Small-Amplitude Disturbances In A Radiating And Scattering Grey Medium Ii. Solutions Of Given Real Wave Number <Emphasis Type="BoldItalic">k</Emphasis>

We study the fundamental modes of radiation hydrodynamic waves arising from one-dimensional small-amplitude initial fluctuations with wave number k in a radiating and scattering grey medium using the Eddington approximation. The dispersion relation analyzed is the same as that of Paper I (Kaneko et al., 2000), but is solved as a quintic in angular frequency ω while a quadratic in k ² in Paper I. Numerical results reveal that wave patterns of five solutions are distinguished into three types of the radiation-dominated and type 1 and type 2 matter-dominated cases. The following wave modes appear in our problem: radiation wave, conservative radiation wave, entropy wave, Newtonian-cooling wave, opacity-damped and cooling-damped waves, constant-volume and constant-pressure diffusion modes, adiabatic sound wave, cooling-damped and drag–force-damped isothermal sound waves, isentropic radiation-acoustic wave, and gap mode. The radiation-dominated case is characterized by the gap between the isothermal sound and isentropic radiation-acoustic speeds within which there is not any acoustic wave propagating with real phase speed. One of the differences between type 1 and type 2 matter-dominated cases is the connectivity of the constant-volume diffusion mode, which originates from the radiative mode in the former case, while from the Newtonian-cooling wave in the latter case. Analytic solutions are derived for all wave modes to discuss their physical significance. The criterion, which distinguishes between radiation-dominated and type 1 matter-dominated cases, is given by Γ₀ = 9, where Γ₀ = C _p ^(tot)/C _V ^(tot) is the ratio of total specific heats at constant pressure and constant volume. Waves in a scattering grey medium are also analyzed, which provides us some hints for the effects of energy and momentum exchange between matter and radiation.     

Small-Amplitude Disturbances in a Radiating and ScatteringGrey MediumI. Solutions of Given Real Frequency ω

We reanalyze the propagation of one-dimensional small-amplitude disturbances of given real frequency ω in a radiating and scattering grey medium using the Eddington approximation, which has been studied previously by us (Kaneko et al., 1976). Numerical results reveals three frequency regimes to be distinguished, and two wave modes always appear in each frequency regime. The governing equations and analytic solutions are derived for all wave modes using Whitham's method modified into quadratic form and approximate methods based on radiation thermodynamics. In the high-frequency regime appear the radiation-wave and adiabatic sound modes, which are damped by opacity and radiative cooling, respectively. Wave patterns in the intermediate-frequency and low-frequency regimes depend critically on the importance of radiation, for which the criterion is given in terms of the ratio of total specific heats at constant pressure and constant volume. When the radition overwhelms the matter (radiation-dominated case), the radiative mode in the intermediate-frequency regime is the constant-volume diffusion mode. When the matter overwhelms the radiation (matter-dominated case), damped radiation-wave and damped radiation-diffusion modes newly appear between the radiation-wave and constant-volume diffusion modes. The acoustic mode in the intermediate-frequency regime is the isothermal sound mode,which is damped by radiative cooling at higher frequencies and by radiation-thermal drag force at lower frequencies. Two modes appearing in the low-frequency regime are the isentropic radiation-acoustic and constant-pressure diffusion modes. The absorption coefficient derived for the former is shown to be a radiation-thermodynamic extension of that of Landau and Lifshitz (1987). The transition frequencies between all adjacent two modes are also derived to discuss the implications of them. This revised version was published online in July 2006 with corrections to the Cover Date.     

Nonlinear stability in the restricted three-body problem with oblate and variable mass

This study investigates the nonlinear stability of the triangular equilibrium points when the bigger primary is an oblate spheroid and the infinitesimal body varies (decreases) it’s mass in accordance with Jeans’ law. It is found that these points are stable for all mass ratios in the range of linear stability except for three mass ratios depending upon oblateness coefficient A and β, a constant due to the variation in mass governed by Jeans’ law.     

Linear waves in a radiating and scattering grey medium

The equation of radiative acoustics is derived by taking into account the effect of a non-transverse magnetic field, and the solutions are schematically represented. The main results shown in Paper I and Paper II are valid even in the presence of a non-transverse magnetic field, and the only difference is that theadiabatic, isothermal, andisentropic speeds of sound and theradiation-acoustic speed in Paper I which respectively correspond to theadiabatic, isothermal, andisentropic magnetoacoustic speeds and theradiation-magnetoacoustic speed in Paper II are replaced by the sets of speeds ofadiabatic, isothermal, isentropic, andradiation-acoustic fast andslow waves, respectively.     

