The Energy Equation in the Lower Solar Convection Zone

As a consequence of the Taylor–Proudman balance, a balance between the pressure, Coriolis and buoyancy forces in the radial and latitudinal momentum equations (that is expected to be amply satisfied in the lower solar convection zone), the superadiabatic gradient is determined by the rotation law and by an unspecified function of r, say, S(r), where r is the radial coordinate. If the rotation law and S(r) are known, then the solution of the energy equation, performed in this paper in the framework of the ML formalism, leads to a knowledge of the Reynolds stresses, convective fluxes, and meridional motions. The ML-formalism is an extension of the mixing length theory to rotating convection zones, and the calculations also involve the azimuthal momentum equation, from which an expression for the meridional motions in terms of the Reynolds stresses can be derived. The meridional motions are expanded as U _r(r,)=P ₂(cos)₂(r)/r ²+P ₄(cos)₄(r)/r ² +..., and a corresponding equation for U (r,). Here is the polar angle, is the density, and P ₂(cos), P ₄(cos) are Legendre polynomials. A good approximation to the meridional motion is obtained by setting ₄(r)=–H₂(r) with H–1.6, a constant. The value of ₂(r) is negative, i.e., the P ₂ flow rises at the equator and sinks at the poles. For the value of H obtained in the numerical calculations, the meridional motions have a narrow countercell at the poles, and the convective flux has a relative maximum at the poles, a minimum at mid latitudes and a larger maximum at the equator. Both results are in agreement with the observations.     

ION–Neutral Collisions Effect on MHD Surface Waves

The effect of ion–neutral collisions on the propagation of MHD waves and surface waves at a single magnetic interface is investigated. The dispersion equations for MHD waves in a partially ionized medium are derived. There are three damped propagating modes in a uniform unbounded medium: an Alfvén mode, and fast and slow modes. The damping of waves depends on both the collisional frequency and the ionization fraction. Wave damping increases as ionization fraction decreases. Surface waves are discussed in three cases: (a) the incompressible limit, (b) the low plasma, and (c) for parallel propagation. The incompressible limit leads to Alfvén surface waves in a partially ionized medium and the dispersion characteristics are similar to those obtained by Uberoi and Datta. In the low plasma of the Earth's auroral F region there are two damped propagating magnetoacoustic surface waves for =/3. There is only one damped surface mode for =/2, but no surface wave is able to propagate for =0°. For the case of parallel propagation (=0°) the results obtained in the absence of ion-neutral collisions are consistent with the results of Jain and Roberts. It is found that a three-mode structure of damped propagating waves occurs owing to ion–neutral collisions for a comparatively high ionization fraction. For the case of the solar photosphere, where the ionization fraction is low, two weakly damped surface waves are found, though the damping is almost negligible. The pattern of propagation is similar to that found in the case discussed by Jain and Roberts, but the wave speeds are lower due to ion–neutral collisions. The strong collisions tie the ion–neutral species together and reduce the damping.     

A New Method for Resolving the 180° Ambiguity in Solar Vector Magnetograms

We present a new method to resolve the 180° ambiguity for solar vector magnetogram measurements. The basic assumption is that the magnetic shear angle (), which is defined as the difference between the azimuth components of observed and potential fields, approximately follows a normal distribution. The new method is composed of three steps. First, we apply the potential field method to determine the azimuthal components of the observed magnetic fields. Second, we resolve the ambiguity with a new criterion: –90°+_mp lele90°+_mp, where _mp is the most probable value of magnetic shear angle from its number distribution. Finally, to remove some localized field discontinuities, we use the criterion B _tB _mt ge0, where B _t and B _mt are an observed transverse field and its mean value for a small surrounding region, respectively. For an illustration, we have applied the new ambiguity removal method (Uniform Shear Method) to a vector magnetogram which covers a highly sheared region near the polarity inversion line of NOAA AR 0039. As a result, we have found that the new ambiguity solution was successful and removed spatial discontinuities in the transverse vector fields produced in the magnetogram by the potential field method. It is also found that our solution to the ambiguity gives nearly the same results, for highly sheared vector magnetograms and vertical current density distributions, of NOAA AR 5747 and AR 6233 as those of other methods. The validity of the basic assumption for an approximate normal distribution is demonstrated by the number distributions of magnetic shear angle for the three active regions under consideration.     

