Protodiscs around hot magnetic rotator stars

We develop equations and obtain solutions for the structure and evolution of a protodisc region that is initially formed with no radial motion and super-Keplerian rotation speed when wind material from a hot rotating star is channelled towards its equatorial plane by a dipole-type magnetic field. Its temperature is around 10⁷ K because of shock heating and the inflow of wind material causes its equatorial density to increase with time. The centrifugal force and thermal pressure increase relative to the magnetic force and material escapes at its outer edge. The protodisc region of a uniformly rotating star has almost uniform rotation and will shrink radially unless some instability intervenes. In a star with angular velocity increasing along its surface towards the equator, the angular velocity of the protodisc region decreases radially outwards and magnetorotational instability (MRI) can occur within a few hours or days. Viscosity resulting from MRI will readjust the angular velocity distribution of the protodisc material and may assist in the formation of a quasi-steady disc. Thus, the centrifugal breakout found in numerical simulations for uniformly rotating stars does not imply that quasi-steady discs with slow outflow cannot form around magnetic rotator stars with solar-type differential rotation.     

Shaping of Planetary Nebulae by Dust Driven AGB Winds

We have investigated the interacting winds model (IWM) in which the shapes of elliptical Planetary Nebulae (PNe) can be explained by the asymmetric mass loss produced by a rotating AGB star. The mass loss mechanism is based on a dust driven wind calculated for stationary situations. Already for small rotation rates of the AGB star we get a significantly angle-dependent mass loss which is concentrated towards the equatorial plane. This pole to equator density variation in the space surrounding the star influences the shape of the later developed PN. We compare these theoretical shapes with observed PNe and for some objects with well known quantities our model can fit the observations quite well. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of rotation on hot star winds

We discuss the structure of a radiatively-driven wind from a rapidly rotating hot star. When the rotation rate is large, there is a region at low latitudes near the stellar surface where the force of gravity is larger than the radiation pressure. Within this region, the streamlines fall toward the equator, and if the rotation rate is large enough, the fluid collides with the flow from the opposite hemisphere of the star. The shock compression and subsequent cooling produces a dense equatorial disk. This wind-compressed disk forms only if the star is rotating fast enough. The rotation threshold for disk formation is about 70% of the break-up speed for B stars and is much higher for O stars. If theoretical calculations of the terminal speed are correct, then the behavior of the disk formation threshold as a function of spectral type potentially explains the frequency distribution of Be stars. The geometry of the wind-compressed disk agrees quite well with observations of Be stars; however, the disk density is a factor of 100 too small to explain the magnitude of the IR excess, optical polarization, and H emission, if current UV mass-loss rates are correct. However, recent X-ray observations indicate that the mass-loss rates of B stars may be much larger than previously thought.     

Differential rotation and giant cell circulation of solar Ca+-network

We report new results obtained from high precision computer controlled tracings of ca. 400 bright Ca⁺-mottles made during summer 1975 in continuation of our 1974 program (Schröter and Wöhl, 1975). In particular, we looked in 1975 for the existence of a giant circulation pattern in the equatorial zone. We find for the differential rotation: = 13.93 – 2.90 sin² B (deg/day, sidereal) when combining the new measurements with those obtained in 1974. Observations from 26th April until June 19th give strong evidence that at that time four giant circulation cells, crossing the solar equator, (i.e. a nonaxisymmetric velocity field pattern with respect to the solar equator) did exist. This yields two more rapid and two slower rotating sectors with v = ±80 m s^–1. These giant cells transport angular momentum towards the equator.     

