Asymptotic theory of the transfer of polarized radiation with resonance scattering in the doppler core of the line

In the first of the series of papers by Ivanov et al. it was shown that the model problem of the transfer of polarized radiation as a result of resonance scattering from two-level atoms in a homogeneous plane atmosphere in the absence of LTE comes down, in the approximation of complete frequency redistribution, to the solution of an integral matrix equation of the Wiener-Hopf type for a (2 × 2) matrix source function S(τ). In the second paper in this series, devoted to the vector Milne problem, complete asymptotic expansions of the matrix I(z) [which is essentially a Laplace transform of the matrix S(τ)] for the case of a Doppler profile of the coefficient of absorption, and the coefficients of asymptotic expansions of S(τ) (τ » 1) are expressed in terms of coefficients of the expansions of I(z). We show that asymptotic expansions of S(τ) can be found directly from an integral matrix equation of the Wiener-Hopf type for S(τ). We give new recursive equations for the coefficients of these expansions, as well as a new derivation of asymptotic expansions of the matrix I, including its second column, which was considered only briefly by Ivanov et al.     

Expansion of the Hamiltonian for the planetary problem into a Poisson series in the heliocentric reference frame

An expansion of the Hamiltonian for the N-planet problem into a Poisson series using a system of modified (complex) Poincare´ canonical elements in the heliocentric coordinate system is constructed. The Lagrangian and Hamiltonian formalisms are used. The first terms in the expansions of the principal and complementary parts of the disturbing function are presented. Estimates of the number of terms in the presented expansions have been obtained through numerical experiments. A comparison with the results of other authors is made.     

Fourier techniques for an analysis of eclipsing binary light curves: IV

The aim of the present paper is to establish two further series expansions (alternative to those given in Demircan, 1979a, b, c), for the observed light changes of eclipsing binary system. The coefficients of these expansions have also been expressed in the form of general series expansions in terms of the eclipse elementsr _1.2,i andL ₁ of the spherical model on which all other distorted models may be based (Kopal, 1975, 1976) in an analysis in the frequency-domain.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, elliptic expansions in terms of the sectorial variables θ _j ⁽ⁱ⁾ introduced in Paper IV (Sharaf, 1982) to regularise highly oscillating perturbation force of some orbital systems will be explored for the first four categories. For each of the elliptic expansions belonging to a category, literal analytical expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations, numerical results are included to provide test examples for constructing computational algorithms.     

Asymptotic expansions for acoustical modes of the homogeneous compressible model

The behavior ofp-modes of high degree and high order in the homogeneous compressilbe model is examined. The second-order differential equation of Pekeris is used to construct asymptotic expansions near the centre and near the surface, which are singular points, and near the turning point of that equation. An equation for the frequencies is obtained by requiring the continuity of the asymptotic solutions and of their first derivatives. Numerical applications are considered.     

Modulation of Jovian middle magnetosphere currents and auroral precipitation by solar wind-induced compressions and expansions of the magnetosphere: initial response and steady state

We present results from a theoretical model which has been used to investigate the modulation of the magnetosphere-ionosphere coupling currents in the Jovian middle magnetosphere by solar wind-induced compressions and expansions of the magnetosphere. We consider an initial system in which the current sheet field lines extend to 50R_J in the equatorial plane, and where the iogenic plasma in the current sheet undergoes steady outward radial diffusion under the influence of the ionospheric torque which tends to maintain corotation with the planet. We show using typical Jovian parameters that the upward-directed field-aligned currents flowing throughout the middle magnetosphere region in this system peak at values requiring the existence of significant field-aligned voltages to drive them, resulting in large precipitating energy fluxes of accelerated electrons and bright ‘main oval’ UV auroras. We then consider the changes in these parameters which take place due to sudden expansions or compressions of the magnetosphere, resulting from changes in the solar wind dynamic pressure. Two cases are considered and compared, these being first the initial response of the system to the change, determined approximately from conservation of angular momentum of the radially displaced plasma and frozen-in field lines, and second the subsequent steady state of steady outward radial diffusion applied to the compressed or expanded system. We show that moderate inward compressions of the outer boundary of the current sheet field lines, e.g. from 50 to 40R_J, are effective in significantly reducing the coupling currents and precipitation in the initial state, the latter then recovering, but only partly so, during the evolution to the steady state. Strong inward compressions, e.g. to 30R_J cause significant super-corotation of the plasma and a reversal in sense of the current system in the initial state, such that bright auroras may then be formed poleward of the usual ‘main auroral oval’ due to the ‘return’ currents. The sense of the currents subsequently reverts back to the usual direction as steady-state conditions are restored, but they are weak, and so is the consequent electron precipitation. For outward expansions of the current sheet, however, the field-aligned currents and electron precipitation are strongly enhanced, particularly at the poleward border mapping to the outer weak field region of the current sheet. In this case there is little evolution of the parameters between the initial expansion and the subsequent steady state. Overall, the results suggest that the Jovian middle magnetosphere coupling currents and resulting ‘main oval’ auroral acceleration and precipitation will be strongly modulated by the solar wind dynamic pressure in the sense of anti-correlation, through the resulting compressions and expansions in the size of the magnetosphere.     

