The restricted problem of n+ν bodies

The simplest, conventional, and original form of the circular restricted problem of three bodies is briefly described in sidereal and synodic systems using dimensional and non-dimensional variables. This dynamical system is generalized to n2 primary bodies (from n=2) with masses M_i, 1in, interacting with arbitrary force laws (instead of only gravitational forces). The number of bodies of small mass mM_i not perturbing the primaries is increased from =1 to 1 where 1 and the minor bodies are allowed to interact with one another under arbitrary force laws. While the minor bodies (m) do not affect the motions of the primaries (M_i), the primaries influence the motions of the minor bodies with arbitrary force laws.For the case where n=2, 1, and only gravitational forces act on the system, an integral of the system is derived. It is shown that the energy integral of the general problem of N bodies and the Jacobian integral of the classical restricted problem of three bodies are limiting cases of this integral. The role of the integral in bounding the motion of the minor bodies is discussed. Several applications of this system are given.     

On the solving of the Hill's differential equation through the method of operational calculus

Using a method previously applied to the treatment of the Mathieu differential equation, we solve the Hill's differential equation of lunar theory through the way of operational calculus, which avoids the cumbersome infinite determinants of the classical procedure. The one-sided Laplace transformation changes it into a difference equation with an infinite number of terms and variable coefficients. When its first member is divided by a suitable factor, this difference equation is the image of an integral equation of the Volterra type which is equivalent to the initial Hill's differential equation. Solution of this Volterra integral equation is unique and it is the general solution of the Hill's differential equation. This solution is a series in the powers of a small dimensionless parameter ² which appears as a factor in the second member of the Hill's differential equation. We reduce it to the sum of its terms of degree 12 with respect to which is the precision usually required in a lunar theory and we write down effectively the coefficients of the terms in ², ( ²)² and the coefficient of the term in ( ²)³ which depends upon the initial valuey(0) of the Hill's differential equation.     

On the occurrence of critical levels in solar magnetohydrodynamics

It is shown that the singular behaviour exhibited by a solution of the magnetoatmospheric wave equation for motion in the presence of a horizontal magnetic field is a special case of the valve type critical level discussed by Acheson (1973), with the difference that the valve effect does not strictly occur; waves are captured as they approach the singular level from either side and are neither reflected or transmitted, but constrained to propagate along the field line. This effect is also likely to occur for purely vertical fields. The possible importance of such critical levels to solar physics is discussed.     

Detecting Very-High-Frequency Relic Gravitational Waves by a Waveguide

The polarization vector (PV) of an electromagnetic wave (EW) will experience a rotation in a region of spacetime perturbed by gravitational waves (GWs). Based on this consideration, Cruise's group has built an annular waveguide to detect GWs. We give detailed calculations of the rotations of polarization vector of an EW caused by incident GWs from various directions and in various polarization states, and then analyze the accumulative effects on the polarization vector when the EW passes n cycles along the annular waveguide. We reexamine the feasibility and limitation of this method to detect GWs of high frequency around 100 MHz, in particular the relic gravitational waves (RGWs). By comparing the spectrum of RGWs in the accelerating universe with the detector sensitivity of the current waveguide, it is found that the amplitude of the RGWs is too low to be detected by the waveguide detectors currently operating. Possible ways of improvements on detection are suggested.     

The relation between brightness fluctuations and polarization of solar metre-wave emission

Solar 80 MHz emission recorded by the Culgoora radioheliograph over a number of years, no matter what its spectral type, has followed a persistent pattern. Emission with average polarization < 40%, even if bursty, shows constant polarization: the polarization remains weak during both lulls and bursts of brightness. By contrast, if emission with average polarization > 40% is bursty, it shows fluctuating polarization: the polarization decreases during the bursts. The effect is discussed in terms of streaming versus gyrating electrons.     

