Source vector in Rayleigh-Cabannes scattering atmosphere: the nonconservative milne problem

We consider the radiative transfer in a nonconservative homogeneous plane-parallel semi-infinite planetary atmosphere where the scattering processes are described by the Rayleigh-Cabannes phase matrix and where the primary sources are in infinitely deep layers. If we use the superposition principle we derive the Cauchy problem for the source vector.As a by-product the external field of radiation for the problem described is obtained using the principle of invariance by Chandrasekhar. The respective formulae for the radiation field in the deep layers and for the extrapolation distance are given. It is shown that the Rubenson degree of polarization even in the case of near-conservative atmospheres reaches the asymptotic regime at rather small values of the optical depth. The-plane reliefs of the characteristic equation, extrapolation distance and the normalized components of the source vector at the boundary are given along with a sample of zeros of the characteristic equation.     

Energy-dependent transport problem with generalized boundary conditions

The slowing-down Boltzmann equation for generalized boundary conditions is considered and transformed to one-speed equation in Laplace space. Exact relations between energy reflection and transmission coefficients for a problem with diffuse reflecting boundary conditions and the albedos for the problem with isotropic boundary conditions are obtained. The Galerkin method is used to calculate the energy reflection coefficient for a finite slab for different thicknesses at different mass ratiosA, target to projectile mass, at different synthetic-scattering kernels. The results for partial heat fluxes for isotropic and anisotropic-scattering dispersive medium are given. The results obtained for isotropic boundary conditions are compared well with the exact results.     

Effect of perturbation on the location of libration point in the robe restricted problem of three bodies

The effect of small perturbation in the Coriolis and centrifugal forces on the location of libration point in the ‘Robe (1977) restricted problem of three bodies’ has been studied. In this problem one body,m ₁, is a rigid spherical shell filled with an homogeneous incompressible fluid of densityϱ ₁. The second one,m ₂, is a mass point outside the shell andm ₃ is a small solid sphere of densityϱ ₃ supposed to be moving inside the shell subject to the attraction ofm ₂ and buoyancy force due to fluidϱ ₁. Here we assumem ₃ to be an infinitesimal mass and the orbit of the massm ₂ to be circular, and we also suppose the densitiesϱ ₁, andϱ ₃ to be equal. Then there exists an equilibrium point (−μ + (ɛ′μ)/(1 + 2μ), 0, 0).     

Numerical treatment of the unsteady hydromagnetic thermal boundary layer problem

This paper presents a suitable numerical method for the treatment of the unsteady hydromagnetic thermal boundary layer problem for flows past an infinite porous flat plate, the motion of which is governed by a general time-dependent law, under the influence of a transverse externally set magnetic field. The normal velocity of suction/injection at the plate is also assumed to be time-dependent. The results obtained on the basis of numerical approximations seem to compare favourably with earlier results (Pandeet al., 1976; Tokis, 1978). Analytical approximations are given for the cases of a plate (i) generally accelerated and (ii) harmonically oscillating. The direct numerical treatment is obviously advantageous since it allows, handling of cases where the known methods for analytical approximations are not applicable. This problem is closely related to the motions and heat transfer occurring locally on the surfaces of stars.     

An intermediate matching technique for solving two point boundary value problems using the perturbation method

The perturbation method, a numerical method for solving two point boundary value problems (TPBVP), is modified to attempt to improve inherent instability and sensitivity problems associated with the method. The desired solution to the TPBVP is divided into two time intervals. The differential equations required to define a solution to the two point boundary value problem are integrated independently over these shorter segments rather than consecutively over the entire trajectory. The independent integration of the differential equations over approximately half of the trajectory instead of the entire trajectory substantially decreases sensitivity and stability properties associated with the numerical integration. The equations for both time segments can be integrated simultaneously. By this procedure, a system of twice the dimension of the original problem is integrated for a period of time equal to half of the time interval for the original problem. To show the effectiveness of the method, two impulse trajectories which minimize the total velocity increment required to transfer a spacecraft from an Earth orbit into a lunar orbit are calculated.     

Secular resonances in the asteroid belt: Theoretical perturbation approach and the problem of their location

In this paper a theoretical perturbation approach to the problem of the dynamics in secular resonance is exposed. This approach avoids any expansion of the main term of the Hamiltonian (linear term in the masses) with respect to the eccentricity or the inclination of the asteroid, in order to achieve results valid for any value of these variables. Moreover suitable action-angle variables are introduced to take properly into account the dynamics related to the motion of the argument of perihelion of the asteroid, which is relevant at high inclination. A class of secular resonances wider than that usually considered is found. An explicit computation of the location of the main secular resonances, estimating also the contribution of the quadratic term in the masses by means of classical series expansion, is reported in the last sections. The accuracy of computations obtained by series expansion is discussed in the paper.     

Effect of perturbation of coriolis and centrifugal forces on the location and linear stability of the libration points in the robe problem

The locations and linear stability of the main libration points in Robe's restricted three-body problem under perturbed Coriolis and centrifugal forces are investigated. The perturbed locations of these points are given. The perturbation magnitude of their locations and linear stability are estimated. The results obtained by Shrivastava and Garain^[10] are improved.     

A note on the Hori-Lie perturbation technique

In this paper we describe how to avoid the introduction of Hori's pseudo time in general planetary theory in the case of two planets.     

