active Kl giant with its rotation synchronized to the 17.8-day orbital period. Our revised lithium abundance of log n = 1.2 places ζ And in the vicinity of Li-rich RGB stars but it is nevertheless a Li-normal chromospherically active binary star. The star seems to undergo its first standard dredge-up dilution. Methods. Four consecutive Doppler images were obtained from a continuous 67-night observing run at NSO-McMath in 1996/97. An additional single image was obtained from a continuous 19-night run at KPNO in 1997/98. These unique data allow to compute a small time series of the evolution of the star's surface structure. All line-profile inversions are done with a modified TempMap version that takes into account the non-spherical shape of the star. Representative test reconstructions are performed and demonstrate the code's reliability and robustness. Results. High and low-latitude spot activity was recovered together with an asymmetric polar cap-like feature. The latter dominated the first half of the two-month time series in 1996/97. The second half showed mostly medium-to-high latitude activity and only a fainter polar spot. The coolest areas were restored with a temperature contrast of about 1000 ± 200 K. Some weaker features at equatorial latitudes were also recovered but these could be partially spurious and appear blurred due to imperfect phase coverage. We use our line profiles to reconstruct an average non-sphericity of R pole /R point = 0.96 which would, if not taken into account, mimic a temperature difference pole-to-equator of ≈ 220 K, especially at the phases of quadrature. Finally, we apply two different methods for restoring surface differential rotation and found a weak solar-type rotation law with a shear ΔΩ ≈ 0.95°/day (α = ΔΩ/Ω eq = +0.049 ± 0.003), i.e. roughly a factor of four weaker at a rotation rate roughly 1.5 times faster than the Sun's.

Twisted‐Torus Equilibrium Structures of Magnetic Fields in Magnetized Stars

We have investigated the of the twisted-torus equilibrium structures of the magnetic fields in deformed stars. The Tomimura-Eriguchi scheme for equilibrium structures of uniformly rotating magnetized stars is extended to treat equilibrium configurations of differentially rotating and meridional circulation-free magnetized polytropes with infinite conductivity. Characteristics of the magnetic fields obtained are summarized as follows: The exterior magnetic fields of the stars behave like dipolar poloidal fields, which decrease as r-3 when r → ∞. The interior magnetic fields are a mixture of poloidal and toroidal fields. Their geometric structures are tori of twisted field lines around the symmetry axis and of untwisted poloidal fields that penetrate the surface of the stars to continue to the exterior fields. It has been found that these structures of magnetic fields in stationary magnetic stars are robust and universal in the sense that they are nearly independent of the compressibility of the matter, rotation laws, and degrees of differential rotation. This universality is a consequence of the natural assumption that the electric current should be confined inside the star and of the integrability condition of the basic equation, i.e., that the electric current in the meridional plane is a function of the flux function, whose behavior is governed by a partial differential equation of the elliptic type.

First direct detection of a Keplerian rotating disk around the Be star $\mathsf{\alpha}$ Arae using AMBER/VLTI

Aims. We aim to study the geometry and kinematics of the disk around the Be star $\alpha$ Arae as a function of wavelength, especially across the Br$\gamma$ emission line. The main purpose of this paper is to answer the question about the nature of the disk rotation around Be stars. Methods. We use the VLTI/AMBER instrument operating in the K band which provides a gain by a factor 5 in spatial resolution compared to previous VLTI/MIDI observations. Moreover, it is possible to combine the high angular resolution provided with the (medium) spectral resolution of AMBER to study the kinematics of the inner part of the disk and to infer its rotation law. Results. We obtain for the first time the direct evidence that the disk is in keplerian rotation, answering a question that occurs since the discovery of the first Be star $\gamma$ Cas by father Secchi in 1866. We also present the global geometry of the disk showing that it is compatible with a thin disk + polar enhanced winds modeled with the SIMECA code. We found that the disk around $\alpha$ Arae is compatible with a dense equatorial matter confined in the central region whereas a polar wind is contributing along the rotational axis of the central star. Between these two regions the density must be low enough to reproduce the large visibility modulus (small extension) obtained for two of the four VLTI baselines. Moreover, we obtain that $\alpha$ Arae is rotating very close to its critical rotation. This scenario is also compatible with the previous MIDI measurements.

