Adiabatic pulsations and convective instability of rotating gaseous masses

Third order virial equations have been used to investigate the oscillations and the stability of the sequence of differentially rotating, compressible Maclaurin spheroids in the presence of toroidal magnetic fields. It is shown that the neutral point occurring at eccentricitye=0.731 13, which is the analogue of the first point of bifurcation along the Dedekind sequence, remains unaffected by the presence of differential rotation or a toroidal magnetic field. The point of onset of dynamical instability corresponding to the third harmonic deformations does, however, depend upon the magnetic field. It is shifted to values higher thane=0.966 96, the value that obtains in the case of uniform rotation; and a sufficiently large magnetic field can suppress this point. Complete frequency spectra (‘Kelvin’ modes belonging to the harmonicsl=3 and compressible modes belonging tol=1) are obtained in two cases of interest: when the equilibrium state is one of equipartition, and when toroidal magnetic and velocity fields (vanishing at the surface) are present in a configuration rotating with a constant angular velocity.     

Magnetically distorted polytropes

The oscillations of a gaseous polytrope with a magnetic field having both a toroidal and a poloidal component are examined using the second-order tensor virial equations on the assumption that the magnetic energy is small compared with the gravitational energy. The frequencies of oscillation of the transverse shear, the toroidal and the coupled pulsation modes are tabulated for polytropic indicesn=1, 1.5, 2, 3 and 3.5. It is found that the magnetic field decreases the frequency of oscillation of (i) the transverse shear mode and (ii) the mode which starts as a radial pulsation in the absence of a magnetic field while it increases the frequency of oscillation of (i) the toroidal mode and (ii) the Kelvin mode. In all cases the shift in frequency decreases with increasingn.     

Magnetically distorted polytropes

The fundamental frequencies of the non-radial mode of oscillation belonging to the second harmonic (l=2) of magnetically distorted polytropic gas spheres are evaluated in the second approximation by a variational method. The magnetic field is assumed to have both the toroidal and the poloidal components. We find that the frequencies of oscillation are increased due to the presence of the magnetic field and that these depend only slightly on the value of , the ratio of the specific heats. We have also determined the value of <1+1/n for the mode of oscillation which exhibits convective instability. This value is lower than the one which is obtained in the absence of a magnetic field.     

Sunspots and physics of magnetic flux tubes overstability and Kelvin-Helmholtz instability in a magnetic field

We have studied the dynamical properties of convective overstability and Kelvin-Helmholtz instability in a vertical magnetic field with a downdraft. The Kelvin-Helmholtz instability and overstability produce the upward downward propagating Alfvén waves depending upon the magnitudes of the kinematic viscosity coefficient (eddy viscosity) ν, and thermometric conductivity κ. It is found that the instability may be driven by the density stratification and the effect of the eddy viscosity is to make the system stable. We discuss also the interaction of the overstability inx<0 and the downdraft inx>0, and the overstability at a vertical boundary of the field.     

The effect of a toroidal magnetic field on the non-radial oscillations of polytropes

Frequencies of non-radial oscillation of polytropic models of stars, belonging to spherical harmonics of ordersl=1, 2 and 3, are evaluated, in a second approximation, by a variational method. Equilibrium configurations in the presence of toroidal magnetic fields are obtained numerically without any restriction on the field strength. The value of the ratio of the specific heats, , is assumed to be equal to 5/3 and only two polytropic indeces,n=1.5 and 3.0, are considered. It is found that a polytropic star stays stable for magnetic fields considerably stronger than expected from the results obtained by the weak field perturbation methods.     

