Equilibrium structure of stars obeying a generalized differential rotation law

In the present paper we have considered the problem of determining the equilibrium structure of differentially rotating stars in which the angular velocity of rotation varies both along the axis of rotation and in directions perpendicular to it. For this purpose, a generalized law of differential rotation of the type ² =b ₀+b ₁ s ²+b ₂ s ⁴+b ₃ z ²+b ₄ z ⁴+b ₅ z ² s ² (here is a nondimensional measure of the angular velocity of a fluid element distants from the axis of rotation andz from the plane through the centre of the star perpendicular to the axis of rotation, andb's are suitably chosen parameters) has been used. Whereas Kippenhahn and Thomas averaging approach has been used to incorporate the rotational effects in the stellar structure equations, Kopal's results on Roche equipotentials have been used to obtain the explicit form of the stellar structure equations, which incorporate the rotational effects up to second order of smallness in the distortion parameters. The method has been used to compute the equilibrium structure of certain differentially rotating polytropes. Certain differentially rotating polytropes. Certain differentially rotating models of the Sun have also been computed by using this approach.     

A large-eddy simulation of turbulent compressible convection: differential rotation in the solar convection zone

We present the results of two simulations of the convection zone, obtained by solving the full hydrodynamic equations in a section of a spherical shell. The first simulation has cylindrical rotation contours (parallel to the rotation axis) and a strong meridional circulation, which traverses the entire depth. The second simulation has isorotation contours about mid-way between cylinders and cones, and a weak meridional circulation, concentrated in the uppermost part of the shell.
We show that the solar differential rotation is directly related to a latitudinal entropy gradient, which pervades into the deep layers of the convection zone. We also offer an explanation of the angular velocity shear found at low latitudes near the top. A non-zero correlation between radial and zonal velocity fluctuations produces a significant Reynolds stress in that region. This constitutes a net transport of angular momentum inwards, which causes a slight modification of the overall structure of the differential rotation near the top. In essence, the thermodynamics controls the dynamics through the Taylor–Proudman momentum balance . The Reynolds stresses only become significant in the surface layers, where they generate a weak meridional circulation and an angular velocity 'bump'.     

Effects of variations of parallel angular velocity and vorticity on the oscillations of compressible homogeneous rotating ellipsoids

Earlier work on the oscillations of an ellipsoid is extended to investigate the behaviour of a nonequilibrium compressible homogeneous rotating gaseous ellipsoid, with the components of the velocity field as linear functions of the coordinates, and with parallel angular velocity and uniform vorticity. The dynamical behaviour of the ellipsoid is obtained by numerically integrating the relevant differential equations for different values of the initial angular velocity and vorticity. This behaviour is displayed by the (a ₁,a ₂) and (a ₁,a ₃) phase plots, where thea _i's (i = 1, 2, 3) are the semi-diameters, and by the graphs ofa ₁,a ₂,a ₃, the volume, and the angular velocity as functions of time.The dynamical behaviour of the nonequilibrium ellipsoid depends on the deviation of the angular momentum from its equilibrium value; for larger deviations, the oscillations are more nonperiodic with larger amplitudes.An initially ellipsoidal configuration always remains ellipsoidal, but it cannot become spheroidal about its rotation axis, though it may become spheroidal instantaneously about either one of the other two principal axes.For an ellipsoid approaching axisymmetry about its axis of rotation, the angular velocity can suddenly increase by a large amount. Thus if an astrophysical object can be modelled by a nonequilibrium ellipsoid, it may occasionally undergo sudden large increases of angular velocity.     

Equilibrium Structures of Differentially Rotating and Tidally Distorted White Dwarf Models of Stars

In this paper we present a method for computing the equilibrium structures and various physical parameters of a primary component of the binary system assuming that the primary is more massive than the secondary and is rotating differentially according to the law of the w² = b₀ + b₁ × s² + b₂ × s⁴, w being the angular velocity of rotation of a fluid element distant s from the axis of rotation and b₀, b₁, b₂ suitably chosen numerical constants. This method utilizes the averaging approach of Kippenhahn and Thomas (1997) and the concept of Roche equipotentials in a manner earlier used by Mohan et al. (1997) to incorporate the effects of rotation and tidal distortions on the equilibrium structures of certain rotationally and tidally distorted stellar models. The use of the method has been illustrated by applying it to obtain the structures and some observable parameters of certain differentially rotating and tidally distorted binary systems whose primary component is assumed to be a white dwarf star.     