Penumbral filaments: A dissipation form of the magnetic field

In this paper we suggest that penumbral filaments are a phenomenon of magnetohydrodynamic instability, developed in a stable and uniform magnetic field of sunspots during a dissipation process. We have solved local magnetohydrodynamic disturbance equations and have obtained the necessary condition for filament instability mode, that the ratio of filament length to width must be larger than the ratio of Alfvén speed to sound speed. We have also obtained correlations between two fluctuations from their phase difference. Although there are two correlations between the fluctuation of temperature (or filament intensity) and (1) the fluctuation of magnetic field, and (2) the fluctuation of the flow during the phases of developing and dissipating of the filament, we cannot distinguish whether the correlation is associated with the light filament or dark filament and we cannot decide whether the phase difference is 0° or 180° from tg() = 0. However, we can make a judgment: if the correlation is associated with a light filament during its development phase, it will be associated with a dark filament during its dissipation phase, andvice versa. In addition, there are no correlations between the fluctuations mentioned above for a stable filament, because the phase difference of the filament is changing with time.The phase differences of filaments are related to the existence of a gravitational field.     

Dynamical evolution of the rotation of Venus

By considering the torque of the bodily tides, the effect of the core-mantle viscous coupling and the torque of the atmospheric tides have been obtained by numerical calculation: the evolution of the spin angular velocity and the obliquity of the Venus are calculated numerically with the step-variable Runge-Kutta method of 7th order; and 7 sets of the probable Cytherean spin evolution have been obtained. It is indicated that the present spin state of Venus is the result of long-term evolution within the reasonable ranges of some disposable parameters. The early spin period is between 7 h to 2 d and the corresponding obliquity is about 90 ° ~ 100 °. The effects of the torques of body and atmospheric tides and the core-mantle viscous coupling of Venus on its spin angular velocity could nearly cancel out each other about a billion years ago. Therefore, Venus could have been captured in a spin-orbit resonant state by the gravitational torque of the Earth on the permanent deformation part of Venus; and this resonant state has lasted up to the present time.     

On the A-transform of the exponential integrals

In this paper, a technique of recursive analysis is developed for the integral transform A of the exponential integral functionsE _n which is denoted as _n (). The main result of this analysis enables us to establish a two-term recurrence formula for _n (0) and a three-term recurrence formula for _n (); 0. A computational algorithm based on these formulae is also constructed and its numerical results forn=2(1)25 are presented to 15-digit accuracy.     

Diffusion approximation in the theory of radiative transfer

The transfer process of hard X-ray radiation passing through a cold plasma is studied in this paper under the diffusion approximation. We call such a particular transfer process a down-Comptonization. A diffusion equation describing this process is derived and its potential applications in astrophysics and radiation physics are also pointed out.     

Sobolev's Φ-function of radiative transfer in planar and spherical scattering media

In recent years Sobolev's -function of radiative transfer has been discussed in connection with the resolvent of Milne's integral equation such that it plays an important role in the determination of the radiation field in semi-infinite (or finite) atmospheres with internal sources (cf. Sobolev, 1963). In the present paper, the part of Sobolev's -function in plane-parallel and spherical, isotropically scattering, atmospheres with internal source distribution is investigated from analytical and numerical aspects. With the aid of invariant imbedding (cf. Bellmanet al., 1968), we computed Sobolev's -function of Milne's integral equation for the planar case by solving the Cauchy system for the auxiliary function and Chandrasekhar'sX- andY-functions. The corresponding -function for the spherical case is readily obtained from the for the planar case.Investigation supported by the National Science Foundation under Grant No. GP 29049, the Atomic Energy Commission, Division of Research under Contract No. AT(04-3)-113, Project 19, and the National Institutes of Health under Grant No. 16197-05.     

Diffraction of light and lens effect of the stellar gravitation field

The purpose of the present communication is to consider the focussing of light by gravitational lenses (stars). Finite estimates of the gainq on the optical axis (r»0) can be obtained with an account of diffraction of light.     

