Solution of the equation of transfer for interlocked multiplets by the method of discrete ordinates with the planck function as a nonlinear function of optical depth

The equation of transfer for interlocked multiplets has been solved by the method of discrete ordinates, originally due to Chandrasekhar, considering nonlinear form of the Planck function to be

Two Sets of Improved Approximate Expressions of the Gyrosynchrotron Radiation

In this paper two sets of improved approximate expressions of emissivity , absorptivity , effective temperature T_eff, and frequency of peak brightness _p of gyrosynchrotron radiation are presented respectively for the ranges from 5 to 10 and 10 to 100 of harmonic numbers s(= /_B). The expressions are designed for the range from 20° to 80° of viewing angle , and the range 2 to 7 of electron energy spectral index . They are expressed by a power-law function in which the indexes are fitted by polynomial expressions of . Their statistical errors are, respectively, 24% and 32% for and for and 28% for . Their accuracies are much better than those of linear fitting of the power-law index.     

Possible semi-annual variation of the Newtonian constant of gravitation

A possible semi-annual variation of the Newtonian constant of gravitationG is established. For the aphelion and perihelion points of the Earth's orbit we find, respectively,

Distribution of Flare Energies Based on Independent Reconnecting Structures

A model is presented to explain the observed frequency distribution of flare energies, based on independent flaring at a number of distinct topological structures (separators) within active-region magnetic fields. The model is a modification and generalization of a recent model due to Craig (2001), and reconciles that model with the observed flare waiting-time distribution, and the observed absence of a flare waiting-time versus energy relationship. The basic assumptions of the model are that flares of energy E ² occur at separators of length , and that the frequency of flaring at a separator is defined by the Alfvén transit time of the structure. To reproduce the observed distribution of flare energies the model requires a probability distribution P( ) ^–1 of separator lengths within active regions. This prediction of the model is in principle testable. A theoretical origin for this distribution is also discussed.     

The motion of the line of apsides of ζ phoenicis

An estimate of the period of the rotation of the line of apsides of the double-star system Phe is obtained by representing the density function as a product of a normal Gaussian distribution and an associated Legendre polynomial .The asymptotic behaviour of this function coincides with the results obtained by Zeldovichet al. (1981).The period of motion of the line of apsides of Phe (about 63 years) obtained in this way comes close to the period determined by an empirical formula for of Batten (1973).     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by Eddington's method

An approximate solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, viz., $$B_v \left( T \right) = b_0 + b_1 e^{ - \beta \tau } $$ is obtained by Eddington's method. is obtained by Eddington's method.     

New perspective about the existence of a supermassive black hole in a quasar with double redshift systems

Research of several years has confirmed the general aspect that the most acceptable model in quasars' nuclei is a rotating Kerr black hole.Assuming the Kerr type potential given by the equation:

The critical and the saturation content of magnetic monopoles in rotating relativistic objects

Both the critical content _c ( N _m /N _B , whereN _m ,N _B are the total numbers of monopoles and nucleons, respectively, contained in the object), and the saturation content _s of monopoles in a rotating relativistic object are found in this paper. The results are:

The dependence of the frequency spectra of the interplanetary magnetic-field fluctuations on the direction of the mean field

The frequency spectra of the interplanetary magnetic field fluctuations are the projection of their wavenumber spectra onto one dimension. Only the frequency spectra can be measured by spacecrafts. It is studied how their measured size depends on the direction of the mean fieldB ₀, which structures the symmetry of the fluctuations relative to the solar wind system. It is specialized for the slab model, Alfvén waves, magneto-acoustic waves and the isotropic case. For the slab model the frequency spectra are proportional to , whereq is the spectral index and the angle betweenB ₀ and the radial direction. For the diffusion coefficientK _TT the relation holds.     