Incompressible MHD Turbulence

The inertial range of incompressible MHD turbulence is most conveniently described in terms of counter propagating waves. Shear Alfvén waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfvén waves. Cascades composed entirely of shear Alfvén waves do not generate a significant measure of slow waves. MHD turbulence is anisotropic with energy cascading more rapidly along k _⊥ than along k _∥. Anisotropy increases with k _⊥ such that the excited modes are confined inside a cone bounded by k _∥∝ k _perp ^2/3. The opening angle of the cone, θ(k _⊥)∝ k _⊥ ^-1/3, defines the scale dependent anisotropy. MHD turbulence is generically strong in the sense that the waves which comprise it are critically damped. Nevertheless, deep inside the inertial range, turbulent fluctuations are small. Their energy density is less than that of the background field by a factor θ²(k _⊥)≪. MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counter propagating wave packets. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfvén waves are responsible for the mapping's shear and slow waves for its dilatation. The former exceeds the latter by θ^-1(k _⊥)≫ 1 which accounts for dominance of the shear Alfvén waves in controlling the cascade dynamics. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effect of <Emphasis Type="Italic">q</Emphasis>-nonextensive distribution of electrons on electron acoustic solitons

Ion acoustic shock waves (IASWs) are studied in a plasma consisting of electrons, positrons and ions. Boltzmann distributed positrons and superthermal electrons are considered in the plasma. The dissipation is taken into account the kinematic viscosity among the plasma constituents. The Korteweg–de Vries–Burgers (KdV–Burgers) equation is derived by reductive perturbation method. Shock waves are solutions of KdV–Burgers equation. It is observed that an increasing positron concentration decreases the amplitude of the waves. Furthermore, in the existence of the kinematic viscosity among the plasma, the shock wave structure appears. The effects of ion kinematic viscosity (η ₀) and the superthermal parameter (k) on the ion acoustic waves are found.     

Nonlinear stability of equilibrium points in the restricted three-body problem with variable mass

The nonlinear stability of the equilibrium points in the restricted three-body problem with variable mass has been studied. It is found that, in the nonlinear sense, the collinear points are unstable for all mass ratios and the triangular points are stable in the range of linear stability except for three mass ratios, which depend upon β, the constant due to the variation in mass governed by Jeans’ law.     

Exact solution to radiation-filled Brans-Dicke closed space cosmology

Under the assumption of a power law (k·R ⁿ=C,C=const.) between the gravitational constantk and the radius of curvatureR of the Universe and forP=1/3 the exact solution is sought for the cosmological equations of Brans and Dicke. The solution turns out to be valid for closed space and the parameter of the scalar-tensor theory is necessarily negative. The radius of curvature increases linearly with respect to the age of the Universe while the gravitational constant grows with the square of the radius of curvature. It has been shown (Lessner, 1974) that in this case (KR ²) the spatial component of the field equations is independent of the remaining equations. However, our solution satisfies this independent equation. This solution for the radiation-dominated era corresponds to the solution for the matter-dominated era found by Dehnen and one of the authors (Dehnen and Obregón, 1971). Our solution, as is the solution previously obtained for the matter-dominated era, is in contradiction to Dirac's hypothesis in which the gravitational constant should decrease with time in an expanding Universe.     

On the formation of void in the Universe

This paper deals with the void with a volume of ～10⁶ Mpc in the Universe. By using spherically-symmetric Einstein field equations the dynamics of void is studied. It seems to be possible that the void will decay in the radiation-dominated era so that the void will amplify and form in the matter-dominated era only.     

The formation of shock waves in the recombination era of the universe

The formation and the evolution of shock waves in a two-component fluid during the recombination era of the Universe is studied. The fluid consists of photons and ionized hydrogen gas with interaction due to Thomson scattering. The analysis uses the expansion procedure in powers of the coordinates near the travelling wave front. Incorporating discontinuities in the first derivatives of the physical quantities, we are able to give the analytical solution which answers the questions as to when the discontinuities appear in the physical quantities themselves. Of the two possible waves (a radiation- and a matter-dominated wave) only the matter-dominated wave shows the shock phenomena. However, the shock is induced by the radiation due to Thomson scattering. Therefore, the time for the shock formation depends on the initial amplitude of the wave (as in usual shock phenomena) and on the collision frequency of the photons.     

Non-radial oscillations and energy transport in rotating solar (stellar) wind

The propagation of non-radial, small amplitude perturbations superposed on a zero-order, stationary, non-magnetic, polytropic, rotating stellar wind is studied in the limit of the local theory, i.e. for k r 1, k being the module of the wave vector and r the characteristic scale of the zero-order flow. The resulting dispersion equation is of the 3rd order in (complex) frequency and the possible modes correspond to two acoustic type waves, and to a gravity-shear wave with strongly anisotropic propagation properties, due to coupling between the internal gravity waves and shear motion. The gravity-shear mode allows velocity differences in the medium to exist with no corresponding density fluctuations and hence with no shock wave formation. It is suggested that this mode corresponds to some of the fast-slow velocity streams observed in the interplanetary medium and may provide means for wave energy being transported outwards with the zero-order flow, with little dissipation in the inner region of the solar wind.     