Heat and mass transfer of an oscillatory flow with Hall current

In the first part of these notes new expressions—simpler than any previously obtained—are presented in integral form for the derivatives of the α _n ⁰ -functions (required for an interpretation of the observed light changes of eclipsing variables) with respect to the fractional radiir _{1, 2} and projected separation δ of their centres in terms of the modified Bessel functionsK _{0, 1} (x) of the second kind; and utilized for establishing new asymptotic formulae for the computation of ‘boundary integrals’ of the formJ _?1 ⁰ ,n(μ). In the second part of this paper, by a resort to bi-polar coordinates, we shall establish a new type of expansions for the α _n ⁰ -functions valid for any type of eclipses, and converging faster than the expansions of the cross-correlation integral of the form (1) for α _n ⁰ that have so far been established.     

Sur de nouvelles séries pour le problème de masses critiques de Routh dans le problème restreint plan des trois corps

The planar restricted 3-body problem, linearized in the neighborhood of Lagrangian equilibriaL ₄ andL ₅, has in general two distinct eigenvalues and their opposites. When they are pure imaginary and not multiples of each other, they generate two families of periodic solutions called long and short periodic families. This is essentially a consequence of the famous theorem of Liapunov (Siegel, 1956). We showed (Roels, 1971b) how to solve the problem when the eigenvalues are multiples of each other in building series with negative exponents instead of the integer expansions of Siegel (Roels and Lauterman, 1970). When the eigenvalues are equal, which is the case for the mass ratio of Routh, the problem was solved by Deprit and Henrard (1968) using formal series in ordinary unnormalized variables. That leads to very complicated series because of the use of variables that are not well adapted to the problem. The convergence of the series was proven by Meyer and Schmidt (1971). In this paper we solve the problem by using normalized variables. This brings us to build expansions with fractional exponents. So in summary, normalized variables generate integer series in the non-resonant cases, series with negative exponents in the case of resonancek≥3, and series with fractional exponents when the resonance is 1.     

One-parameter representation of the daily averaged solar-wind velocity

An empirical formula was found to describe the dependence V(S) of the daily average solar-wind velocity V on the coronal-hole area S on the visible side of the Sun in the form of first-and second-order Taylor expansions. The results can be used for approximate evaluation of the solar-wind velocity at the Earth’s orbit from coronal-hole observations.     

Recurrence formulae for the Hansen's developments

In this paper we derive some recurrence formulae which can be used to calculate the Fourier expansions of the functions (r/a) ⁿ cosmv and (r/a) ⁿ sinmv in terms of the eccentric anomalyE or the mean anomalyM. We also establish a recurrence process for computing the series expansions for alln andm when the expansions of two basic series are known. These basic series were given in explicit form in the classical literature. The recurrence formulae are linear in the functions involved and thus make very simple the computation of the series.This work was supported by NASA contract No. NASr 54(06).—The paper was presented at the AIAA/AAS meeting, Princeton University, August 1969.     

Study of Bianchi I anisotropic model in f(R,T) gravity

We discuss the Bianchi type I model with perfect fluid as matter content in f(R,T) gravity, where R is the Ricci scalar and T is the trace of the energy-momentum tensor. We obtain exact solutions of the field equations employing the anisotropic feature of spacetime for two expansion laws namely exponential and power expansions. The physical and kinematical quantities are examined for both cases in future evolution of the universe. We also explore the validity of null energy condition and conclude that our solutions are consistent with the current observations.     