Simulations of cosmic-ray particle diffusion

The diffusion of charged particles in a stochastic magnetic field (strengthB) which is superimposed on a uniform magnetic fieldB ₀ k is studied. A slab model of the stochastic magnetic field is used. Many particles were released into different realizations of the magnetic field and their subsequent displacements z in the direction of the uniform magnetic field numerically computed. The particle trajectories were calculated over periods of many particle scattering times. The ensemble average was then used to find the parallel diffusion coefficient . The simulations were performed for several types of stochastic magnetic fields and for a wide range of particle gyro-radius and the parameterB/B ₀. The calculations have shown that the theory of charged particle diffusion is a good approximation even when the stochastic magnetic field is of the same strength as the uniform magnetic field.     

On the influence of gradients in the angular velocity on the solar meridional motions

If fluctuations in the density are neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations and where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, , and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example, where is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived from the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell flow in terms of : the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral indicative of a transport of angular momentum towards the equator.With the help of a formalism developed earlier we evaluate, for solid body rotation, the velocity correlations and for several values of an arbitrary parameter, D, left unspecified by the theory. The most striking result of these calculations is the increase of with D. Next we calculate the turbulent viscosity coefficients defined by whereC _ro ⁰ and C _o ⁰ are the velocity correlations for solid body rotation. In these calculations it was assumed that ₂ was a linear function of r. The arbitrary parameter D was chosen so that the meridional flow vanishes at the surface for the rotation laws specified below. The coefficients v _ro ⁱ and v _0o ⁱ that allow for the calculation of C _ro and C _0o for any specified rotation law (with the proviso that ₂ be linear) are the turbulent viscosity coefficients. These coefficients comply well with intuitive expectations: v _ro ¹ and –v _0o ³ are the largest in each group, and v _0o ³ is negative.The equations for the meridional flow were first solved with ₀ and ₂ two linear functions of r ( ₀ ¹ = – 2 × 10 ^–12 cm ^–1) and ( ₂ ¹ = – 6 × 10 ¹² cm ^–1). The corresponding angular velocity increases slightly inwards at the poles and decreases at the equator in broad agreement with heliosismic observations. The computed meridional motions are far too large ( 150m s^–1). Reasonable values for the meridional motions can only be obtained if _o (and in consequence ), increase sharply with depth below the surface. The calculated meridional motion at the surface consists of a weak equatorward flow for gq < 29° and of a stronger poleward flow for > 29°.In the Sun, the Taylor-Proudman balance (the Coriolis force is balanced by the pressure gradient), must be altered to include the buoyancy force. The consequences of this modification are far reaching: is not required, now, to be constant along cylinders. Instead, the latitudinal dependence of the superadiabatic gradient is determined by the rotation law. For the above rotation laws, the corresponding latitudinal variations of the convective flux are of the order of 7% in the lower SCZ.     

Stability of motion of the restricted circular and charged three-body problem

Stabiliity is applied to characterize type of motion in which the moving body is confined to certain limited regions and in this sense we may say that the motion of the body in question is stable. This method has been used in the past chiefly in connection with the classical restricted problem of three bodies.In this paper we consider a dynamical system defined by the Lagrangian

The light curved in the CM field

In this paper we introduce the CM field in Sections 2 and 3 based on the paper by Wang and Peng (1985), and calculate the light curved in the CM field in Section 4. The result shows thatP makes _CM larger than _C at , and smaller at . Under a special circumstance which source, CM lens, and observer are in the same line, if we get | ₀₌₀ , and | _=/2 , we can determine theP(M) andQ(M) of the CM lens,M is the mass of the CM lens.     

Ion cyclotron instability in the solar wind

An ion cyclotron instability, arising because of the relative drift between the beam and the main components of the proton distribution function in the solar wind at 1 AU, is studied. The instability is excited in a bounded range of wave numbers provided the relative drift exceeds a certain minimum value called instability threshold. For 1, the instability threshold is smaller than or equal to the threshold of magnetosonic and Alfvén instabilities. The growth rates are enhanced by increasing relative drift and ratio of beam to main proton number density and by decreasing the wave numbers.     

