Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.     

Effects of mass transfer,free-convection currents and heat sources on the Stokes' problem for an infinite vertical plate

An exact analysis of the effects of heat sources, mass transfer and free-convection currents on the flow past an impulsively started vertical plate is investigated. Closed-form solutions to the velocity and temperature field have been derived by using Laplace transform and expressions are given for the Skin-friction and the rate of heat transfer. Variations of the above quantities are presented graphically, and the paper is concluded with a quantitative discussion.     

Free convection effects on the stokes problem for an infinite vertical limiting surface with constant suction

This paper studies the effects of free convection currents on the flow field of an incompressible viscous fluid past an impulsively started infinite vertical porous limiting surface when the fluid is subjected to a suction with uniform velocity. An exact analysis of the problem is obtained with the aid of the Laplace transform technique and the expressions for the velocity field, temperature field and for their related quantities of skin friction and rate of heat transfer are given. The results thus obtained are discussed quantitatively for air and water.     

Gravitational instability

In this paper we review the gravitational instability theory of galaxy formation. Closed-form expressions for the rate of growth of density perturbations in the regime prior to recombination are presented in special cases in which the cosmological constant is non-zero. After recombination, the fate of perturbations in Lemaître universes is studied with the aid of exact solutions to the density perturbation equation orginally derived by Bonnor.     

Complete closed-form solutions of the Stark problem

Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direction. For both the two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions.     

Exact low-energy expansion of the Rayleigh scattering cross-section by atomic hydrogen

We present an exact low-energy expansion redward of Lyα of the Rayleigh scattering cross-section by atomic hydrogen, which is given by the Kramers–Heisenberg formula. The expansion is expressed as a power series of  (ω/ω_l)  , where ω_l and ω are the angular frequencies corresponding to the Lyman limit and the incident radiation, respectively. The expansion coefficients are represented as infinite sums over all the intermediate states and they can also be expressed as the regular particular solutions of inhomogeneous differential equations, which is known as the Dalgarno and Lewis method. In this paper, using a software capable of symbolic calculations, we obtain the exact values of these coefficients. We provide brief discussions on the accuracies of approximate expressions for Rayleigh scattering by a hydrogen atom found in the literature.     

Orbits of fluid elements accreting onto a black hole

An exact analysis of the unsteady free and forced convection flow of an incompressible viscous fluid is presented in the presence of thermal diffusion effect. The Laplace transform technique is used to obtain the expressions for velocity, leading edge effects, and skin-friction. During the course of the discussion, the effects ofS (thermal diffusion parameter), Pr (Prandtl number), andt (time parameter) on velocity, leading edge, and skin-friction are extensively discussed.     

Gravitational potential energy of interpenetrating spherical galaxies in Hernquist’s model

Hernquist’s (1990) mass model for spherical galaxies and bulges described by the deVaucouleur’s profile gives analytical expressions for the density profile and the potential. These have been used to derive a simple and exact analytical expression for the gravitational potential energy of a pair of interpene-trating spherical galaxies represented by this model. The results are compared with those for polytropic and Plummer models of galaxis.     

Radius of the Roche equipotential surfaces

In literature, there is no exact analytical solution available for determining the radius of Roche equipotential surfaces of distorted close binary systems in synchronous rotation. However, Kopal (Roche Model and Its Application to Close Binary Systems, Advances in Astronomy and Astrophysics, Academic Press, New York 1972) and Morris (Publ. Astron. Soc. Pac. 106:154, 1994) have provided the approximate analytical solutions in the form of infinite mathematical series. These series expressions have been commonly used by various authors to determine the radius of the Roche equipotential surfaces, and hence the equilibrium structures of rotating stars and stars in the binary systems. However, numerical results obtained from these approximating series expressions are not very accurate. In the present paper, we have expanded these series expressions to higher orders so as to improve their accuracy. The objective of this paper is to check, whether, there is any effect on the accuracy of these series expressions when the terms of higher orders are considered. Our results show that in most of the cases these expanded series give better results than the earlier series. We have further used these expanded series to find numerically the volume radius of the Roche equipotential surfaces. The obtained results are in good agreement with the results available in literature. We have also presented simple and accurate approximating formulas to calculate the radius of the primary component in a close binary system. These formulas give very accurate results in a specified range of mass ratio.     