Electron cyclotron maser emission from solar flares

Energetic electrons, with energies 10–100 keV, accelerated during the impulsive phase of solar flares, sometimes encounter increasing magnetic fields as they stream towards the chromosphere. A consequence of the conservation of their magnetic moment is that the electrons with large initial pitch angle will be reflected at different heights from the atmosphere. Energetic electrons reflected below the transition zone will lose most of their energy to collisions and will never return to the corona. Thus, electrons reflected above the transition zone form a loss-cone velocity distribution which can be unstable to Electron Cyclotron Maser (ECM). The interaction of quasi-perpendicular shocks with the ambient coronal plasma will form a ‘ring’ or ‘hollow beam’ velocity distribution upstream of the shock. ‘Ring’ velocity distributions are also unstable to the ECM instability. A review of the recent results on the theory of ECM will be presented. We will focus our discussion on the questions: (a) What are the characteristics of the linear growth rate of the ECM during solar flares? (b) How does the ECM saturate and what is its efficiency? (c) How does the ECM generated radiation modify the flare environment? Finally we will review the outstanding questions in the theory of ECM and we will relate the theoretical predictions to current observations.     

Observations and interpretation of the close binary system CG Cyg

Photometric observations of the short-period (RS CVn-type) eclipsing binary system CG Cyg have been presented. Two sets of results, obtained from an analysis of theB, V andR light curves, represent ‘occultation’ and ‘transit’ solutions. The occultation solution is preferred as it gives a better fit to the colour curve. This hypothesis may also offer a more promising explanation of the observed peculiarities such as period changes and the light variation outside eclipses.     

Energy changes of cosmic rays in the interplanetary region

A clarification and discussion of the energy changes experienced by cosmic rays in the interplanetary region is presented. It is shown that the mean time rate of change of momentum of cosmic rays reckoned for a fixed volume in a reference frame fixed in the solar system is 〈p〉 =p V·G/3 (p=momentum,V is the solar wind velocity andG=cosmic-ray density gradient). This result is obtained in three ways:

1. by a rearrangement and reinterpretation of the cosmic-ray continuity equation;
2. by using a scattering analysis based on that of Gleeson and Axford (1967);
3. by using a special scattering model in which cosmic-rays are trapped in ‘magnetic boxes’ moving with the solar wind.

The third method also gives the rate of change of momentum of particles within a moving ‘magnetic box’ as 〈p〉_ad = ?p ?·V/3, which is the adiabatic deceleration rate of Parker (1965). We conclude that ‘turnaround’ energy change effects previously considered separately are already included in the equation of transport for cosmic rays.     

The cooling of a sunspot

This paper considers the recent criticism by Mullan (1973) of sunspot models and the cooling mechanism which I have proposed in Papers I, II and III of this series. The discussion of the cooling produced by an idealized flow cycle has been extended to include vertical temperature gradients which are consistent with a convectively unstable atmosphere. This leads to an expression for Mullan's parameter f (the ratio in which estimates of the energy flux based on an idealized Carnot cycle should be reduced) which is appropriate to this situation. It is shown that, for a cycle similar to that of Paper III, f = 0.82, while for one which has a vertical extent of order 5 Mm, f= 0.4. Hence the energy flux which, in principle, can be transported away from a sunspot by such a cycle is conservatively estimated to be 1.1 × 10²⁹ erg s^?1 compared with a typical sunspot energy deficit of 2.2 × 10²⁹ erg s^?1. Other criticisms relating to the magnetic field amplification and the ‘cool one’ model are discussed. It is concluded that the essential features of these models remain valid and that the modifications suggested by Mullan's criticism greatly increase their applicability to the sunspot problem.     

Oscillations of a polytrope with a toroidal magnetic field

The radial and the non-radial (l=2) modes of oscillation of a gaseous polytrope with a toroidal magnetic field are examined using a variational principle. It is found that the frequencies of oscillation of the radial mode and the Kelvin mode (l=2) decrease due to the presence of the magnetic field. The shift in the frequency of the Kelvin mode may be split up into two parts, viz. the shift in frequency due to the magnetic field on the unperturbed sphere [(1²)m, say] and the shift in frequency due to the distortion of the structure by the magnetic field [(1²)s, say]. In the first order calculations using one parameter trial function, it is found that (1²)m is indeed positive but is overweighed by a negative (1²)s. In the second order calculations using a trial function with two variational parameters, we find that the general behaviour of (1²)m and (1²)s is unchanged but that (1²)m becomes negative for polytropic indicesn1.5.In Appendix I we study the effect of a small rotation and toroidal magnetic field on the structure of a polytrope. It is found that the resulting configuration is a prolate spheroid, a sphere or an oblate spheroid according as respectively. Here denotes the magnetic energy andT the kinetic energy due to rotation andq is a constant which depends on the polytropic indexn. The values ofq are given in Table I.     

