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.     

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.     

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.     

Expanding and shearing magneto-viscous fluid cosmological model in general relativity

The object of this paper is to investigate the behavior of a magnetic field in a viscous fluid cosmological model. It has been assumed that the expansion () is proportional to the eigenvalue ₁ of the shear tensor _i ^j and the coefficient of shearing viscosity is proportional to the scalar of expansion. The paper also discusses the behavior of the model when the magnetic field tends to zero and comments on some other physical properties.     

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.     

Geometrical dynamics: A new approach to periodic orbits aroundL 4

Geometrical dynamics is the study of the geometry of the orbits in configuration space of a dynamical system without reference to the system's motion in time.Generalized coordinates for the circular restricted problem of three bodies are taken as polar coordinatesr, centered at the triangular libration pointL ₄. A time-independent nonlinear second order ordinary differential equation forr as a function of is derived. Approximations to periodic solutions are obtained by perturbations and Fourier series.     

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.     

Time-dependent nucleation theory and the formation of interstellar grains

The theory of homogeneous nucleation is discussed and applied to the formation of interstellar grains. Special attention is given to the thermodynamics of very small clusters (N10); it is found that, at least for carbon and metals, the free energies ofN-mers may be conveniently expressed in terms of a parameter _N which is nearly independent ofN. Estimates of _N from the Lothe-Pound theory disagree with the experimental observations forN<10; it is preferable to simply approximate _N 1/2 ( is known from bulk properties). The kinetic process of nucleation and cluster growth is essentially just a function of two dimensionless parameters; /T and . The domain in which nucleation theory should be valid is delineated. Approximate formulae are given for the critical supercooling, the critical cluster size, and the mean final cluster size; more accurate numerical results are presented graphically. Nucleation of grains in planetary nebulae, novae, and red giant outflows is discussed. The origin of the Pt-rich nuggets found in the Allende meteorite is considered; it is shown that the nuggets could have been formed by homogeneous nucleation in the primordial solar nebula, whereas they couldnot have formed in a supernova explosion.Invited contribution to the Proceedings of a Workshop onThermodynamics and Kinetics of Dust Formation in the Space Medium, held at the Lunar and Planetary Institute, Houston, 6–8 September, 1978.     

Effects of mass transfer on the unsteady free convection flow past an accelerated vertical porous plate with variable suction

An exact analysis of the effects of mass transfer on the flow of a viscous incompressible fluid past an uniformly accelerated vertical porous and non-porous plate has been presented on taking into account the free convection currents. The results are discussed with the effects of the Grashof number Gr, the modified Grashof number Sc, the Schmidt number Sc, and the suction parametera for Pr (the Prandtl number)=0.71 representating air at 20°C.Nomenclature a suction parameter - C species concentration - C species concentration at the free stream - g acceleration due gravity - Gc modified Grashof number (vg^*(C –C )/U ₀ ³ ) - Pr Prandtl number (C _p/K) - T temperature of the fluid near the plate - T dimensionless temperature near the plate ((T-T )/(T -T )) - U(t) dimensionless velocity of the plate (U/U ₀) - v normal velocity component - v ₀ suction/injection velocity - x, y coordinate along and normal to the plate - v kinematic viscosity (/gr) - C _p specific heat at constant pressure - C _w species concentration at the plate - C non-dimensional species concentration ((C-C )/(C _w -C )) - Gr Grashof number (g(T _w -T )/U ₀ ³ ) - D chemical molecular diffusivity - K thermal conductivity - Sc Schmidt number (/D) - T _w temperature of the plate - T free stream temperature - t time variable - t dimensionless time (tU ₀ ² /) - U ₀ reference velocity - u velocity of the fluid near the plate - u non-dimensional velocity (u/U ₀) - v dimensionless velocity (v/U ₀) - v ₀ non-dimensionalv ₀ (v ₀ /U₀)=–at^–1/2 - y dimensionless ordinate (yU ₀/) - density of the fluid - coefficient of viscosity     

Čerenkov radiation in anisotropic plasmas

Following our series of works on anisotropic radiation, we analyze the erenkov condition in magnetized plasmas in this paper. We have discovered that the usual erenkov condition cos =1/n isnot satisfied at a far field point in anisotropic media, implying that when a charge is moving in a magnetized plasma, a linear shock wave front does not form. Thus we can calculate the power received at a far field per unit time in such a medium — this quantity could not be evaluated according to previous theory. Numerical examples are presented to show various relevant characteristics of erenkov radiation in model plasmas.     

An extended investigation of Helios 1 and 2 observations: The interplanetary magnetic field between 0.3 and 1 AU

Helios-1 and 2 spacecraft allowed a detailed investigation of the radial dependence of the interplanetary magnetic field components between 0.3 and 1 AU. The behaviour of the radial component B _ris in a very good agreement with Parker's model (B _r r ^-2) and the azimuthal component B also shows a radial dependence which is close to theoretical predictions (B r ^-1). Experimental results for the normal component B and for the field magnitude B are consistent with those from previous investigations. The relative amplitude of the directional fluctuations with periods less than 12 hr is essentially independent of heliocentric distance, while their power decreases approximately as r ^–3 without any appreciable difference between higher and lower velocity regimes.Also at Laboratorio Plasma nello Spazio, CNR, Frascati.     