Surface differential rotation and photospheric magnetic field of the young solar-type star HD 171488 (V889 Her)

[Abstract]: We present spectropolarimetric observations of the young, single early G-dwarf HD 171488. These observations were obtained over a five-night period in 2004 September at the 3.9-m Anglo-Australian Telescope using the SEMPOL spectropolarimeter visitor instrument. Using the technique of least-squares deconvolution to increase the signal-to-noise ratio of the data, we have applied Zeeman Doppler imaging to reconstruct brightness and magnetic surface topologies of the star. The brightness image shows a large polar spot with weaker low- to mid-latitude features, confirming an earlier Doppler imaging observation. The reconstruction of the surface magnetic field shows regions of radial field at all latitudes (except near the pole) and regions of azimuthal field predominantly at high latitudes (60°-70°), with the azimuthal field almost forming a ring around the polar regions. We have incorporated a solar-like differential rotation law into the imaging process to determine the surface differential rotation of cool spots on HD 171488. This gives an equatorial rotation rate of 1.313 ± 0.004 d and a surface shear of dΩ= 0.402 ± 0.044 rad d−1. This means that the equator of HD 171488 laps the poles every ∼16 ± 2 d and that HD 171488 has a photospheric shear approximately seven times the solar value. This is the largest measurement of surface differential rotation yet obtained using the Doppler imaging method and is over twice the value of previously observed early G-dwarfs.

Modeling the Neutral Hydrogen Interstellar Medium: A Better Kinematic Distance Tool

An advanced approach to the kinematic distance method is developed. The method is applicable to second- and third-quadrant Galactic objects with known velocities. It is based on fitting a model of the density and velocity features in an isothermal H I disk to observed H I data. The velocity field of the gas is modeled with a power law for basic circular rotation, underlying noncircular motions from a two-armed density wave pattern. With a reasonable number of adjustable parameters and constraints the model reproduces observations toward many Galactic objects, and accurate distances are found from the modeled velocity field. High-resolution H I spectral line data from the Canadian Galactic Plane Survey (CGPS; Taylor et al.) are used to discriminate clouds from the intercloud medium (the stratum) for which the model is intended. The ability of the model to reproduce these data is demonstrated in one- [Tb(v)] and two- [Tb(l, v)] dimensional fits. Distances to 22 H II regions and SNRs calculated by the fitted velocity field compare extremely well with other kinematically independent distances.

Magneto-rotational instability in differentially rotating liquid metals

We study the stability of Couette flow between two cylinders in the presence of axial magnetic field in local WKB approximation. We find the analytical expression of the critical angular velocity minimized over the wave number and the imposed magnetic field as a function of the measure of deviation of the rotation law from the Rayleigh line. The result found is in a good agreement with the previously known numerical results based on the global analysis. We perform a minimization of the critical Reynolds number over the wave number at fixed magnetic field both analytically and numerically. We show that a compromise between resistive suppression of magneto-rotational instability at weak magnetic field and the increase of the critical Reynolds number with the increase of magnetic field is possible. It takes place at moderate values of magnetic field of order 3 × 10 2 gauss giving the critical Reynolds number of order 4 × 10 4 .

Solar Rotation in the 17th century

Sunspot drawings made by Galileo Galilei in 1612 are used to derive the law of differential rotation at that time. The main interest of the work is during the time of observations, just at the beginning of telescopic observations and some decades before the Maunder Minimum (1645 – 1715), a period where the sunspots almost disappeared from the solar surface. For this purpose we have carried out careful corrections of the different sources of errors derived from the observing technique. By comparing with other results of the same century, a significant difference is only detected by comparing with data corresponding to the deep Maunder Minimum (Paris Observatory drawings). The characteristics of the solar differential rotation, and extrapolating the behavior of solar activity, did not differ before or after the Maunder Minimum. We also include an analysis of hitherto ignored sunspot drawings by N. Bion made in October and November 1672.

On rotation of solar filaments

The study of the differential rotation law is an important issue in solar physics. Earlier studies of filaments found formulae for their differential rotation at stable existence stage. This paper relies on Big Bear Solar Observatory H-alpha filtergrams to estimate angular velocities of quiescent filaments at different stages of existence. Angular velocities are shown to cover a wide range of values. Likely causes of this velocity scatter are discussed.