Magnetoacoustic surface waves at a single interface

The occurrence of magnetoacoustic surface waves at a single magnetic interface one side of which is field-free is explored for the case of non-parallel propagation. Phase-speeds and penetration depths of the waves are investigated for various Alfvén speeds, sound speeds and angles of propagation to the applied field. Both slow and fast magnetoacoustic surface waves can exist depending on the values of sound speeds and propagation angle. The fast waves penetrate more than the slow waves.The parallel propagation of fast and slow magnetoacoustic surface waves on a magnetic-magnetic interface is investigated. The slow surface wave is unable to propagate below a critical sound speed. In a low -plasma, only the fast mode exists (0 0).     

On the triangular libration points in photogravitational restricted three-body problem with variable mass

This paper investigates the triangular libration points in the photogravitational restricted three-body problem of variable mass, in which both the attracting bodies are radiating as well and the infinitesimal body vary its mass with time according to Jeans’ law. Firstly, applying the space-time transformation of Meshcherskii in the special case when q=1/2, k=0, n=1, the differential equations of motion of the problem are given. Secondly, in analogy to corresponding problem with constant mass, the positions of analogous triangular libration points are obtained, and the fact that these triangular libration points cease to be classical ones when α≠0, but turn to classical L ₄ and L ₅ naturally when α=0 is pointed out. Lastly, introducing the space-time inverse transformation of Meshcherskii, the linear stability of triangular libration points is tested when α>0. It is seen that the motion around the triangular libration points become unstable in general when the problem with constant mass evolves into the problem with decreasing mass.     

Linear wave conversion processes in the theory of the proton whistler

The linear coupling between the different kinds of waves propagating in a warm plasma inhomogeneous along thex direction is investigated in order to locate the regions (,k) space where two of the roots of the characteristic equation coalesce. Firstly, using the approximation of geometrical optics the differential equation is derived and wave propagation at fixed wave numberk _z is studied in these special cases for which the characteristic equation reduces to a biquadratic. When the density gradient is parallel to the magnetic field, a detailed analysis of the different properties of the waves shows that the mechanism proposed by Gurnett and others to explain the characteristics of the proton whistler is unlikely to operate, even if a wave coupling occurs at the so called cross over frequency for small incidence angles. The only relevant process occurs when the density gradient is perpendicular to the magnetic field for waves propagating at small incidence angles. It is shown that, close to a coalescence point, but within the limit of the geometrical optics approximation, one of the WKB solutions is a mixed (transverse-longitudinal) mode which becomes purely longitudinal in the limit of large wave numbers. Consequently, as this wave has E almost parallel tok, coalescence implies that the waves are nearly longitudinal at the singular point, in agreement with other results. Next, application of the theory is made to some relevant space observations. It is shown that the proton whistler could be the result of a linear coupling between the extraordinary and the slow ion cyclotron waves close to the Buchsbaum resonance in ionospheric regions above 300 to 400 km where the H⁺ density begins to grow. Transformation conditions are given which favour the coupling mechanism in regions of strong latitudinal gradients. Finally, a comparison is made with experiment which confirms the principal features of the present theory.     

Spatial damping of linear compressional magnetoacoustic waves in quiescent prominences

We study the spatial damping of magnetoacoustic waves in an unbounded quiescent prominence invoking the technique of MHD seismology. We consider Newtonian radiation in the energy equation and derive a fourth order general dispersion relation in terms of wavenumberk. Numerical solution of dispersion relation suggests that slow mode is more affected by radiation. The high frequency waves have been found to be highly damped. The uncertainty in the radiative relaxation time, however, does not allow us to conclude if the radiation is a dominant damping mechanism in quiescent prominence.     

Ammonia in the atmospheres of Jupiter and Saturn: Absorption coefficients

Spectral values (with 1 nm spectral resolution) of the product of γk _ν and γ′k _ν (where k _ν is the monochromatic coefficient of ammonia absorption and γ and γ′ are the relative (with respect to 0.85/0.15 hydrogen-helium mixture and methane, respectively)) concentrations of ammonia for the absorption bands at λλ = 552, 604, 645, 787, and 932 nm in thermal conditions of Jupiter’s and Saturn’s atmospheres are determined ().     

Shock waves on neutron star cores

A numerical model of the transient behavior of a radiation-dominated shock was calculated in order to demonstrate the relatively large initial escape of internal energy that takes place when the opacity law is a positive power of temperature, as for neutrinos,K∼T ². Attention was given to the consequences of diffusive transport of radiation versus local production of internal energy by duplicating the calculation with and without artificial viscosity. It is concluded that a shock formed on the neutron star core of an imploding supernova may radiate its internal energy in electron neutrinos more effectively than had hithertofore been considered.     