Cosmological viscous fluid universe in slow rotation

The problem of slowly rotating cosmological viscous fluid universe in a homogeneous and isotropic models has been investigated by considering the perturbation in the metric rotation function to the first order of smallness associated with certain physical restrictions imposed on the metric rotation function and matter angular velocity. Some more general solutions for the metric rotation function have been obtained and physical interpretation of the solutions have been investigated.     

Equilibrium Structures of Differentially Rotating Primary Components of Binary Stars

In this paper a method is proposed for computing the equilibrium structures and various other observable physical parameters of the primary components of stars in binary systems assuming that the primary is more massive than the secondary and is rotating differentially about its axis. Kippenhahn and Thomas averaging approach (1970) is used in a manner earlier used by Mohan, Saxena and Agarwal (1990) to incorporate the rotational and tidal effects in the equations of stellar structure. Explicit expressions for the distortional terms appearing in the stellar structure equations have been obtained by assuming a general law of differential rotation of the typeω² = b ₀+b ₁ s ²+b ₂ s ⁴, where ω is the angular velocity of rotation of a fluid element in the star at a distance s from the axis of rotation, and b ₀, b ₁, b ₂ are suitably chosen numerical constants. The expressions incorporate the effects of differential rotation and tidal distortions up to second order terms. The use of the proposed method has been illustrated by applying it to obtain the structures and observable parameters of certain differentially rotating primary components of the binary stars assuming the primary components to have polytropic structures. This revised version was published online in July 2006 with corrections to the Cover Date.     

Hydrodynamic simulations of rotating molecular jets

Molecular outflows and the jets which may drive them can be expected to display signatures associated with rotation if they are the channels through which angular momentum is extracted from material accreting on to protostars. Here, we determine some basic signatures of rapidly rotating flows through three-dimensional numerical simulations of hydrodynamic jets with molecular cooling and chemistry. We find that these rotating jets generate a broad advancing interface which is unstable and develops into a large swarm of small bow features. In comparison to precessing jets, there is no stagnation point along the axis. The greater the rotation rate, the greater the instability. On the other hand, velocity signatures are only significant close to the jet inlet since jet expansion rapidly reduces the rotation speed. We present predictions for atomic, H₂ and CO submillimetre images and spectroscopy including velocity channel maps and position–velocity diagrams. We also include simulated images corresponding to Spitzer IRAC band images and CO emission, relevant for APEX and eventual ALMA observations. We conclude that protostellar jets often show signs of slow precession but only a few sources display properties which could indicate jet rotation.     

Extension of Cassini's Laws

We investigate the Cassini's laws which describe the rotational motion in a 1:1 spin-orbit resonance. When this rotational motion follows the conventional Cassini's laws, the figure axis coincides with the angular momentum axis. In this case we underline the differences between the rotational Hamiltonian for a 'slow rotating' body like the Moon and for a 'fast rotating' body like Phobos. Then, we study a more realistic rotational Hamiltonian where the angle J between the figure axis and the angular momentum axis could be different from zero. This Hamiltonian has not been studied before. We have found a new particular solution for this Hamiltonian which could be seen as an extension of the Cassini's laws. In this new solution the angle J is constant, which is not zero, and the precession of the angular momentum plane is equal to the mean motion of the argument of pericenter of the rotating body. This type of rotational motion is only possible when the orbital eccentricity of the rotating body is not zero. This new law enables describing in particular, the Moon mean rotational motion for which the mean value of the angle J is found to be equal to 103.9±0.7 s of arc.     

On the angular momentum in star formation

We discuss the rotation of interstellar clouds which are in a stage immediately before star formation. Cloud collisions seem to be the principal cause of the observed rotation of interstellar clouds. The rotational motion of the clouds is strongly influenced by turbulence.Theories dealing with the resolution of the angular momentum problem in star formation are classified into five major groups. We develop the old idea that the angular momentum of an interstellar cloud passes during star formation into the angular momentum of double star systems and/or circumstellar clouds.It is suggested that a rotating gas cloud contracts into a ring-like structure which fragments into self-gravitating subcondensations. By collisions and gas accretion these subcondensations accrete into binary systems surrounded by circumstellar clouds. Using some rough approximations we find analytical expressions for the semi-major axis of the binary system and for the density of the circumstellar clouds as a function of the initial density and of the initial angular velocity of an interstellar cloud. The obtained values are well within the observational limits.     

Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque

New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

Evolutionary status of the primary component of YZ Cassiopeiae

The models of non-rotating and rotating 2.31M _\ stars of Population I composition have been calculated, starting at the threshold of stability. A 2.31M _\ star was chosen to compare the results with the observational parameters of the primary component of the well-known detached binary YZ Cassiopeiae. The effects of rotation on the internal structure during the evolution of the star were studied by constructing sequences of axisymmetric rotating models under the assumption that angular momentum was conserved according to a predetermined angular velocity distribution depending on the structure of the star.The first section of this paper deals with effects of rotation on the evolutionary behaviours of the 2.31M _\ star through the pre-Main-Sequence evolution as well as the zero-age Main Sequence.In the second section of this paper, the evolutionary studies have been extended up to near-hydrogen exhaustion phase in order to obtain a theoretical model corresponding to the given mass and radius of the primary component of YZ Cassiopeiae. The theoretical models were found to be in a good agreement with observational parameters. The computed rotating models of the primary of YZ Cassiopeiae indicates that its evolutionary age is 6.01×10⁸ years; and the central hydrogen content 0.183 — which means that about 75% of its original value was depleted.     

Evolution of collapse nonuniformity for rotating magnetic interstellar clouds

We investigate the formation and evolution of isothermal collapse nonuniformity for rotating magnetic interstellar clouds. The initial and boundary conditions correspond to the statement of the problem of homogeneous cloud contraction from a pressure equilibrium with the external medium. The initial uniform magnetic field is collinear with the angular velocity. Fast and slow magnetosonic rarefaction waves are shown to be formed and propagate from the boundary of the cloud toward its center in the early collapse stages. The front of the fast rarefaction wave divides the gas mass into two parts. The density, angular velocity, and magnetic field remain uniform in the inner region and have nonuniform profiles in the outer region. The rarefaction wave front surface can take both prolate and oblate shapes along the rotation axis, depending on the relationship between the initial angular velocity and magnetic field. We derive a criterion that separates the two regimes of rarefaction wave dynamics with the dominant role of electromagnetic and centrifugal forces. Based on analytical estimations and numerical calculations, we discuss possible scenarios for the evolution of collapse nonuniformity for rotating magnetic interstellar clouds.     

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.     

A General Relativistic Calculation of Boundary Layer and Disc Luminosity for Accreting Non-magnetic Neutron Stars in Rapid Rotation

We calculate the disc and boundary layer luminosities for accreting rapidly rotating neutron stars with low magnetic fields in a fully general relativistic manner. Rotation increases the disc luminosity and decreases the boundary layer luminosity. A rapid rotation of the neutron star substantially modifies these quantities as compared with the static limit. For a neutron star rotating close to the centrifugal mass shed limit, the total luminosity has contribution only from the extended disc. For such maximal rotation rates, we find that well before the maximum stable gravitational mass configuration is reached, there exists a limiting central density, for which particles in the innermost stable orbit will be more tightly bound than those at the surface of the neutron star. We also calculate the angular velocity profiles of particles in Keplerian orbits around the rapidly rotating neutron star. The results are illustrated for a representative set of equation of state models of neutron star matter.     

Fragmentation of a Molecular Cloud Core versus Fragmentation of the Massive Protoplanetary Disk in the Main Accretion Phase

We present three-dimensional numerical simulations on binary formation through fragmentation. The simulations follow gravitational collapse of a molecular cloud core up to growth of the first core by accretion. At the initial stage, the gravity is only slightly dominant over the gas pressure. We made various models by changing initial velocity distribution (rotation speed, rotation law, and bar-mode perturbation). The cloud fragments whenever the cloud rotates sufficiently slowly to allow collapse but faster enough to form a disk before first-core formation. The latter condition is equivalent to Ω₀ t _ff ? 0.05, where Ω₀ and t _ff f denote the initial central angular velocity and the freefall time measured from the central density, and the condition is independent of the initial rotation law and bar-mode perturbation. Fragmentation is classified into six types. When the initial cloud rotates rigidly the cloud collapses to form a adiabatic disk supported by rotation. When the bar-mode perturbation is very minor, the disk deforms to a rotating bar, and the bar fragments. Otherwise, the adiabatic disk evolves into a central core surrounded by a circumstellar disk, and the the circumstellar disk fragments. When the initial cloud rotates differentially, the cloud deforms to a ring or bar in the isothermal collapse phase. The ring fragments into free or more cores, while the bar fragments into only two cores. In the latter case, the core merges due to low orbital angular momentum and new satellite cores form in the later stages.     