An integrable case of a rotational motion analogous to that of Lagrange and Poisson for a gyrostat in a Newtonian force field

The scope of the present paper is to provide analytic solutions to the problem of the attitude evolution of a symmetric gyrostat about a fixed point in a central Newtonian force field when the potential function isV ⁽²⁾.We assume that the center of mass and the gyrostatic moment are on the axis of symmetry and that the initial conditions are the following: (t ₀)=₀, (t ₀)=₀, (t ₀)=(t ₀)=₀, ₁(t ₀)=0, ₂(t ₀)=0 and ₃(t ₀)= ₃ ⁰ .The problem is integrated when the third component of the total angular momentum is different from zero (B ₁ 0). There now appear equilibrium solutions that did not exist in the caseB ₁=0, which can be determined in function of the value ofl ₃ ^r (the third component of the gyrostatic momentum).The possible types of solutions (elliptic, trigonometric, stationary) depend upon the nature of the roots of the functiong(u). The solutions for Euler angles are given in terms of functions of the timet. If we cancel the third component of the gyrostatic momentum (l ₃ ^r =0), the obtained solutions are valid for rigid bodies.     

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.     

Compton scattering in sources with synchrotron reabsorption

By considering the consecutive effects of synchrotron reabsorption, Compton scattering and other kinds of energy losses of relativistic electrons, it may be possibile to form a universal distribution of electrons in the region of reabsorption (synchrotron reactor). This will be either a power law with a power index of the energy spectrumn _r=3–5, or a relativistic Maxwell distribution with an electron temperatureT _e=4T _b(1+), where is the ratio of Compton (or other losses) to synchrotron losses, andT _bis the brightness temperature of the radiation. Since the total energy losses of electrons in the reactor is equal to zero, this ensures the continuous existence and accumulation of relativistic electrons in the region of reabsorption and their associated hard scattered radiation. Multiple Compton scattering produces a specific stepped power distribution of scattered radiation by which we can identify the reactor. In the nuclei of quasars W _Hand, therefore,n _r=3; hence the spectral index of scattered radiation in the corresponding ranges (optical, UV, X- and -ray) is .Consideration of other kinds of losses and gains of energy by electrons can lead to the dependencen =3–5(E) — where (E) may have either positive or negative values—which, in turn, leads to the frequency dependence of the spectral index of scattered radiation = 1 – (), |()| < 1, |(E)| < 1.Within the framework of the model being considered, the physical parameters of the nucleus of quasar 3C 273 are calculated.     

The spectrum of the Bp star HD 36916

The spectrum of the peculiar Bp star HD 36916 has been studied on plates with dispersion 9.7 A/mm and compared with those of Aur, B3V andi And, B8V. A comparison of the observed contours of H, H and of the Balmer discontinuity D with those computed by Mihalas gives the following results: HD 36916: _e=0.347, logg=3.8;i And: _e=0.383, logg=3.2; Aur: _e=0.302, logg=3.6. HD 36916 presents the characteristics of the Si-4200 stars: strong deficiency in helium and probably also in oxygen and nitrogen, strong excess of silicon and strontium; 3984 Hgii is present. Moreover this star also has characteristics which are not common to Si-4200 stars but rather to Mn stars: excess by a factor of ten of the elements of the iron group and an excess larger, probably of the order of 100, for manganese. The star is a member of the Sword subgroup of the Orion association.     

The investigation of the solar wind transsonic region at meter wavelengths

Radio astronomy experiments designed to probe the interplanetary plasma close to the Sun have been carried out at the wavelength = 2.92 m using the occultation method. The experiments are based on a modification of the occultation method by which the sources, in this case quasars, are used to measure the radial dependence of the apparent source size, the scattering angle (R). The radial dependence of this source size (R) reveals that an anomalous enhancement in the scattering appears at radial distances from about 16 to 30 R , which is associated with the solar wind transsonic region. It is shown by analysis of the theoretical equations that the radial profiles of both the source size (R) and the scintillation index m(R) are consistent in the near solar interplanetary medium (R 40 R ). Combining these two independent applications of the occultation method extends the range of the investigation and provides a powerful new diagnostic of the solar wind transsonic region.     

Effect of Krein signatures on the stability of relative equilibria

We consider the linear stability of the equilibrium points of the generic rotating potentials U(r), U(r, ), U(r, and U(r, , . The stability analysis is performed using the concept of Krein's signature. This signature is calculated for all eigenvalues of the above potentials. Thereby, the Lagrangian points of the restricted three-body problem and the synchronous satellites of oblate and prolate planets are also studied. We find also the new positions of the eigenvalues for perturbations of the original L ₄ and L ₅ points of Mars, Jupiter, Saturn, Uranus and Neptune. Finally, we study the problem with the mass ratio µ close to the critical value and the stability of geostationary satellites perturbed by the Moon.     

Observed oddities in the lines H,K, b and Hβ

We compare microphotometer intensity traces perpendicular to dispersion in simultaneous spectrograms of good spatial resolution traced at various 's in each of the lines. Cross correlations between the different traces show the following: (a) For each _K there is a corresponding _{b 1}at which the coefficient of correlation, r, is a maximum, usually > 0.8. (b) No such high correlations are found between H and H. (c) Comparison of traces in the continuum and at all observed 's in K, H, b₁, b₂ show a range of 's in each line over which r is very significantly negative, while H shows no such peculiarity.     

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.