A theory of differntial rotation based on the discussion of turbulent transport of angular momentum

This paper deals with the theory of the solar rotational law. We assume the turbulence to be of the largest influence compared with the momentum flux caused by molecular viscosity and meridional circulation. Firstly we use heuristical forms for the needed cross correlations Qrφ (turbulent radial momentum flux) and Qνφ (turbulent latitudinal momentum flux): Qrφ = −α₀r ϑΩ/ϑr · sin ν + Q₀ sin ν + Q₂ sin³ ν, Qνφ = −δ₀ ϑΩ/ϑν· sin ν + P₂ sin²θ cos θ. It is shown that a radial dependence of the angular velocity Ω is given by Q₀. Furthermore, the observed equatorial acceleration occurs in the case of non-negativity of Q₂ and/or P₂. Because of the spatial dependence of the solar angular velocity the coefficients of Q and P are unfortunately not to be measured. Secondly, we determine the coefficients with a theory founded upon the hypothesis that a rotating stochastical force field — independent from Ω — maintains an anisotropic turbulence. The global fast rotation produces, indeed, finite cross correlations Q₂ and P₂. It is suggested that horizontally directed turbulent motions with not too small radial correlations lengths and time scales of about 2 weeks could be responsible for the solar differential rotation. Finally, we show that also short-living turbulent horizontal modes provide the observed equatorial acceleration if they occur preferably at the equatorial region.     

Differential Rotation Driven by Precession

The tidal force in the Earth–Moon system exerted on the Earth's equatorial bulge results in the Earth's precession. It was proposed a long time ago that the strong shear flow driven by the precession of the Earth may power the Earth's dynamo in its liquid core. We present a nonlinear analytical study investigating how the Poincaré force in a rotating, precessing spherical system drives a large-amplitude differential rotation which plays a major role in the modern theory of the geodynamo. The analysis is based on a perturbation approach in terms of the small Poincaré force parameter. It is found that the amplitude of the precession-driven differential rotation is consistent with that estimated from the geomagnetic secular variation.     

2-D hydrodynamical simulations of wind compressed disks

We present results of 2-D hydrodynamical simulations of a radiatively driven stellar wind from a rapidly rotating Be-star. These generally confirm predictions of the semi-analytic Wind-Compressed-Disk model recently proposed by Bjorkman and Cassinelli to explain the circumstellar disks inferred observationally to exist around such rapidly rotating stars. However, our numerical simulations are able to incorporate several important effects not accounted for in the simple model, including a dynamical treatment of the outward radiative driving and gas pressure, as well as a rotationally distorted, oblate stellar surface. This enables us to model quantitatively the compressed wind and shock that forms the equatorial disk. The simulation results thus do differ in several important details from the simple model, showing, for example, an inner diskinflow not possible in the heuristic approach of assuming a fixed outward velocity law. There is also no evidence for the predicted detachment of the disk that arises in the fixed outflow picture. The peak equatorward velocity in the dynamical models is furthermore about a factor of two smaller than the analytically predicted value of 50% the stellar equatorial rotation speed. As a result, the dynamical disks are somewhat weaker than predicted, with a wider opening angle, lower disk/pole density ratio, and smaller shock velocity jump (each by roughly the same factor of two).     

Convection in a rotating deep compressible spherical shell: Application to the Sun

In this paper we study the interaction of rotation with convection in a deep compressible spherical shell as the Sun's convection zone. We examine how the energy transport and the large scale motions can be affected by rotation. In particular we study how a large scale meridional circulation can give rise to variations of angular velocity with latitude and depth.It is assumed that the energy transport is only due to convection and that the mixing-length theory gives an adequate representation of it. Furthermore we assume that rotation acts as a perturbation of the turbulent convective flux through its transport coefficient.The equations involved in the model are integrated numerically in the limit of large viscosity and slow rotation. After having expanded all physical quantities to the first order in terms of Legendre polynomials, the fitting with the observed solar differential rotation gives the expansion parameter, which represents the coupling constant between rotation and convection.The results show a three-cell circulation extending from the poles to the equator. The first one is located in the lower half of the convection zone with the fluid rising at the equator and sinking at the poles. In the second one the direction of the motion is reversed while the third one, located in a thin upper layer, shows the same characteristics of the first one. The meridional velocities at the surface are directed towards the poles and are about 20 cm s^-1. In the other cells the meridional velocities are typically of a few cm s^-1 while the radial velocities are of the order of a few tenths of cm s^-1.The heat flux relative variation at the surface is about 10^-4 (3 × 10^-3 at the bottom) with a polar excess. The temperature variation at the surface is of the same order, with an equatorial excess however. The convection seems to be stabilized stronger at the equator. The angular velocity increases inwards and varies about 6% between the surface and the bottom of the convection zone.An attempt is made for explaining the picture which emerges. In particular the negligible flux and temperature variations at the surface are explained in terms of equalization by the particular structure of the latitudinal flow. This configuration of large scale circulation is attributed to the high stratification of the convection zone with depth.     

