恒星线性非绝热非径向脉动问题的退耦化处理方法

本文发展了一种解恒星线性非绝热非径向脉动问题的退耦化方法。这个方法把非绝热非径向脉动问题的六阶线性微分方程,分解为由一个代数方程联系起来的一个四阶线性微分方程和另一个二阶线性微分方程进行数值求解。这样的一个退耦处理,有利于克服以前在数值解这类问题时常常遇到的收敛域小和收敛速度慢等困难,并且为数值解方程时所采用的Henyey方法提供了一个自然和方便的初始猜测解。     

A linear solution of the equations of motion of an earth-orbiting satellite based on a Lie-series

Using Hill's variables, an analytical solution of a canonical system of six differential equations describing the motion of a satellite in the gravitational field of the earth is derived. The gravity field, expanded into spherical harmonics, has to be expressed as a function of the Hill variables. The intermediary is chosen to include the main secular terms. The first order solution retains the highly practical formal structure of Kaula's linear solution, but is valid for circular orbits and provides of course a spectral decomposition of radius vector and radial velocity. The resulting eccentricity functions are much simpler than the Hansen functions, since a series evaluation of the Kepler equation is avoided. The present solution may be extended to higher order solutions by Hori's perturbation method.     

The vector approach to the problem of physical libration of the Moon. II. The nonlinear problem

By the new vector method in a nonlinear setting, a physical libration of the Moon is studied. Using the decomposition method on small parameters we derive the closed system of nine differential equations with terms of the first and second order of smallness. The conclusion is drawn that in the nonlinear case a connection between the librations in a longitude and latitude, though feeble, nevertheless exists; therefore, the physical libration already is impossible to subdivide into independent from each other forms of oscillations, as usually can be done. In the linear approach, ten characteristic frequencies and two special invariants of the problem are found. It is proved that, taking into account nonlinear terms, the invariants are periodic functions of time. Therefore, the stationary solution with zero frequency, formally supposing in the linear theory a resonance, in the nonlinear approach gains only small (proportional to e) periodic oscillations. Near to zero frequency of a resonance there is no, and solution of the nonlinear equations of physical libration is stable. The given nonlinear solution slightly modifies the previously unknown conical precession of the Moon’s spin axis. The character of nonlinear solutions near the basic forcing frequency Ω₁, where in the linear approach there are beats, is carefully studied. The average method on fast variables is obtained by the linear system of differential equations with almost periodic coefficients, which describe the evolution of these coefficients in a nonlinear problem. From this follows that the nonlinear components only slightly modify the specified beats; the interior period T ≈ 16.53 days appears 411 times less than the exterior one T ≈ 18.61 Julian years. In particular, with such a period the angle between ecliptic plane and Moon orbit plane also varies. Resonances, on which other researches earlier insisted, are not discovered. As a whole, the nonlinear analysis essentially improves and supplements a linear picture of the physical libration.     

Ekman层风速随高度分布方程的数值解试验

根据大气水平运动方程推导出Ekman层风速随高度分布方程,该分布可用二阶线性微分方程表示,用有限差分法求解该二阶线性微分方程的边值问题的数值解,并给出计算程序,输出结果。将结果与经典解析解比较,讨论了经典解的正确性和适用范围。     

On an alternative method for approximate solution of linear stochastic differential equations

An approximate method for solving formal linear stochastic differential equations of first order is proposed. On the basis of the Reynold averaging technique, the stochastic differential equation is transformed into an infinite hierarchical system. This infinite system is cut off in such a manner that uncorrelated (totally randomly) and totally correlated stochastic processes are exactly included. By this the applicability of the Reynold method is extended.     

Dealing with uncertainties in angles-only initial orbit determination

A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map uncertainties from the observation space to the state space. When a minimum set of observations is available, DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available, high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology.     

The growth of density perturbations in Friedmann model universes with a decoupled radiation field

We examine the linear growth of density perturbations in homogeneous isotropic (Friedmann) model universes, including the effect of a decoupled radiation pressure field in the modelling. Amplification factors for density perturbations in all models are derived numerically, and it is shown that the effect of radiation pressure is to decelerate the growth of such condensations, thus requiring larger inhomogeneities to be produced at radiation decoupling in order to produce protogalaxies.     

A Simplified Kinetic Element Formulation for the Rotation of a Perturbed Mass-Asymmetric Rigid Body

Euler's equations, describing the rotation of an arbitrarily torqued mass asymmetric rigid body, are scaled using linear transformations that lead to a simplified set of first order ordinary differential equations without the explicit appearance of the principal moments of inertia. These scaled differential equations provide trivial access to an analytical solution and two constants of integration for the case of torque-free motion. Two additional representations for the third constant of integration are chosen to complete two new kinetic element sets that describe an osculating solution using the variation of parameters. The elements' physical representations are amplitudes and either angular displacement or initial time constant in the torque-free solution. These new kinetic elements lead to a considerably simplified variation of parameters solution to Euler's equations. The resulting variational equations are quite compact. To investigate error propagation behaviour of these new variational formulations in computer simulations, they are compared to the unmodified equations without kinematic coupling but under the influence of simulated gravity-gradient torques.     