Observations of horizontal transport effects on high latitude metastable O(1D), N(2D) auroral emissions

An analysis is presented of photometric measurements of the NI (λ = 520nm),OI(λ = 630nm)and other emissions made at Nord, where the invariant latitude is Λ = 80°4. The time variations of the intensities are interpreted in the following way by comparison with simultaneous ground based or satellite measurements.The N(²D) atoms formed in the dayside cleft are carried by the neutral wind in a plume across the polar cap, so that the ratio of λ(630 nm) to λ(520 nm) intensities decreases along the plume with increasing distance from the source region.In the polar cap, but outside the plume region, 630 nm emission is produced by electron impact of polar rain and by substorms that reach high latitudes. Ionization produced at the same time, especially by the substorms, will produce further 630 nm emission through dissociative recombination. In any case, the region outside the plume may be regarded as a source region, with a high value of the ratio $\text{I(630)}\text{I(520)}$. This explains in part the diurnal variations, since this ratio is depressed as Nord crosses the dayside plume.The electron energy along the oval increases progressively from the dayside to the nightside. The intensity ratio increases with increasing electron energy because N(²D) is quenched more rapidly than O(¹D). Thus the ratio rises progressively from noon to midnight.An effect of the interplanetary magnetic field is superimposed on this pattern : as its North-South component B_z increases, the oval contracts so that Nord becomes nearer from the cleft source and the intensity ratio increases on the dayside. The inverse effect is also observed. On the nightside, negative B_z is associated with substorms that produce poleward expansions of the poleward oval boundary, that brings more energetic precipitation to Nord. This causes the intensity ratio to increase with decreasing B_z in a way that is opposite to that for the dayside.     

Expansion of the principal functions of Keplerian motion using complex variables

A special system of canonical variables is considered. An algorithm for expanding the principal functions of Keplerian motion in new elements is presented. The advantage of the proposed system is a relatively small number of terms in the classical expansions of the unperturbed two-body problem. A method for expanding the time derivatives of the rectangular coordinates is proposed. Some estimates of the number of terms in the presented expansions have been obtained through numerical experiments.     

Navigating to the Moon along low-energy transfers

This paper presents a navigation strategy to fly to the Moon along a Weak Stability Boundary transfer trajectory. A particular strategy is devised to ensure capture into an uncontrolled relatively stable orbit at the Moon. Both uncertainty in the orbit determination process and in the control of the thrust vector are included in the navigation analysis. The orbit determination process is based on the definition of an optimal filtering technique that is able to meet accuracy requirements at an acceptable computational cost. Three sequential filtering techniques are analysed: an extended Kalman filter, an unscented Kalman filter and a Kalman filter based on high order expansions. The analysis shows that only the unscented Kalman filter meets the accuracy requirements at an acceptable computational cost. This paper demonstrates lunar weak capture for all trajectories within a capture corridor defined by all the trajectories in the neighbourhood of the nominal one, in state space. A minimum Δv strategy is presented to extend the lifetime of the spacecraft around the Moon. The orbit determination and navigation strategies are applied to the case of the European Student Moon Orbiter.     

Long-term variations of the Earth's orbital elements

Critical analysis of theories of the long-term variations of the ecliptical elements of the Earth leads to the following conclusions, regarding the influence of different terms on the accuracy of the expansions used:

1. further improvement in planetary masses will not have significant influence:
2. for the (e, π) system, terms depending upon the second order as to the disturbing masses are more important than ones coming from the third degree with respect to the planetary eccentricities and inclinations;
3. for the (i, Ω) system, the latter terms have highly significant influence, whereas additional terms in masses are negligible. The same conclusion can be drawn for (ε,Ψ _g ). Using these results, a new solution for the long-term variations of the Earth's orbital elements is obtained. The results fore, π,i, Ω include terms depending upon the second power as to the disturbing masses and to the third degree with respect to the planetarye's andi's. For the obliquity ε and the annual general precession in longitudeΨ _g , a Laplace series is proposed where amplitudes, mean rates and phases are computed from those of the (i, Ω) system.

     

Direct conversion of los acceleration data to vertical gravity anomalies: A new approach

Most of the existing lunar and planetary gravity data are in the form of LOS (line-of-sight) components which cannot be used for conventional geophysical modelling. Current methods to invert LOS data yield non-unique or poorly constrained results or results of low spatial resolution. An alternate method presented here promises to produce unique, detailed and more reliable results. It utilizes the fact that three non-coplanar LOS acceleration vectors determined at different times at some point of observation uniquely define the total acceleration vector at that point. Vector analysis shows that the local cartesian componentsg _i of the total gravity anomaly vector may be obtained by inversion of the system $$a_{ij} g_j = A_i^2 $$ where thea _ij andA _i 's are, respectively, the local cartesian components and scalar magnitudes of the three required LOS acceleration vectors. In principle, the method is applicable to lunar as well as planetary LOS data.     