Inhomogeneous convection and the equatorial acceleration of the sun

The interaction of rotation and turbulent convection is assumed to give rise to an inhomogeneous, but isotropic, latitude dependent turbulent energy transport, which is described by a convective conduction coefficient _c which varies with latitude. Energy balance in the convective zone is then possible only with a slow meridian circulation in the outer convective zone of the sun. The angular momentum transported by this circulation is balanced in a steady state by turbulent viscous transport down an angular velocity gradient. A detailed model is constructed allowing for the transition from convective transport to radiative transport at the boundaries of the convective zone, by using a perturbation analysis in which the latitude variation of _c is small. The solution for a thin compressible shell gives equatorial acceleration and a hotter equator than pole, assuming that the convection is preferentially stabilised at the equator. For agreement with the sun's equatorial acceleration the model predicts an equatorial temperature excess of 70 K and a surface meridional velocity of 350 cm/sec from pole to equator.     

The angular momentum transport in rotating turbulent convection

The angular momentum transport in rotating turbulent convection is simulated with the NIRVANA code for Taylor numbers up to 10⁶. The box consists of an unstable layer embedded in two stable overshoot layers. We find the expected anisotropies in the rotating anisotropic turbulence field: 〈u^′2_r〉 exceeds 〈u^′2_ϕ〉, and 〈u^′2_ϕ〉 exceeds 〈u^′2_θ〉. The resulting radial angular momentum transport is directed inwards and peaks in the middle of the convective layer. The resulting latitudinal angular momentum transport is equatorwards, peaks at the surface and is highly concentrated to the equatorial region. Apart from a factor of 2–3 the total amplitudes of the cross‐correlations are of the same order of magnitude. In the lower overshoot region (‘tachocline’) the cross‐correlations are negative. It is argued that the concentration of the latitudinal angular momentum transport towards the surface and towards the equator does not too strongly reduce its potential to produce rotation laws with accelerated equators. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamical History of the Earth–Moon System

Differential equations describing the tidal evolution of the earth's rotation and of the lunar orbital motion are presented in a simple close form. The equations differ in form for orbits fixed to the terrestrial equator and for orbits with the nodes precessing along the ecliptic due to solar perturbations. Analytical considerations show that if the contemporary lunar orbit were equatorial the evolution would develop from an unstable geosynchronous orbit of the period about 4.42 h (in the past) to a stable geosynchronous orbit of the period about 44.8 days (in the future). It is also demonstrated that at the contemporary epoch the orbital plane of the fictitious equatorial moon would be unstable in the Liapunov's sense, being asymptotically stable at early stages of the evolution. Evolution of the currently near-ecliptical lunar orbit and of the terrestrial rotation is traced backward in time by numerical integration of the evolutional equations. It is confirmed that about 1.8 billion years ago a critical phase of the evolution took place when the equatorial inclination of the moon reached small values and the moon was in a near vicinity of the earth. Before the critical epoch t _cr two types of the evolution are possible, which at present cannot be unambiguously distinguished with the help of the purely dynamical considerations. In the scenario that seems to be the most realistic from the physical point of view, the evolution also has started from a geosynchronous equatorial lunar orbit of the period 4.19 h. At t < t _cr the lunar orbit has been fixed to the precessing terrestrial equator by strong perturbations from the earth's flattening and by tidal effects; at the critical epoch the solar perturbations begin to dominate and transfer the moon to its contemporary near-ecliptical orbit which evolves now to the stable geosynchronous state. Probably this scenario is in favour of the Darwin's hypothesis about originating the moon by its separation from the earth. Too much short time scale of the evolution in this model might be enlarged if the dissipative Q factor had somewhat larger values in the past than in the present epoch. Values of the length of day and the length of month, estimated from paleontological data, are confronted with the results of the developed model.     