The transformational behaviour of perturbation theories

The transformational behaviour of Hori's noncanonical perturbation theory (Hori 1971) as well as that of the theory of Krylov-Bogoliubof-Mitropolsky is studied. An integration procedure of the perturbation equations is based on the transformation properties that have been established.     

The depth of the convective boundary layer on Mars

We are using observations obtained with Mars Express to explore the structure and dynamics of the martian lower atmosphere. We consider a series of radio occultation experiments conducted in May-August 2004, when the season on Mars was midspring of the northern hemisphere. The measurements are widely distributed in latitude and longitude, but the local time remained within a narrow range, 17.0-17.2 h. Most of the atmospheric profiles retrieved from these data contain a distinct, well-mixed convective boundary layer (CBL). We have accurately determined the depth of the CBL and its spatial variations at fixed local time through analysis of these profiles. The CBL extends to a height of 3-10 km above the surface at the season and locations of these measurements. Its depth at fixed local time is clearly correlated with variations in surface elevation on planetary scales, with a weaker dependence on spatial variations in surface temperature. In general, the CBL is deep (8-10 km) where the surface elevation is high, as in Tharsis Montes and Syrtis Major, and shallow (4-6 km) where the surface elevation is low, as in Amazonis and Utopia. This variability results from the combined effects of conditions near the surface and in the atmosphere above the CBL. Convection arises from solar heating of the ground, and the impact of this heat source on thermal structure is largest where the surface pressure and atmospheric density are smallest, at high surface elevations. The vertical extent of the CBL is in turn constrained by the static stability of the overlying atmosphere. These results greatly reduce the long-standing uncertainty concerning the depth of the CBL.     

A second-order solution to the two-point boundary value problem for rendezvous in eccentric orbits

A new second-order solution to the two-point boundary value problem for relative motion about orbital rendezvous in one orbit period is proposed. First, nonlinear differential equations to describe the relative motion between a chaser and a target are presented considering the second-order terms in the gravity. Then, by regarding the second-order terms as external accelerations, we establish second-order state transition equations. Moreover, the J2 perturbations effects can also be considered in the state transition equations. Last, the initial relative velocity to fulfill a rendezvous is determined by solving the state transition equations. Numerical simulations show that the new second-order state transition equations are accurate. The second-order solution to the two-point boundary value problem on eccentric orbits is valid even if the relative range is farther than 500 km.     

A numerical study of the pre-ejection, magnetically-sheared corona as a free boundary problem

A class of magnetostatic equilibria with axial symmetry outside a unit sphere in the presence of plasma pressure and an r ^–2 gravitational field is constructed. The structure contains a localized current-carrying region confined by a background bipolar potential field, and the shape of the region changes subject to the variation of the electric current. The continuity requirement for the magnetic field and plasma pressures at the outer boundary of the cavity defines a free boundary problem, which is solved numerically using a spectral boundary scheme. The model is then used to study the expansion of the current-carrying region, caused by the buildup of magnetic shear, against the background confining field. The magnetic shear in our model is induced by the loading of an azimuthal field, accompanied by a depletion of plasma density.We show that due to the additional effect of confinement by the dense surrounding plasma, the energy of the magnetic field can exceed the energy of its associated open field, presumably a necessary condition for the onset of coronal mass ejections. (However, the plasma beta of the confining fluid is higher than that in the outer boundary of a realistic helmet-streamer structure.) Furthermore, under the assumption that coronal mass ejections are driven by magnetic buoyancy, the result from our model study lends further support to the notion of a suspended magnetic flux rope in the low-density cavity of a helmet-streamer as a promising pre-ejection configuration.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Encke's special perturbation technique associated with the KS regularized variables

In this paper of the series, a special perturbation technique of Encke-type associated with the KS regularized variables will be developed for satellite motions in the Earth's gravitational field with axial symmetry. Its computational algorithm is of recursive nature and could be applied for any perturbed conic motion whatever the number of the zonal harmonic coefficients may be. Applications of the algorithm are also included.Now at the Department of Mathematics, Girls College of Education, Jeddah, Saudi Arabia.     

The critical inclination problem with small eccentricity

A qualitative solution is presented of the critical inclination problem in artificial satellite theory for motions in which the orbits are nearly circular. The effects of all the zonal harmonics are taken into account, and bothshallow anddeep resonance regimes are considered. An investigation of the (e sing,e cosg)-plane reveals that six fundamentally different types of phase-plane portraits exist. These portraits illustrate the long-term behaviour of the eccentricity and line of apsides.     

The restricted three-body problem with logarithm potential

We present a study of the restricted three body problem with logarithm potential. We discuss equilibria, stability, Hill’s regions of motion and the families of periodic orbits near equilibria. Moreover, we show that equilibria and some periodic orbits continue in the logarithm three body problem.     

The planar Hill problem with oblate primary

The regularized equations of motion of the planar Hill problem which includes the effect of the oblateness of the larger primary body, is presented. Using the Levi-Civita coordinate transformation as well as the corresponding time transformation, we obtain a simple regularized polynomial Hamiltonian of the dynamical system that corresponds to that of two uncoupled harmonic oscillators perturbed by polynomial terms. The relations between the synodic and regularized variables are also given. The convenient numerical computations of the regularized equations of motion, allow derivation of a map of the group of families of simple-periodic orbits, free of collision cases, of both the classical and the Hill problem with oblateness. The horizontal stability of the families is calculated and we determine series of horizontally critical symmetric periodic orbits of the basic families g and g'. This revised version was published online in July 2006 with corrections to the Cover Date.