εлектроны космических лучей радиогало галактики

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О колебаниях хвоста магнитосферы в приБлижении квазигидро динамики

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Возможный механизм радиоизлучения пульсаров

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Bianchi type-VI0 perfect fluid cosmological models with magnetic field

In this paper we discuss Bianchi type-VI₀ cosmological models with perfect fluid distribution and magnetic field directed along the axial direction. Einstein's equations have been solved for the condition that the expansion scalar bears a constant ratio to the anisotropy in the direction of a space-like unit vector ⁱ.     

Critical inclinations in planetary problems

The general conception of the critical inclinations and eccentricities for theN-planet problem is introduced. The connection of this conception with the existence and stability of particular solutions is established. In the restricted circular problem of three bodies the existence of the critical inclinations is proved for any values of the ratio of semiaxes . The asymptotic behaviour of the critical inclinations as 1 is investigated.

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Expanding and superdense astrophysical systems in general relativistic hierarchical cosmology

A semi-continuous hierarchy, (i.e., one in which there are galaxies outside clusters, clusters outside superclusters etc.), is examined using an expression of the field equations of general relativity in a form due to Podurets, Misner and Sharp. It is shown (a) that for a sufficiently populous hierarchy, the thinning factor( _i+1/ _i [r _i /r _i+1] is approximately equal to the exponentN in a continuous density law (=aR ^–N) provided (r _i /r _i+1)^3-1; (b) that a hierarchical Universe will not look decidedly asymmetric to an observer like a human being because such salient observers live close to the densest elements of the hierarchy (viz stars), the probability of the Universe looking spherically symmetric (dipole anisotropy0.1 to such an observer being of order unity; (c) the existence of a semi-continuous or continuous hierarchy (Peebles) requires that 2 if galaxies, not presently bound to clusters were once members of such systems; (d) there are now in existence no less than ten arguments for believing 2, though recent number counts by Sandageet al. seem to be in contradiction to such a value; (e) Hubble's law, withH independent of distance, can be proved approximately in a relativistic hierarchy provided (i)N=2, (ii)2GM(R)/c ² R1; (iii)Rc (iv)M0 in a system of massM, sizeR (f) Hubble's law holds also in a hierarchy with density jumps; (g)H100 km s^–1 Mpc^–1; (h) objects forming the stellar level of the hierarchy (in a cosmology of the Wilson type) must once have had 2GM/c ² R1; (i) there is a finite pressurep=2Ga in all astrophysical systems (a=R ^N ,N2); (j) for the Galaxy, theory predictsp _G7×10^–12 dyn cm^–2, observation givesp _G5×10^–12 dyn cm^–2; (k) if the mass-defect (or excess binding energy) hypothesis is taken as a postulate, all non-collapsed astrophysical systems must be non-static, and any non-static, p0 systems must in any case be losing mass; (1) the predicted mass-loss rate from the Sun is 10¹² g s^–1, compared to 10¹¹ g s^–1 in the observed solar wind; (m) the mass-loss rates known by observation imply timescales of 5×10⁹ years for the Sun and 10¹⁰ years for other astrophysical systems; (n) degenerate superdense objects composed of fermions must haveN-2 if they were ever at their Schwarzschild radii and comprised a finite numberN _B of baryons; (o)N _B10⁵⁷N for degenerate fermion and boson systems; (p)285-4; (q) the metric coefficients for superdense bodies give equations of motion that imply equal maximum luminosities for all evolving superdense bodies (L _max10⁵⁹ erg s^–1); (r) larger bodies have longer time-scales of energy radiation atL _max (10^–5 s for stars,1 h for QSO's) (s) expansion velocities are c soon after the initial loss of equilibrium in a superdense object; (t) if the density parametera(t) in aR ^–N isa=a (non-atomic constants of physicsc, G, A), andA, thenN=2; (u) N2 is necessary to giveMM at the stellar level of the hierarchy;(v) systems larger than, and including, galaxies must have formed by clumping of smaller systems and not (as advocated by Wertz and others) in a multiple big bang.     