The elimination of the critical terms of a first order Uranus-Neptune theory by Hori's method

In this part we determine the value ofS ₁, and in terms of the canonical variables of H. Poincaré. A complete solution of the auxiliary system of equations generated by the Hamiltonian is presented.     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by the method of discrete ordinates

A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz., $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of discrete ordinates originally due to Chandrasekhar.     

Invariant properties of families of moon-to-earth trajectories

Analytical techniques are employed to demonstrate certain invariant properties of families of moon-to-earth trajectories. The analytical expressions which demonstrate these properties have been derived from an earlier analytical solution of the restricted three-body problem which was developed by the method of matched asymptotic expansions. These expressions are given explicitly to orderµ ^1/2 where is the dimensionless mass of the moon. It is also shown that the inclusion of higher order corrections does not affect the nature of the invariant properties but only increases the accuracy of the analytic expressions.The results are compared with the work of Hoelker, Braud, and Herring who first discovered invariant properties of earth-to-moon trajectories by exact numerical integration of the equations of motion. (Similar properties for moon-to-earth trajectories follow from the principle of reflection). In each instance the analytical expressions result in properties which are equivalent, to orderµ ^1/2, with those found by numerical integration. Some quantitative comparisons are presented which show the analytical expressions to be quite accurate for calculating particular geometrical characteristics.

Nomenclature

Orbital Elements near the Moon energy - angular momentum - semi-major axis - eccentricity - inclination - argument of node - argument of pericynthion Orbital Elements near the Earth h _e energy - l _e angular momentum - i inclination - argument of node - argument of perigee - t _f time of flight Other symbols parameters used in matehing - U a function of the energy near the earth - a function of the angular momentum near the earth - r _p perigee radius - perincynthion radius - radius at node near moon - true anomaly of node near moon - initial angle between node near moon and earth-moon line - a function ofU, , andi - earth phase angle - dimensionless mass of the moon - U ₀, U₁ U=U ₀+U ₁ - i ₀, i_1/2, i₁ i=i ₀+µ ^1/2 i _1/2+µ i ₁ - ₀, _1/2, ₁ = ₀+µ ^1/2 i _1/2+µ i ₁ - _p longitude of vertex line - _n latitude of vertex line - R _o ,S _o ,N _o functions ofU ₀ and - a function ofU ₀, and      

Epstein weight functions for non-radial oscillations of polytropes

Weight functions for the determination of the periods of linear adiabatic non-radial oscillations have been calculated in the same manner as Epstein's classic treatment of purely radial oscillations. Quadrupole (l=2) oscillations for thef and lower orderp andg-modes were considered. One group of static models were polytropes in the range 1.0n4.0 with ; thus included were configurations that were convectively stable, unstable and neutrally stable throughout. Another group consisted ofn=3.0 polytropes with convective shells or convective cores; ₁ was set at different values in each region in order to produce stability ( ) or instability ( ). The weight function provides a pictorial means for assessing the relative importance of each region of a given static model with respect to generating a given non-radial mode.     

Propagating of sound and thermal waves in a reacting fluid

The propagation of linear sound and thermal waves in a reacting fluid, in which the heating and cooling processes can be represented by a heat-loss functionL(, T, is studied. A complex dispersion relation is found, from which the phase velocity and the scale length for damping (or amplification), of the above two-wave mode are calculated Wave amplification may occur in reacting locally stable fluids. Results are applied to a hydrogen plasma model assumed to be heated at a non-specified constant rate and cooled by recombination, excitation, and ionization by collisions, and free-free transitions. The phase velocity , the scale-length for damping , and the relevant relaxation times are calculated as functions of the dimensionless frequency , for temperatures ranging from those at which the hydrogen plasma is neutral to those at which it becomes completely ionized.     

The spherically-symmetrical trajectories of spin particles in the Schwarzschild field

The motion of spin particles in the Schwarzschild field is examined in this paper. It is shown that Mathisson-Papapetrou equations under additional conditions , where is the Lie derivative of the Killing vector _j , permit only radial motion, motion in the equatorial plane and in the equilibrium points. All these types of motion are considered more in detail.