The Crab Nebula: Interpretation of Chandra observations

We interpret the observed X-ray morphology of the central part of the Crab Nebula (torus + jets) in terms of the standard theory by Kennel and Coroniti (1984). The only new element is the inclusion of anisotropy in the energy flux from the pulsar in the theory. In the standard theory of relativistic winds, the Lorentz factor of the particles in front of the shock that terminates the pulsar relativistic wind depends on the polar angle as γ = γ₀ + γ _m sin² θ, where γ₀∼200 and γ_m∼4.5×10⁶. The plasma flow in the wind is isotropic. After the passage of the pulsar wind through the shock, the flow becomes subsonic with a roughly constant (over the plerion volume) pressure P=1/3;n∈ where n is the plasma particle density and ∈ is the mean particle energy. Since ∈∼γmc ², a low-density region filled with the most energetic electrons is formed near the equator. A bright torus of synchrotron radiation develops here. Jet-like regions are formed along the pulsar rotation axis, where the particle density is almost four orders of magnitude higher than that in the equatorial plane, because the particle energy there is four orders of magnitude lower. The energy of these particles is too low to produce detectable synchrotron radiation. However, these quasijets become comparable in brightness to the torus if additional particle acceleration takes place in the plerion. We also present the results of our study of the hydrodynamic interaction between an anisotropic wind and the interstellar medium. We compare the calculated and observed distributions of the volume emissivity of X-ray radiation.     

The 27-Day Cosmic Ray Intensity Variations During Solar Minimum 23/24

We have studied the 27-day variations and their harmonics in Galactic cosmic ray (GCR) intensity, solar wind velocity, and interplanetary magnetic field (IMF) components during the recent prolonged solar minimum 23/24. The time evolution of the quasi-periodicity in these parameters connected with the Sun’s rotation reveals that the synodic period of these variations is ≈?26?–?27 days and is stable. This means that the changes in the solar wind speed and the IMF are related to the Sun’s near-equatorial regions in considering the differential rotation of the Sun. However, the solar wind parameters observed near the Earth’s orbit provide only the conditions in the limited local vicinity of the equatorial region in the heliosphere (within ±?7^° in latitude). We also demonstrate that the observed period of the GCR intensity connected with the Sun’s rotation increased up to ≈?33?–?36 days in 2009. This means that the process that drives the 27-day GCR intensity variations takes place not only in the limited local surroundings of the equatorial region but in the global 3-D space of the heliosphere, covering also higher latitude regions. A relatively long period (≈?34 days) found for 2009 in the GCR intensity gives possible evidence of the onset of cycle 24 due to active regions at higher latitudes and rotating slowly because of the Sun’s differential rotation. We also discuss the effect of differential rotation on the theoretical model of the 27-day GCR intensity variations.     

Meridional flow in the solar wind in the presence of latitudinally dependent boundary conditions

The influence of latitudinally dependent boundary conditions on the large radius values of meridional flow in the distant solar wind is examined through a double perturbation expansion of the magnetohydrodynamic equations. A general result is derived for the meridional velocity which allows arbitrary specification of radial velocity, radial magnetic field, and mass flux, as a function of colatitude at some coronal reference surface. Three specific examples are treated, including the model of Pneuman and Kopp (1971). The latter example indicates that there may be flow toward the equator at large radii, as opposed to the pure equatorial divergence of internally generated motion due to a flow which is latitudinally uniform at the reference radius. A solar cycle effect most probably averages the boundary conditions so that only the equatorial divergence from an average spherically symmetric corona is seen in comet-tail observations. This may also explain the off-and-on-again nature of the meridional gradient in the radial velocity of the solar wind as seen in radio scintillation observations.     