Радиоизлучение Быстрых космических ионов

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Study of the nonradial directional property of the rays of the streamer belt and chains in the solar corona

Based on analyzing corona images taken by the LASCO C1, C2, and C3 instruments, a study is made of the behavior of the streamer belt spanning one half of the 1996–2001 cycle of solar activity, from minimum to maximum activity, in the absence of coronal mass ejections. It is shown that: (1) The position of the streamer belt relative to the solar equator is generally characterized by two angles: _o and _E, where _o is the latitudinal position (near the solar surface) of the middle of the base of the helmet, the top of which gradually transforms to a ray of the streamer belt with a further distance from the Sun, and _E is the latitude of this ray for R>5–6 R from the Sun's center where the ray becomes radial. (2) Only rays lying at some of the selected latitudes _o retain their radial orientation (_oE) throughout their extent. Namely: _o0° (equator), _o±90° (north and south poles), and the angle _o lying in the range ±(65°–75°) in the N- and S-hemispheres. (3) A deviation of rays from their radial orientation in the direction normal to the surface of the streamer belt occurs: for latitudes _o<|65°–75°| toward the equator (>0°) reaching a maximum in the N and S hemispheres, respectively, when _OM40°, and _OM–42° for latitudes _o>|65°–75°| toward the pole (<0°). The regularities obtained here are a numerical test which can be used to assess of the validity of the theory for describing the behavior of the Sun's quasi-stationary corona over a cycle of solar activity.     

The orientation of magnetic fields in quiescent prominences

We have measured the longitudinal component, B, of the magnetic field in quiescent prominences and obtained a relationship between B and , where is the angle between the long axis of the prominence and the north-south direction on the sun. From this relationship we deduce a distribution function for the magnetic field vector in quiescent prominences in terms of the angle between the field and the long axis of the prominence. The mean angle, , for our data is small, - 15°, indicating that the magnetic field traverses quiescent prominences under a small, but finite angle.On leave from Max-Planck Institut für Physik und Astrophysik, München.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Implications of a strongly peaked polar magnetic field

Using the flux-transport equation in the absence of sources, we study the relation between a highly peaked polar magnetic field and the poleward meridional flow that concentrates it. If the maximum flow speed _m greatly exceeds the effective diffusion speed /R, then the field has a quasi-equilibrium configuration in which the poleward convection of flux via meridional flow approximately balances the equatorward spreading via supergranular diffusion. In this case, the flow speed () and the magnetic field B() are related by the steady-state approximation () (/R)B()/B() over a wide range of colatitudes from the poles to midlatitudes. In particular, a general flow profile of the form sin ^p cos ^q which peaks near the equator (q p) will correspond to a cos ⁿ magnetic field at high latitudes only if p = 1 and _m = n /R. Recent measurements of n 8 and 600 km² s^–1 would then give _m 7 m s^–1.     

On symmetrical planetary corotations

Exact corotations are equilibrium points in the phase space of the asteroidal elliptic restricted problem of three bodies averaged over the synodic period, at a mean-motions resonance. If the resonant critical angle is =(p+q) _jup –p–q, exact corotations are double resonant motions defined by the conditionsd/dt=0 andd(– _jup )/dt=0. The first condition is characteristic of the periods resonance(p + q) : p and the second one is a secular resonance equivalent to that usually known as thev ₅-resonance. This paper presents the symmetric solutions =0 (mod ), = _jup (mod ). Corotations have a coherence property which is unique in non-collisional Celestial Mechanics: An elementary calculation shows that, in the neighbourhood of these solutions, the motions cluster aroundp independent longitude values and are, in each cluster, as close together as and are close to the equilibrium values.     