Pinch instabilities in Taylor-Couette flow

The linear stability of the dissipative Taylor-Couette flow with an azimuthal magnetic field is considered. Unlike ideal flows, the magnetic field is a fixed function of a radius with two parameters only: a ratio of inner to outer cylinder radii, eta, and a ratio of the magnetic field values on outer and inner cylinders, muB. The magnetic field with 0<muB<1/eta stabilizes the flow and is called a stable magnetic field. The current free magnetic field (muB=eta) is the stable magnetic field. The unstable magnetic field, which value (or Hartmann number) exceeds some critical value, destabilizes every flow including flows which are stable without the magnetic field. This instability survives even without rotation. The unstable modes are located into some interval of the axial wave numbers for the flow stable without magnetic field. The interval length is zero for a critical Hartmann number and increases with an increasing Hartmann number. The critical Hartmann numbers and length of the unstable axial wave number intervals are the same for every rotation law. There are the critical Hartmann numbers for m=0 sausage and m=1 kink modes only. The sausage mode is the most unstable mode close to Ha=0 point and the kink mode is the most unstable mode close to the critical Hartmann number. The transition from the sausage instability to the kink instability depends on the Prandtl number Pm and this happens close to one-half of the critical Hartmann number for Pm=1 and close to the critical Hartmann number for Pm=10(-5). The critical Hartmann numbers are smaller for kink modes. The flow stability does not depend on magnetic Prandtl numbers for m=0 mode. The same is true for critical Hartmann numbers for both m=0 and m=1 modes. The typical value of the magnetic field destabilizing the liquid metal Taylor-Couette flow is approximately 10(2) G.

Differential Rotation in Self Gravitating Filaments

The effect of differential rotation on the equilibria of self-gravitating filaments has been examined for isothermal and logatropic equations of state (EOS). Parametric regions for existence of solutions and upper limits for \(\frac{T}{{| W |}}\) (the ratio of the rotational kinetic energy to the gravitational potential energy) have been worked out for the different laws of rotation. Effect of magnetic field has also been discussed.

Rotation- and temperature-dependence of stellar latitudinal differential rotation

More than 600 high resolution spectra of stars with spectral type F and later were obtained in order to search for signatures of differential rotation in line profiles. In 147 stars, the rotation law could be measured, 28 of them are found to be differentially rotating. Comparison to rotation laws in stars of spectral type A reveals that differential rotation sets in at the convection boundary in the HR-diagram; no star that is significantly hotter than the convection boundary exhibits the signatures of differential rotation. Four late A-/early F-type stars close to the convection boundary and at vsini~100 km/s show extraordinarily strong absolute shear at short rotation periods around one day. It is suggested that this is due to their small convection zone depth and that it is connected to a narrow range in surface velocity. Detection frequencies of differential rotation were analyzed in stars with varying temperature and rotation velocity. Measurable differential rotation is more frequent in late-type stars and slow rotators. The strength of absolute shear and differential rotation are examined as functions of the stellar effective temperature and rotation period. The strongest shear is found at rotation periods between two and three days. In slower rotators, the strongest shear at a given rotation rate is given approximately by DOmega_max ~ P^{-1}. In faster rotators, alpha_max and DOmega_max diminish less rapidly. A comparison with differential rotation measurements in stars of later spectral type shows that F-stars exhibit stronger shear than cooler stars do, the upper boundary in absolute shear DOmega with temperature is consistent with the temperature scaling law found in Doppler Imaging measurements.

Modelling the behaviour of an embankment on soft clay with different constitutive models

The paper investigates the effect of constitutive models on the predicted response of a simplified benchmark problem, an embankment on soft soil. The soft soil is assumed to have the properties of POKO clay from Finland and five different constitutive models are used to model the deposit. Two of the models are isotropic models, i.e. the Modified Cam Clay model and the Soft-Soil model. The other models are recently proposed constitutive models that account for plastic anisotropy. The S-CLAY1 and S-CLAY1S models are embedded in a standard elasto-plastic framework and account for anisotropy via a rotational hardening law. In addition, the S-CLAY1S model accounts for bonding and destructuration. In contrast, the Multilaminate Model for Clay (MMC) accounts for plastic anisotropy by utilizing so-called multilaminate framework. The results of numerical simulations show that accounting for anisotropy results in notable differences in the predicted settlements and horizontal movements compared to the predictions using the isotropic models. There are also significant differences in the K0 predictions by the different constitutive models and this has a significant impact on the results. Copyright © 2006 John Wiley & Sons, Ltd.

Gravity-darkening exponents in semi-detached binary systems from their photometric observations. II.