Escape Of Planetary Atmospheres Including The Effects Of Rotation And Orbital Motion Of The Planet

The problem of escape of planetary atmospheres is discussed taking into account the effects of planet's rotation and planet's orbital momentum. The expressions for the rates of loss of mass and angular momentum per unit area and the total rates of loss of mass and angular momentum for the whole planetary surface are obtained. These expressions are, then, applied to study the escape of the constituents of the atmospheres of Mars and Venus. This revised version was published online in July 2006 with corrections to the Cover Date.     

Central configurations of four bodies with an axis of symmetry

A complete solution is given for a symmetric case of the problem of the planar central configurations of four bodies, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry. The positions of the bodies on the axis of symmetry are described by angle coordinates with respect to the outside bodies. The solution is such, that giving the angle coordinates, the masses for which the given configuration is a central configuration, can be computed from simple analytical expressions of the angles. The central configurations can be described as one-parameter families, and these are discussed in detail in one convex and two concave cases. The derived formulae represent exact analytical solutions of the four-body problem.     

On the atmosphere with exponential turbulence

We present an exact calculation for the photon-average bending angle and intensity produced during occultation of a point source by an isothermal atmosphere with an exponential height dependence of turbulent power. The calculation is performed via an expansion in powers of the gradients of average refractivity and turbulent power. Conditions for the validity of the expansion are discussed. We compare results with other expressions, for the same distribution of refractivity, recently presented by V. R. Eshleman and B. S. Haugstad. Discrepancies are noted. Certain properties of photon averages, which may be of importance for the reduction of occultation data, are discussed.     

Scintillation of Solar Radio Sources: Implication for Spikes

Possible scattering regimes of the emission from a solar radio source due to dielectric permitivity fluctuations of an extended coronal plasma co-rotating with the Sun are discussed. The exact and approximate expressions are given for the spectrum of temporal intensity fluctuations in the regime of weak scattering. The frequency, at which the spectrum shows a bend, is determined by the location of the effective scattering screen if the source size is not too large. In the regime of strong scattering of the emission from a broadbanded nonimpulsive radio source, the formation of random intensity spikes, namely millisecond, narrowbanded spike bursts is a possibility. Their apparent size can be quite significant. However, the sources with very small true sizes are required in order to produce strong spikes.     

Modulation patterns of astronomical imaging systems based on rotating grids

Various astronomical imaging systems use rotating grids as modulators and operate in the optical, X-ray, and gamma-ray regions. The position and strength of each measured source are encoded in a modulation pattern, produced by the imaging system. Here we present the mathematical expressions of the modulation patterns which permit efficient computer implementation and a simple approximate expression is also given which facilitates the understanding of the basic properties of the modulation patterns. Although various modulation devices strongly differ in appearance, their modulation patterns show a remarkable similarity. This suggests that a unified approach is possible by analysing the output signals of these devices.     

On the radiative and thermodynamic properties of the cosmic radiations using COBE FIRAS instrument data: I. Cosmic microwave background radiation

Using the explicit form of the functions to describe the monopole and dipole spectra of the Cosmic Microwave Background (CMB) radiation, the exact expressions for the temperature dependences of the radiative and thermodynamic functions, such as the total radiation power per unit area, total energy density, number density of photons, Helmholtz free energy density, entropy density, heat capacity at constant volume, and pressure in the finite range of frequencies v ₁≤v≤v ₂ are obtained. Since the dependence of temperature upon the redshift z is known, the obtained expressions can be simply presented in z representation. Utilizing experimental data for the monopole and dipole spectra measured by the COBE FIRAS instrument in the 60–600 GHz frequency interval at the temperature T=2.72548 K, the values of the radiative and thermodynamic functions, as well as the radiation density constant a and the Stefan-Boltzmann constant σ are calculated. In the case of the dipole spectrum, the constants a and σ, and the radiative and thermodynamic properties of the CMB radiation are obtained using the mean amplitude T _amp=3.358 mK. It is shown that the Doppler shift leads to a renormalization of the radiation density constant a, the Stefan-Boltzmann constant σ, and the corresponding constants for the thermodynamic functions. The expressions for new astrophysical parameters, such as the entropy density/Boltzmann constant, and number density of CMB photons are obtained. The radiative and thermodynamic properties of the Cosmic Microwave Background radiation for the monopole and dipole spectra at redshift z≈1089 are calculated.     