Radio emission of fast cosmic ions

It is usually assumed that the ions of cosmic rays contribute nothing to the observable electromagnetic radiation. However, this is true only when these ions are moving in a vacuum or a quiet (nonturbulent) plasma. In the case of fast ions in a turbulent plasma, there is an effective nonlinear mechanism of radiation which is discussed in this paper. The fast ion (relativistic or nonrelativistic) moving in the plasma creates a polarization cloud around itself which also moves with the particles. The turbulent plasma waves may scatter on the moving electric field of this polarization cloud. In the process of this scattering an electromagnetic wave with frequency (2.7) is generated. Let ₁ and k₁ be the frequency and wave vector of turbulent plasma waves,V is the velocity of the ion, and is the angle between the wave vector of electromagnetic radiation and the direction of the ion velocity. The method of calculating the probability of the conversion of plasma waves (k₁) into electromagnetic waves (k) by scattering on an ion with velocityV is described in detal in Section 2 (Equation (2.14)).The spectral coefficients of spontaneous radiation in the case of scattering of plasma waves on polarization clouds created by fast nonrelativistic ions are given in (3.6) for an ion energy distribution function (3.4) and in (3.8) for more general evaluations. The Equations (3.9)–(3.13) describe the spectral coefficients of spontaneous emission for different modes of plasma turbulence (Langmuir (3.9), electron cyclotron in a weak (3.10) or strong (3.11) magnetic field and ion acoustic (3.12)–(3.13) waves). The coefficients of reabsorption or induced emission are given by Equations (3.14) and (3.16)–(3.19). There is a maser effect in the case of scattering of plasma waves on a stream of ions. The effective temperature of the spontaneous emission is given by Equation (3.15). The spectral coefficients of radiation due to scattering of plasma waves on relativistic ions are calculated in the same manner (Equations (4.14)–(4.15)). The total energy loss due to this radiation is given in Equations (4.23)–(4.25). The coefficients of induced emission are given in (4.26)–(4.28).The results are discussed in Section 5. It is shown that the loss of energy by nonlinear plasma radiation is much smaller than the ionization loss. However, the coefficients of synchrotron radiation of electrons and nonlinear radiation of ions under cosmic conditions may be comparable in the case of a weak magnetic field and fairly low frequencies (5.5)–(5.6). Usually the spectrum of nonlinear plasma radiation is steeper than in the case of synchroton radiation. Equation (5.10) gives the condition for nonlinear radiation to prevail over thermal radiation.Translated by D. F. Smith.     

A New Look at Mode Conversion in a Stratified Isothermal Atmosphere

Recent numerical investigations of wave propagation near coronal magnetic null points (McLaughlin and Hood: Astron. Astrophys. 459, 641, 2006) have indicated how a fast MHD wave partially converts into a slow MHD wave as the disturbance passes from a low-β plasma to a high-β plasma. This is a complex process and a clear understanding of the conversion mechanism requires the detailed investigation of a simpler model. An investigation of mode conversion in a stratified, isothermal atmosphere with a uniform, vertical magnetic field is carried out, both numerically and analytically. In contrast to previous investigations of upward-propagating waves (Zhugzhda and Dzhalilov: Astron. Astrophys. 112, 16, 1982a; Cally: Astrophys. J. 548, 473, 2001), this paper studies the downward propagation of waves from a low-β to high-β environment. A simple expression for the amplitude of the transmitted wave is compared with the numerical solution.     

Magnetic Power Spectra Derived from Ground and Space Measurements of the Solar Magnetic Fields

We study magnetic power spectra of active and quiet regions by using Big Bear Solar Observatory and SOHO/MDI measurements of longitudinal magnetic fields. The MDI power spectra were corrected with Gaussian Modulation Transfer Function. We obtained reliable magnetic power spectra in the high wave numbers range, up to k=4.6 Mm⁻¹, which corresponds to a spatial scale l=1.4 Mm. We find that the occurrence of the spectral discontinuity at high wave numbers, k≥3 Mm⁻¹, largely depends on the spatial resolution of the data and it appears at progressively higher wave numbers as the resolution of the data improves. The spectral discontinuity in the raw spectra is located at wave numbers about 3 times smaller than wave numbers, corresponding to the resolution of the data, and about 1.5–2.0 times smaller in the case of the noise- and-resolution corrected spectra. The magnetic power spectra for active and quiet regions are different: active-region power spectra are described as ∼k ^−1.7, while in a quiet region the spectrum behaves as ∼k ^−1.3. We suggest that the difference can be due to small-scale dynamo action in the quiet-Sun photosphere. Our estimations show that the dynamo can generate more than 6% of the observed magnetic power.