From the light curve analysis of several semi-detached close binary systems, the exponent of the gravity-darkening (GDE) for the Roche lobe filling components has been empirically estimated. The analysis, based on Roche geometry, has been made using the latest improved version of our computer programme. The present method of the light-curve analysis enables simultaneous estimation of the systems' parameters and the gravity-darkening exponents. The reliability of the method has been confirmed by its application to the artificial light curves obtained with a priori known parameters. Further tests with real observations have shown that in the case of well defined light curves the parameters of the system and the value of the gravity- darkening exponent can be reliably estimated. This first part of our analysis presents the results for 9 of the examined systems, that could be briefly summarised as follows: 1) For four of the systems, namely: ZZ Cru, RZ Dra, XZ Sgr and W UMi, there is a very good agreement between empirically estimated and theoretically predicted values for purely radiative and convective envelopes. 2) For the rest five, namely: TT Aur, V Pup, TV Cas, LT Her, and VV UMa, the estimated values of the gravity-darkening exponents were deduced to be larger than theory expects for purely radiative or convective envelopes, respectively. Moreover, it is worthwhile to mention that these values -although greater than theoretical predictions- are: a) Smaller compared to those found by others. b) In two cases, (TV Cas & LT Her), they are normal if some kind of solar type activity (i.e. by employing the Roche model involving a spotted area on the surface of the secondary star) is assumed. 3) The large values derived for the two early type systems TT Aur & V Pup, are very possibly connected with the eects of rotation laws, and in some degree with the large contribution of the radiative pressure in the total potential, leading by that to considerable deviations of the stellar surfaces from the assumed classical Roche geometry. 4) Finally, in the VV UMa case, the estimated value of the gravity-darkening exponent - although almost double than the expected for stars with convective envelopes - still it is low in comparison with that estimated by others who have analysed the same data. But, since for this particular binary the anomalous GDE is not supported by recent simultaneous uvby observations (Lazaro et al. 2002), the estimated higher value may be due to the old observational material used.

Core‐Collapse Supernovae with Nonuniform Magnetic Fields

We perform two-dimensional numerical simulations on the core collapse of a massive star with strong magnetic fields and differential rotations using the numerical code ZEUS-2D. Changing field configurations and laws of differential rotation parametrically, we compute 14 models and investigate the effects of these parameters on the dynamics. In our models we do not solve the neutrino transport but instead employ a phenomenological parametric EOS that takes into account the neutrino emissions. As a result of the calculations, we find that the field configuration plays a significant role in the dynamics of the core if the initial magnetic field is large enough. Models with initially concentrated fields produce more energetic explosions and more prolate shock waves than the uniform field. Quadrupole-like fields produce a remarkably collimated and fast jet, which might be important for gamma-ray bursts (GRBs). The Lorentz forces exerted in the region where the plasma β is less than unity are responsible for these dynamics. The pure toroidal field, on the other hand, does not lead to any explosion or matter ejection. This suggests that the presupernova models, in which toroidal fields are predominant are disadvantageous for the magnetorotation-induced supernova considered here. Models with initially weak magnetic fields do not lead to explosion or matter ejection, either. In these models magnetic fields play no role, as they do not grow on the timescale considered in this paper and the magnetic pressure could be comparable to the matter pressure. This is because the exponential field growth as expected in MRI is not seen in our models. The magnetic field is amplified mainly by field compression and field wrapping in our simulations.

Preferred sunspot longitudes: non-axisymmetry and differential rotation

As recently found, the distribution of sunspots is non-axisymmetric and spot group formation implies the existence of two persistent active longitudes separated by 180 degrees. Here we quantitatively study the non-axisymmetry of sunspot occurrence. In a dynamic reference frame inferred from the differential rotation law, the raw sunspot data show a clear clustering around the persistent active longitudes. The differential rotation describing the dynamic frame is quantified in terms of the equatorial angular velocity and the differential rotation rate, which appear to be significantly different from those for individual sunspots. This implies that the active longitudes are not linked to the depth of sunspot anchoring. In order to quantify the observed effect, we introduce a measure of the non-axisymmetry of the sunspot distribution. The non-axisymmetric component is found to be highly significant, and the ratio of its strength to that of the axisymmetric one is roughly 1:10. This provides additional constraints for solar dynamo models.

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

AbstractThe 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 &lt; 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 ν0.5core, where νcore is the viscosity beneath the convection zone. The penetration is very small if the tachocline is laminar. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)