In situ acceleration and gradients of charged particles in the outer solar system observed by the voyager spacecraft

The probable connection between cosmic rays and the electromagnetic state of the interplanetary medium was recognized by Hannes Alfvén as early as 1949 (Alfvén, 1949, 1950); he pointed out that the properties of cosmic rays necessitate a mechanism, external to Earth but within the solar system, capable of accelerating particles to extremely high energies. In advocating the view of local origin for part of the cosmic-ray spectrum, Alfvén and his colleagues developed a very general type of acceleration mechanism called magnetic pumping. The unique data set of the two Voyagers extends over an entire decade (1977–1987) and is most suitable to explore the problem of acceleration of charged particles in the heliosphere. The energy coverage of the Low Energy Charged Particle (LECP) experiment covers the range 30 keV to several hundred MeV for ions and 22 keV to several MeV for electrons. Selected observations of interplanetary acceleration events from 1 to 25 AU are presented and reviewed. These show frequent acceleration of ions to several tens of MeV in association with shocks; highest energies (220 MeV oxygen) were measured in the near-perpendicular ( _Bn 87.5°) shock of January 5, 1978 at 1.9 AU, where electron acceleration was also observed. Examples of ion acceleration in association with corotating interaction regions are presented and discussed. It is shown that shock structures have profound effects on high-energy (70 MeV) cosmic rays, especially during solar minimum, when a negative latitudinal gradient was observed after early 1985 at all energies from 70 MeV down to 30 keV. By early 1987, most shock acceleration activity in the outer heliosphere (25 to 30 AU) had ceased both in the ecliptic (Voyager-2) and at higher (30°) ecliptic latitudes (Voyager-1). The totality of observations demonstrate that local acceleration to a few hundred MeV, and as high as a few GeV is continually present throughout the heliosphere. It should be noted that in 1954 when Alfvén suggested local acceleration and containment of cosmic rays within the solar system, no one treated his suggestion seriously, at any energy. The observations reviewed in this paper illustrate once more Alfvén's remarkable prescience and demonstrate how unwise it is to dismiss his ideas.Paper dedicated to Professor Hannes Alfvén on the occasion of his 80th birthday, 30 May 1988.     

Large-scale structure of the interplanetary medium

We propose that the coronal source longitude and latitude of solar wind plasma can be estimated within 10°. Previous writers have argued that the solar wind in the ecliptic should originate near the equator and that a quasi-radial hypervelocity (QRH) approximation (constant radial flow) is valid beyond the magnetohydrodynamic critical points. We demonstrate that an extension of the QRH approximation (as if the solar wind flowed radially with constant velocity from the center of the Sun) yields a proper estimate of the high coronal source location at the release zone where the solar wind makes its transition to radial interplanetary flow. This extrapolated QRH (or EQRH) approximation succeeds because the two main corrections to this source estimate, coronal corotation and interplanetary acceleration, tend to cancel (the former correcting the source location eastward, the latter westward). Although this ideal spiral approximation was first suggested by Snyder and Neugebauer (1966), only recently has it been demonstrated that it relates a wide range of interplanetary plasma, magnetic field and energetic particle data to observed coronal magnetic structure. We estimate quantitatively the error in the EQRH approximation by comparison with steady-state streamlines predicted by azimuthally independent and dependent theoretical solutions to the steady-state plasma equations. We find the error in both cases 10° in longitude and therefore suggest that the EQRH approximation offers the means to relate observed solar initial conditions in the release zone directly to interplanetary measurements. If, in addition, the EQRH approximation also leads to agreement with low coronal structure, then there should be a straightforward correspondence to otherwise unobservable high coronal structure.     

On the coupling of lunisolar resonances for Earth satellite orbits

The resonance C1 occurs when the longitude of the perigee measured from the equinox becomes a slow angle in the doubly averaged equations of motion. This resonance is one of the critical inclination family with I 46°. For prograde Earth satellite orbits, up to five critical points can be identified. Only simple pitchfork bifurcations occur for the single resonance C1. A two degrees of freedom system is studied to check how a coupling of two lunisolar resonances affects the results furnished by the analysis of an isolated resonance case. In the system with two critical angles (g+h and h,+2 , seven types of critical points have been identified. The critical points arise and change their stability through 11 bifurcations. If the initial conditions are selected close to the critical points, the system becomes chaotic as shown in Poincaré maps.     

Coalescing neutron stars as gamma ray bursters?