The location and X-ray emission of shocks around isolated be stars

All hot stars are observed to have X-ray emission: O stars haveL _X /L _bol 10^–7, whilst B stars' emission drops off with spectral subtype. Dynamical instability of OB star radiatively driven winds generates shocked regions which may be responsible for the bulk of the X-rays observed. The wind-compressed disc model of Bjorkman & Cassinelli (1993) presents another site for X-ray emission. The disc formed in the equatorial plane of a fast rotating Be star from equatorward drift of wind streamlines is confined on both sides by a shock which may also generate X-rays. As the X-ray emission originating from the wind shocking is ubiquitous amongst B and Be stars then the wind-compressed disc model näively predicts that Be stars should generate more X-rays than B stars of equivalent spectral subtype.The X-ray emission from the shocks confining compression discs has been calculated and compared to a limited set of observations. The excess X-ray emission from the Be star disc shocks is found to be undetectable over the inherent wind shocking emission.     

Nonlinear boussinesq convective model for large scale solar circulations

We present extensive numerical calculations for a model of thermal convection of a Boussinesq fluid in an equatorial annulus of a rotating spherical shell. The convection induces and maintains differential rotation and meridian circulation. The model is solved for an effective Prandtl number P = 1, with effective Taylor number T in the range 10² <T <10⁶, and effective Rayleigh number R between the critical value for onset of convection, and a few times that value. With = 2.6 × 10^–6 s^–1, d = 1.4 × 10¹⁰ cm (roughly the depth of the solar convection zone) the range of Taylor number is equivalent to kinematic viscosities between 10¹⁴ and 10¹² cm² s^–1, which encompasses eddy viscosities estimated from mixing length theory applied to the Sun.The convection does generally make equatorial regions rotate faster, the more so as T is increased, but local equatorial deceleration near the surface is also produced at intermediate T for large enough R above critical. The differential rotation is maintained primarily through momentum transport in the cells up the gradient, rather than by meridian circulation. Differential rotation energy increases relative to cell energy with increasing T, surpassing it near T = 3 × 10⁴. The differential rotation tends to stretch out the convective cells, analogously to what is thought to happen to solar magnetic regions. Differential rotation and meridian circulation energies are nearly equal for T = 10³, but the meridian circulation energy falls off relative to differential rotation like T ^–1 for larger T. The meridian circulation is always toward the poles near the surface, contrary to models of Kippenhahn, Cocke, Köhler, and Durney and Roxburgh. The radial shear produced in the differential rotation is almost always positive, as in the Köhler model, but contrary to the assumptions made by Leighton for his random walk solar cycle model.Solutions in the neighborhood of T = 3 × 10⁴ seem to compare best with various solar observations including differential rotation amplitude, cell wavelength, tilted structure, horizontal momentum transport, and weak meridian circulation. The local equatorial deceleration (equatorward of 10–15°) has not been observed, although the techniques of data analysis may not have been sensitive to it. The most important deficiency of the model is that all the solutions with T 10³ show the vertical heat transport a rather strong function of latitude, with a maximum at the equator, no evidence of which is seen at the solar surface.The National Center for Atmospheric Research is sponsored by The National Science Foundation.     