Optical depth effects on the formation of spectral lines in rotating and expanding spherical atmospheres

We have investigated the effects of increasing optical depths on spectral lines formed in a rotating and expanding spherical shell. We have assumed a shell whose outer radius is 3 times the inner radius, with the radial optical depths equal to 10, 50, 100, 500. We have employed a constant velocity with no velocity gradients in the shell. The shell is assumed to be rotating with velocities varying as 1/_ρ, where_ρ is the perpendicular distance from the axis of rotation, implying the conservation of angular momentum. Two expansion (radial) velocities are treated: (1)V = 0 (static case) and (2)V = 10 mean thermal units. The maximum rotational velocities areV _rot = 0, 5, 10 and 20. In the shell where there are no radial motions, we obtain symmetric lines with emission in the wings forV _rot = 0 and 5 while forV _rot ≥ 10 we obtain symmetric absorption lines. In the case of an expanding shell, we obtain lines with central emission.     

Calculations of the effects of angular momentum on the early evolution of Jupiter

The planet Jupiter is assumed to have formed as a subcondensation in the solar nebula. The initial phase of its evolution is one of hydrostatic contraction with radiative energy transport. Calculations of evolutionary sequences through this phase are presented, including the effects of angular momentum. The calculations are carried out in two space dimensions under the assumptions of axial symmetry, constancy of angular velocity on cylindrical surfaces about the rotation axis, a pressure-density relation given by the polytrope of index 3, conservation of angular momentum, and a homogeneous composition. The results show that under certain physically reasonable initial distributions of density and angular momentum the formation of a central planet and a rotating circumplanetary envelope is possible, while under assumptions a point of instability is reached that probably results in the breakup of the condensation by fission into two or more parts. The models are discussed with reference to the present angular momenta of Jupiter and its regular satellites.     

On the REYNOLDS stresses in mean-field hydrodynamics III. Two-dimensional turbulence and the problem of differential rotation

It is well-known that, in a rotating star, a meridional circulation directed from pole to equator contributes a latitude dependence to the law of rotation, as it is observed on the Sun. It is also known that such a circulation is produced by a radial dependence of the original angular velocity if the outer parts of the convective zone possess a higher angular velocity than the inner parts. In this paper it is shown that a two-dimensional turbulence with velocity vectors perpendicular to the radial direction, necessarily leads to the required relation dω₀/dr > 0. This also holds when there is an additional three-dimensional homogeneous and isotropic turbulent field. The characteristic lengths of the two turbulences would, however, have to have different orders of magnitude whenever the horizontal turbulence should not be strictly two-dimensional but posses correlation lengths finite in all directions. The application of this to explaining also the phenomenon of the superrotation of the Earth's upper atmosphere is suggested. In the final chapter the possibility of the occurrence of negative viscosity on the Sun is discussed.     

Regular and chaotic orbits in a self-consistent triaxial stellar system with slow figure rotation

We created a self-consistent triaxial stellar system through the cold disipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. The resulting system rotates slowly even though its total angular momentum is zero, i.e., it offers an example of figure rotation. The potential of the system was subsequently approximated with interpolating formulae yielding a smooth potential stationary in the rotating frame. The Lyapunov exponents could then be computed for a randomly selected sample of 3,472 of the bodies that make up the system, allowing the recognition of regular and partially and fully chaotic orbits. The regular orbits were Fourier analyzed and classified using their locations on the frequency map. A comparison with a similar non-rotating model showed that the fraction of chaotic orbits is slightly but significantly enhanced in the rotating model; alternatively, there are no significant differences between the corresponding fractions neither of partially and fully chaotic orbits nor of long axis tubes, short axis tubes, boxes and boxlets among the regular orbits. This is a reasonable result because the rotation causes a breaking of the symmetry that may increase chaotic effects, but the rotation velocity is probably too small to produce any other significant differences. The increase in the fraction of chaotic orbits in the rotating system seems to be due mainly to the effect of the Coriolis force, rather than the centrifugal force, in good agreement with the results of other investigations.     

Non-axisymmetric solar magnetic fields

A simple non-linear, non-axisymmetric mean field dynamo model is applied to a differentially rotating spherical shell. Two approximations are used for the angular velocity, to represent what is now believed to be the solar rotation law. In each case, stable solutions are found which possess a small non-axisymmetric field component. Although the model has a number of obvious shortcomings, it may be relevant to the problem of the solar active longitudes.