Polynomial expansions of single-mode motions around equilibrium points in the circular restricted three-body problem

In this work, the single-mode motions around the collinear and triangular libration points in the circular restricted three-body problem are studied. To describe these motions, we adopt an invariant manifold approach, which states that a suitable pair of independent variables are taken as modal coordinates and the remaining state variables are expressed as polynomial series of them. Based on the invariant manifold approach, the general procedure on constructing polynomial expansions up to a certain order is outlined. Taking the Earth–Moon system as the example dynamical model, we construct the polynomial expansions up to the tenth order for the single-mode motions around collinear libration points, and up to order eight and six for the planar and vertical-periodic motions around triangular libration point, respectively. The application of the polynomial expansions constructed lies in that they can be used to determine the initial states for the single-mode motions around equilibrium points. To check the validity, the accuracy of initial states determined by the polynomial expansions is evaluated.     

Drift theory of charged particles in electric and magnetic fields

In this paper we review the drift theory of charged particles in electric and magnetic fields. No new physical interpretations are added to this classical topic, but through an alternative, simplified derivation of the guiding centre velocity, several complexities are eliminated and possible misconceptions of the theory are clarified. It is shown that:

1. The curvature/gradient drift velocity in the magnetic field, averaged over a particle distribution function is to lowest order in the direction of?×B/B ², while the average particle velocity is in the direction ofB×? P withP the scalar particle pressure.
2. These drift directions are correct for first-order expansions of the particle distribution function, and only second-order or higher expansions change these directions.
3. The?×B/B ² drift, which is the standard gradient plus curvature drift, and which is usually considered as a ‘single particle’ drift, need not be ‘reconciled’ with theB×? P, or ‘macroscopic, collective’ drift, as is often asserted in the literature. They are in fact related per definition and we show how.
4. When viewed in fixed momentum intervals (p,p+dp), the so-called Compton-Getting factor enters into the electric field (E×B)/B ² drift term.
5. The results are independent of the scale length of variation ofE andB, in contrast to existing drift theory. We discuss the implications of this result for three important cases.

     

The lunar conductivity profile and the nonuniqueness of electromagnetic data inversion

A review of the theory for the electromagnetic functional used to date to determine the lunar conductivity profile from spectral analyses of lunar magnetometer data is presented. The “hard” boundary condition used by Sonett et al. (1971a, b) and others appears to be a good approximation for the sunlit lunar hemisphere. The use of only the first spherical harmonic in the electromagnetic functional is not justified; further, there are certain classes of lunar models where the transverse magnetic modal response may not be neglected.     

Automatic Detection and Extraction of Coronal Dimmings from SDO/AIA Data

The volume of data anticipated from the Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA) highlights the necessity for the development of automatic-detection methods for various types of solar activity. Initially recognized in the 1970s, it is now well established that coronal dimmings are closely associated with coronal mass ejections (CMEs), and they are particularly noted as a reliable indicator of front-side (halo) CMEs, which can be difficult to detect in white-light coronagraph data. Existing work clearly demonstrates that several properties derived from the analysis of coronal dimmings can give useful information about the associated CME. The development and implementation of an automated coronal-dimming region detection and extraction algorithm removes visual observer bias, however unintentional, from the determination of physical quantities such as spatial location, area, and volume. This allows for reproducible, quantifiable results to be mined from very large data sets. The information derived may facilitate more reliable early space-weather detection, as well as offering the potential for conducting large-sample studies focused on determining the geo-effectiveness of CMEs, coupled with analysis of their associated coronal dimming signatures. In this paper we present examples of both simple and complex dimming events extracted using our algorithm, which will be run as a module for the SDO/Computer Vision Centre. Contrasting and well-studied events at both the minimum and maximum of solar cycle 23 are identified in Solar and Heliospheric Observatory/Extreme ultra-violet Imaging Telescope (SOHO/EIT) data. A more recent example extracted from Solar and Terrestrial Relations Observatory/Extreme Ultra-Violet Imager (STEREO/EUVI) data is also presented, demonstrating the potential for the anticipated application to SDO/AIA data. The detection part of our algorithm is based largely on the principle of operation of the NEMO software, namely the detection of significant variation in the statistics of the EUV image pixels (Podladchikova and Berghmans in Solar Phys. 228, 265?–?284, 2005). As well as running on historic data sets, the presented algorithm is capable of detecting and extracting coronal dimmings in near real-time.