Orientation of galaxies and a magnetic ‘urfield’

Brown's results (1964, 1968) concerning the distribution of orientation angles of spiral galaxies in different areas of the sky are discussed and a graphical statistical test is applied. The deviations from randomness are found to be significant. It seems difficult to ascribe them to selection effects. It is shown that the observed distributions can be explained, if the angular momenta in greater aggregations of galaxies are distributed at random on congruent precession cones with parallel axes. This hypothesis may apply if the following cosmological conditions hold:

1. the matter in the universe was reheated after the recombination at an epoch, when most of the angular momentum was already transferred to the protogalaxies by tidal interaction, and the angular momenta of the protogalaxies in greater aggregations were predominantly parallel at the epoch of reheating;
2. the ‘magnetic’ model of the universe is valid and the ‘urfield’ was uniform at least at the epoch of reheating. Under these assumptions, the ‘frozen-in’ magnetic field will give rise to forces, which — apart from slowing down the rotation of the protogalaxies — will cause precession of their angular momenta around the direction of the ‘urfield’.

For a rigid body approximation the equations of motion are derived and solved numerically. Approximate analytic solutions are also given. The precession period is in the range of 10⁴ to 10⁸ yr for plausible values of the parameters of the problem. The observed distributions in the four regions of the sky investigated are — via the precession hypothesis — compatible with a direction of the ‘urfield’ indicated by the work of Sofueet al. (1969) and Reinhardt and Thiel (1970) ofl ^II≈280°,b ^II≈+30° tol ^II≈100°,b ^II≈?30°.     

Kink instability of a non-static plasma column

Starting with MHD equations we study the linear theory of stability of a plasma column with flow. From the dispersion equation derived, we calculate the dispersion curves and thereby estimate the effect of a flow in the linear theory. We find that, like the toroidal component of the magnetic field, a flow promotes instability: the rate of growth of instability may be increased by one or two orders of magntiude and the wavelength range for instability is also increased. When the flow velocity is large, the m=o and m=1 modes may appear almost together. Finally, a qualitative interpretation of three typical solar events is given on the basis of our results.     

Nonlinear Kelvin-Helmholtz magnetohydrodynamic instability of a rotating plasma

Nonlinear analysis for Kelvin-Helmholtz instability of an incompressible, inviscid, rotating fluid with infinite conductivity in the presence of gravity and surface tension has been discussed. The unperturbed magnetic field on two sides of the interface is taken to be uniform. The nonlinear Schrödinger equation for the time variation of amplitude of small perturbations with wave number around the neutral stability is derived. It is found that stability of a magnetised K-H rotating configuration depends on the density ratio, surface tension, and discontinuity of velocity and magnetic field. The effect of an aligned magnetic field and rotation on the non-linear instability of a rotating conducting plasma has been discussed in certain important limiting cases.     

Dynamical and secular instability of a self-gravitating rotating cylinder in a toroidal magnetic field

The oscillations and stability of a homogeneous self-gravitating rotating cylinder in a toroidal magnetic field are investigated. It is assumed that the field is proportional to the distance to the axis of the cylinder. We show the existence of four infinite discreta spectra of magnetic (or rotational) modes. Rotation stabilizes the magneticm=1 instability. The magnetic field decreases the growth rate of rotational instability and reduces the interval of unstable wavenumbers. Ifm=1, instability always occurs with the exception of the equipartition state. Ifm>1, the instability can be suppressed by a sufficiently large magnetic field. Resistivity decreases the growth rate of magnetic instability, but increases the growth rate of rotational instability. For zero wavenumber perturbations secular instability occurs due to the action of resistivity before a neutral point is attained where a second secular instabiliity initiates due to the action of resistivity.     