The stability parameters of three-body periodic orbits

Formulae containing the elements of the variational matrix are obtained which determine the linear iso-energetic stability parameters of periodic orbits of the general three-body problem. This requires the numerical integration of the variational equations but produces the stability parameters with the effective accuracy of the numerical integration. The procedure is applied for the determination of horizontally critical orbits among the members of sets of vertical-critical periodic orbits of the threebody problem. These critical-critical orbits have special importance as they delimit the regions in the space of initial conditions which correspond to possibly stable three-dimensional periodic motion of low inclination.     

An element formulation for perturbed motion about the center of mass

The perturbed motion of a rigid body about its center of mass, is formulated in terms of the six elements:l, the magnitude of the angular momentum vector;h, the total energy; and , two linear functions of the independent variable; and ₁ and ₁, two Euler angles that orientate the inertial frame with respect to the unperturbed solution. Solutions from the element formulation and the original Euler equations are numerically compared using shuttle-type data. For applied torques smaller than a given magnitude, the element formulation produced the following results: (1) larger step sizes in the numerical integration of the differential equations, resulting in an overall computational time-saving, and (2) more significant figures of accuracy in the computation of the variables describing the state of the rigid body.     

Linear analysis of the light curves of eclipsing variables

In this paper, general model analysis of the form c _{i i} () for the fractional loss of lightf() exhibited by a close binary system will be considered in the sense of the least-squares criterion. A general recursive method will be established for constructing the normal equations for the most useful functions _i (), and an economical storage of the method on a digital computer, with its computational steps, will also be given. Moreover, a full recursive computational algorithm for the least-squares approximation will also be established. By means of this algorithm, all the solution vectors, the variance for different orders of fit and the corresponding variance-covariance matrix could be computed once and for all and, moreover, recursively. The economical storage of the algorithm and its computational steps will be given. Finally, some of the practical difficulties encountered in the application of the least-squares criterion will be analysed, and some techniques for detecting and controlling these difficulties are also given. Numerical examples on the Algol system using a Fourier cosine series of the form c _i cos [(j - 1) /] will be given for illustration.     

A simplified model solar atmosphere

A simplified representation of the temperature distribution in the solar photosphere is proposed: ( ₀) = ₀ - ₁ log ₀. An expression is derived for the emergent continuous spectrum from the simple model. The limitations and applications of the simple model are discussed.     

Generalized Expanding and Shearing Anisotropic Bianchi Type I Magnetofluid Cosmological Model in General Relativity

The behaviour of magnetic field in anisotropic Bianchi type I cosmological model for perfect fluid distribution in General Relativity, is investigated. The distribution consists of an electrically neutral perfect fluid with an infinite electrical conductivity. It is assumed that the component ¹ ₁ of shear tensor ^j _i is proportional to the expansion () which leads to A = (BC)ⁿ. The other physical and geometrical aspects of the model are also discussed, Bali (1986) obtained the cosmological model for n = 1 in presence of magnetic field. We have investigated the model for general values of n and discussed the particular case and general behaviour of the model.     

The implications of intrinsic luminosity evolution on the value of the density parameter (Ω) and the evolution of radio sizes of radio galaxies and quasars

The implications of the intrinsic luminosity evolution with cosmological epoch on the value of the density parameter () and evolution of radio sizes of extragalactic radio sources have been considered. It is shown that a power law evolution model of the sortP (1 +z) can be used to contrain the value of . In the presence of a strong luminosity evolution, the model yields an upper limit of 0.5.It is also shown that the angular diameter redshift ( – z) relation for quasars can be interpreted in terms of the assumed luminosity evolution combined with a luminosity-linear size correlation with little or no linear size evolution required. On the other hand, strong linear size evolution is needed to explain the – z data for radio galaxies independent of luminosity.