We investigate the dynamics and evolution of coalescing neutron stars. The three-dimensional Newtonian equations of hydrodynamics are integrated by the Piecewise Parabolic Method on an equidistant Cartesian grid. The code is purely Newtonian, but does include the emission of gravitational waves and their back-reaction. The properties of neutron star matter are described by the equation of state of Lattimer and Swesty (1991). Energy loss by all types of neutrinos and changes of the electron fraction due to the emission of electron neutrinos and antineutrinos are taken into account by an elaborate neutrino leakage scheme. We simulate the coalescence of two identical, cool neutron stars with a baryonic mass of 1.6M and a radius of 15 km and with an initial center-to-center distance of 42 km. The initial distributions of density and electron concentration are given from a model of a cold neutron star in hydrostatic equilibrium. We investigate three cases which differ by the initial velocity distribution in the neutron stars. The orbit decays due to gravitational-wave emission and after one revolution the stars are so close that dynamical instability sets in. Within 1 ms the neutron stars merge into a rapidly spinning (P 1 ms), high-density body ( 10¹⁴ g/cm³) with a surrounding thick disk of material with densities 10¹⁰ – 10¹² g/cm³ and orbital velocities of 0.3-0.5 c. The peak emission of gravitational waves has a maximum luminosity of a few times 10⁵⁵ erg/s and is reached for about 1 ms. The amplitudes of the gravitational waves are close to 3 10^–23 at a distance of 1 Gpc and the typical frequency is near the dynamical value of the orbital motion of the merging neutron stars of 2 KHz. In a post-processing step, the rate of neutrino-antineutrino annihilation is calculated from the neutrino luminosities generated during the hydrodynamical simulations. We find the integral annihilation rate to be a few 10⁵⁰ erg/s during the phase of strongest neutrino emission, which is too small to generate the observed bursts considering the fact that the merged object of about 3M will most likely collapse to a black hole within milliseconds.     

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.     

Kepler's problem in constant curvature spaces

In this article the generalization of the motion of a particle in a central field to the case of a constant curvature space is investigated. We found out that orbits on a constant curvature surface are closed in two cases: when the potential satisfies Iaplace-Beltrami equation and can be regarded as an analogue of the potential of the gravitational interaction, and in the case when the potential is the generalization of the potential of an elastic spring. Also the full integrability of the generalized two-centre problem on a constant curvature surface is discovered and it is shown that integrability remains even if elastic forces are added.     

A Low-Sidelobe, High Frequency Corrugated Feed Horn for CMB Observations

We have produced a prototype broadband, low-sidelobe conical corrugated feed horn suitable for measurements of the Cosmic Microwave Blackground (CMB) radiation in the frequency band 120–150 GHz. The antenna is a first prototype for the Low Frequency Instrument array in ESA's PLANCK mission, a space project dedicated to CMB anisotropy mesurements in the 30–900 GHz range. We describe the fabrication method, based on silver electro-formation, and present the two-dimensional antenna beam pattern measured at 140 GHz with a milimeter-wave automated scalar test range. The beam has good symmetry in the E and H planes with a far sidelobe level approaching –60 dB at angles 80°. An upper limit to the return loss was measured to be –21 dB.     

Production of the solar magnetic fine-structure by convection

Motion of the gas in supergranular convection will produce twisted flux tubes. A simple energy calculation shows that for a wide variety of assumptions a tube of field strength 100 G will be given about one twist and will have a diameter 1000–2000 km. This agrees with observations of magnetic fine-structure on the sun.     

On long-term periodicities in the sunspot cycle

Occurrences of interplanetary shock waves near the Earth after the powerful isolated flares of 1957–1978 are investigated. The close connection between the occurrences of shock waves and the positions of magnetic axes of bipolar groups of sunspots is suggested on the basis of a statistical study. The shock waves are principally observed when the Earth finds itself near the planes that are projected through the flares in parallel to the appropriate magnetic axes of the nearest bipolar groups. This regularity is interpreted as an indirect argument for a three-dimensional geometry for the interplanetary shock waves which, when projected on these flattened to corresponding planes, are traces of large circular arcs. The typical angular scales of isolated interplanetary shock waves are estimated as 150° and 30° parallel and perpendicular, respectively, to the magnetic axes correspondingly.