The Tayler instability of toroidal magnetic fields in a columnar gallium experiment

The nonaxisymmetric Tayler instability of toroidal magnetic fields due to axial electric currents is studied for conducting incompressible fluids between two coaxial cylinders without endplates. The inner cylinder is considered as so thin that the limit of R_in → 0 can be computed. The magnetic Prandtl number is varied over many orders of magnitudes but the azimuthal mode number of the perturbations is fixed to m = 1. In the linear approximation the critical magnetic field amplitudes and the growth rates of the instability are determined for both resting and rotating cylinders. Without rotation the critical Hartmann numbers do not depend on the magnetic Prandtl number but this is not true for the corresponding growth rates. For given product of viscosity and magnetic diffusivity the growth rates for small and large magnetic Prandtl number are much smaller than those for Pm = 1. For gallium under the influence of a magnetic field at the outer cylinder of 1 kG the resulting growth time is 5 s. The minimum electric current through a container of 10 cm diameter to excite the instability is 3.20 kA. For a rotating container both the critical magnetic field and the related growth times are larger than for the resting column (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibrational stability of stars in thermal imbalance: A solution in terms of asymptotic expansions

The solution of the partial differential equation describing the ‘non-isentropic’ oscillations of a star in thermal imbalance has been obtained in terms of asymptotic expansions up to the first order in the parameterII/t _s, whereII is the adiabatic pulsation period for the fundamental mode andt _s , a secular time scale of the order of the Kelvin-Helmholtz time. Use has been made of the zeroth order ‘isentopic’ solution derived in I. The solution obtained allows one to derive unambiguously a general integral expression for the coefficient of vibrational stability for arbitrary stellar models in thermal imbalance. The physical interpretation of this stability coefficient is discussed and its generality and its simplicity are stressed. Application to some simple analytic stellar models in homologous and nonhomologous contraction enables one to recover, in a more straightforward manner, results obtained by Coxet al. (1973). Aizenman and Cox (1974) and Davey (1974). Finally, we emphasize that the inclusion of the effects of thermal imbalance in the stability calculations of realistic evolutionary sequences of stellar models, not considered up to now by the other authors, is quite easy and straightforward with the simple formula derived here.     

Does a ‘two-stream’ flow model apply to wakes of large bodies in space?

Structural patterns of ion and electron currents observed on the wake axes of the Ariel J and the Gemini 10 space vehicles are re-examined, together with relevant theoretical and laboratory ‘simulation’, studies. Some insight into existingin-situ data is provided. The possibility that ‘converging-stream’ models describe structural features of current enhancements in the wake region of large spacecraft is discussed.     

Main sequence models for massive Zero-metal stars

Zero-age main-sequence models for stars of 20, 10, 5 and 2M _⊙ with no heavy elements are constructed for three different possible primordial helium abundances:Y=0.00,Y=0.23, andY=0.30. The latter two values ofY bracket the range of primordial helium abundances cited by Wagoner. With the exceptions of the two 20M _⊙ models that contain helium, these models are found to be self-consistent in the sense that the formation of carbon through the triple-alpha process during pre-main sequence contraction is not sufficient to bring the CN cycle into competition with the proton-proton chain on the ZAMS. The zero-metal models of the present study have higher surface and central temperatures, higher central densities, smaller radii, and smaller convective cores than do the population I models with the same masses. If galaxies containing the zero-metal stars were formed as recently as one third the Hubble time, they would likely appear very blue today — perhaps bluer even that most known quasars — and their redshifted effective temperatures could range as high as 3×10⁴ K to 4×10⁴ K.