The deep wind structure of the giant planets: Results from an anelastic general circulation model

The giant gas planets have hot convective interiors, and therefore a common assumption is that these deep atmospheres are close to a barotropic state. Here we show using a new anelastic general circulation model that baroclinic vorticity contributions are not negligible, and drive the system away from an isentropic and therefore barotropic state. The motion is still aligned with the direction of the axis of rotation as in a barotropic rotating fluid, but the wind structure has a vertical shear with stronger winds in the atmosphere than in the interior. This shear is associated with baroclinic compressibility effects. Most previous convection models of giant planets have used the Boussinesq approximation, which assumes the density is constant in depth; however, Jupiter's actual density varies by four orders of magnitude through its deep molecular envelope. We therefore developed a new general circulation model (based on the MITgcm) that is anelastic and thereby incorporates this density variation. The model's geometry is a full 3D sphere down to a small inner core. It is nonhydrostatic, uses an equation of state suitable for hydrogen-helium mixtures (SCVH), and is driven by an internal heating profile. We demonstrate the effect of compressibility by comparing anelastic and Boussinesq cases. The simulations develop a mean state that is geostrophic and hydrostatic including the often neglected, but significant, vertical Coriolis contribution. This leads to modification of the standard thermal wind relation for a deep compressible atmosphere. The interior flow organizes in large cyclonically rotating columnar eddies parallel to the rotation axis, which drive upgradient angular momentum eddy fluxes, generating the observed equatorial superrotation. Heat fluxes align with the axis of rotation, and provide a mechanism for the transport of heat poleward, which can cause the observed flat meridional emission. We address the issue of over-forcing which is common in such convection models and analyze the dependence of our results on this; showing that the vertical wind structure is not very sensitive to the Rayleigh number. We also study the effect of rotation, showing how the transition from a rapidly to a slowly rotating system affects the dynamics.     

The influence of neutral winds on He+ distributions in the equatorial ionosphere

The effects of F-region neutral winds on the distribution of He⁺ in the equatorial ionosphere have been examined using a theoretical model and an observational data set. It is shown by the model that components of neutral wind in the magnetic meridian up to only 50 m s^? can produce He⁺ gradients in the northern and southern sectors of a flux tube that differ by more than 80%. This is associated with interhemisphere transport velocities of He⁺ as large as 15 m s^?1 at 800 km. A substantial latitude gradient in the He⁺ distribution across the dip equator also results from the redistribution of He⁺ The changes in the He⁺ concentration at the dip equator and the latitude distribution of He⁺ in response to different neutral wind components is determined from the model and used to construct longitude distributions of He⁺ to compare with observations made at equinox. Good agreement between the calculations and observations is obtained both at the geographic and geomagnetic equators using the relationship between neutral winds, interhemispheric transport velocity and He⁺ concentration derived from the model. If these relationships can be extrapolated to accommodate the different conditions expected during solstice, we can also discuss the He⁺ distributions expected during this season.     

Anchor depths of flux elements and depths of flux sources in relation to the two rotation profiles of the sun's surface magnetic fields

It is known for over two decades now that the rotation of the photospheric magnetic fields determined by two different methods of correlation analysis leads to two vastly differing rotation laws - one the differential and the other rigid rotation. Snodgrass and Smith (2001) reexamining this puzzle show that the averaging of the correlation amplitudes can tilt the final profile in favour of rigid rotation whenever the contribution of the rigidly rotating large-scale magnetic structures (the plumes) to the correlation dominates over that of the differentially rotating small-scale and mesoscale features. We present arguments to show that the large-scale unipolar structures in latitudes >40 deg, which also show rigid rotation (Stenflo, 1989), are formed mainly from the intranetwork magnetic elements (abbreviated as IN elements). We then estimate the anchor depths of the various surface magnetic elements as locations of the Sun's internal plasma layers that rotate at the same rate as the flux elements, using the rotation rates of the internal plasma layers given by helioseismology. We infer that the anchor depths of the flux broken off from the decay of sunspot active regions (the small-scale and mesoscale features that constitute the plumes) are located in the shallow layers close to the solar surface. From a similar comparison with helioseismic rotation rates we infer that the rigid rotation of the large-scale unipolar regions in high latitudes could only be coming from plasma layers at a radial distance of about 0.66–0.68 R _⊙ from the Sun's centre. Using Stenflo's (1991) ‘balloon man’ analogy, we interpret these layers as the source of the magnetic flux of the IN elements. If so, the IN flux elements seem to constitute a fundamental component of solar magnetism.