ON THE POSSIBLE NUMBER AND MASS OF FRAGMENTS FROM PUŁTUSK METEORITE SHOWER, 1868

The earlier efforts to approximate the total number and mass of fragments from Pu?tusk meteorite shower 1868 have been found unsatisfactory With the use of the field data of Samsonowicz and the sorting equation after Frost, differential expressions approximating the mass and number of fragments are derived. These expressions numerically integrated over the elliptical strewnfield area lead to the estimates ～ 1.8 times 10⁵ and ～ 2 times 10⁶ g for the total number and mass of fragments respectively. The meaning of the obtained results is discussed.     

Source Region of the Decameter–Hectometric Type II Radio Burst: Shock–Streamer Interaction Region

D–H type II radio bursts are widely thought to be caused by coronal mass ejections (CMEs). However, it is still unclear where the exact source of the type IIs on the shock surface is. We identify the source regions of the decameter–hectometric (D–H) type IIs based on imaging observations from SOHO/LASCO and the radio dynamic spectrum from Wind/Waves. The analysis of two well-observed events suggests that the sources of these two events are located in the interaction regions between shocks and streamers, and that the shocks are enhanced significantly in these regions.     

Emden-Chandrasekhar axisymmetric,solid-body rotating polytropes

The basic theory on polytorpes is revisited and EC polytropes are defined. The first-order approximation theory of Chandrasekhar (1933a, b, c) and Chandrasekhar and Lebovitz (1962) is reviewed, refined and extended in such a way that better results are obtained without involving hard analytical or numerical techniques. A more precise equation is given in defining non-outer equipotential surfaces, and a new method is adopted in determining the explicit expression of the gravitational potential. This method essentially consists in equating the expression of the gravitational potential and its first radial derivative determined by accounting for the equilibrium condition, with the corresponding expression of the gravitational potential and its first radial derivative determined by accounting for mass distribution. Such expressions are to be calculated at convenient points — for instance, at the centre and at the pole of the system. In this way, an infinity of exact solutions is derived for the special casesn=0 andn=1, and we then have the problem: ‘Which of the infinite number of solutions available leads to the most stable configuration?’ The simplest of these solutions is taken into account in detail for bothn=0 andn=1; results relative to the latter case allow us to solve the Kopal (1937) problem. EC polytropes withn=5 are found to consist of an inner massive non-rotating component and an outer zero-density rotating atmosphere. It is seen that they are equivalent in some respects to Roche systems, and the corresponding exact solution is derived. Explicit expressions for characteristic physical parameters are also determened in the general case, relative to sequences of equilibrium states characterized by constant masses and angular momenta. Detailed results are given for the special casesn=0, 1 and 5. Finally, some properties of both EC polytropes and R polytropes withn=0 (i.e., generalized Roche systems) are reported and discussed. The conclusions of this paper make it highly desirable to have an extension of the method used here to general values ofn.     

Dynamics of cosmic current layers with localized lower-hybrid-drift turbulence

A two-dimensional magnetohydrodynamic model of the dynamics of tail-like current layers caused by anomalous electrical resistivity in a plasma with lower-hybrid-drift (LHD) turbulence is considered. Additionally to the LHD-resistivity, a resistivity pulse in the magnetic neutral sheet is given initiating a magnetic reconnection process. Then the temporal and spatial evolution of the magnetic and electric fields, the plasma convection and the anomalous resistivity are obtained numerically. Taking into account more exact expressions for the LHD-resistivity in the current layer as done in former works, the LHD-turbulence is found to be excited farther from the neutral sheet, and thus, with the time, secondary current sheets are obtained in the plasma-magnetic field system. It is shown that the inductive electric field moving from the magnetic neutral sheet to the current layer periphery during the reconnection process may be considered as indicator of the plasma disturbances.