A perturbative solution of the transfer equations for the Stokes parameters in a magnetic field

The transfer equations for the Stokes parameters, as derived under the assumption of LTE and in the general case of anomalous Zeeman multiplet, are solved by a perturbative method that reduces the system of four coupled linear differential equations to an iterative series of well known linear differential equations. The perturbation parameter turns out to be of the order of magnitude of the ratio of the Larmor frequency to the line Doppler half-width, so that the method appears to be particularly suitable for small magnetic fields.Some qualitative results on the importance of Faraday rotation and the orders of magnitude of the line profiles of the Stokes parameters are obtained.     

Nonlinear ion-acoustic solitons in a warm plasma with adiabatic positive and negative ions and hot non-isothermal electrons

The propagation of an ion-acoustic soliton in a collisionless plasma with adiabatic positive and negative ions (with equal ion temperature) and hot non-isothermal electrons is studied by use of the renormalization method introduced by Kodama and Taniuti in the reductive perturbation method. The basic set of fluid equations describing the system is reduced to a Korteweg-de Vries (K-dV)-type equation for the first-order perturbed potential and to a linear inhomogeneous differential equation to the second-order of the perturbed potential. A stationary solution of the coupled equations is obtained.     

Theory of the motion of an artificial Earth satellite

An improved analytical solution is obtained for the motion of an artificial Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational model includes zonal harmonics throughJ ₄, and the atmospheric model assumes a nonrotating spherical power density function. The differential equations are developed through second order under the assumption that the second zonal harmonic and the drag coefficient are both first-order terms, while the remaining zonal harmonics are of second order.Canonical transformations and the method of averaging are used to obtain transformations of variables which significantly simplify the transformed differential equations. A solution for these transformed equations is found; and this solution, in conjunction with the transformations cited above, gives equations for computing the six osculating orbital elements which describe the orbital motion of the satellite. The solution is valid for all eccentricities greater than 0 and less than 0.1 and all inclinations not near 0^o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions.     

Vector constants of the motion in spherically symmetric fields of force

The determination of the explicit form of vector constants of the motion for a point mass moving in an arbitrary spherically symmetric time-independent potential is reduced to the solution of an ordinary second-order linear differential equation. The vectors to be determined are assumed to be orthogonal to the angular momentum. The differential equation is solved for some particular fields of force and the corresponding vectors are constructed.     

Determination of the potential in a synodic system

Determination of the potential field in a fixed (inertial) system may be accomplished by the solution of a homogeneous linear partial differential equation when a family of orbits of a body moving in the field is given. This partial differential equation was presented and thoroughly analyzed earlier. The present paper discusses the same problem in a rotating system where the centrifugal and Coriolis effects render the pertinent partial differential equation in general non-homogeneous and non-linear. A linear, though non-homogeneous, partial differential equation for the determination of the synodic potential is obtained only in the special case of iso-energetic families of orbits.     

Nonlinear analytic solution for perturbed relative motion using differential equinoctial elements

This paper develops a nonlinear analytic solution for satellite relative motion in J₂-perturbed elliptic orbits by using the geometric method that can avoid directly solving the complex differential equations. The differential equinoctial elements (DEEs) are used to remove any singularities for zero-eccentricity or zero-inclination orbits. Based on the relationship between the relative states and the DEEs, state transition tensors (STTs) for transforming the osculating DEEs and propagating the mean DEEs have been derived. The formulation of these STTs has been split into a set of vector and matrix operations, which avoids directly expanding the complex second-order terms, and thus, the obtained STTs could be easy-to-understand and easy-to-code. Numerical results show that the proposed nonlinear solution is valid for zero-eccentricity and zero-inclination reference orbit and is more accurate than the previous linear or nonlinear methods for the long-term prediction of satellite relative motion.     

Bounds on the ideal MHD stability of line-tied 2-D coronal magnetic fields

A tractable method for investigating the linear stability of line-tied 2-D coronal magnetic fields is introduced. It is based on the Bernstein et al. (1958) energy principle and can be applied to non-isothermal equilibria with gravity, having a translational invariance. The perturbed potential energy integral is manipulated to produce either necessary conditions for stability to localized modes or sufficient conditions for stability to global modes. Each condition only requires the solution of a set of ordinary differential equations, integrated along the magnetic field lines. The tests are employed to two different classes of equilibria. A linear force-free field is shown to be completely stable, regardless of the shear. The role of pressure gradients, footpoint displacements, line-tying and stratification on an isothermal magneto-hydrostatic equilibrium is assessed.     

Application of the stroboscopic method to the stellar three-body problem

The stellar three-body problem has been approached by directly integrating the equations of motion in the orbital elements. The problem is set up in a barycentric chain with the orbits as perturbed ellipses.The integration was performed using the semi-analytical stroboscopic method, which is particularly useful for solving differential systems that depend on several slow variables and one fast, angular variable. Perturbation theory is applied and the solution for a particular order is obtained by way of successive approximations on the fundamental period of the fast variable.     

Secular effects in the orbits of the Galilean satellites

This paper presents the results of an investigation into the secular behavior of the orbits of the Galilean satellites of Jupiter. Kamel's perturbation method is used to remove all the explicitly periodic variables from the differential equations that describe the long period behavior of the orbits to third order in the masses, and the resulting differential equations for the secular behavior are then solved. Several numerical examples are given to illustrate the sensitivity of the solution to variations in the masses of the satellites.     

Radially accelerated optimal feedback orbits in central gravity field with linear drag

The solution of a feedback optimal control problem arising in orbital mechanics is addressed in this paper. The dynamics is that of a massless body moving in a central gravitational force field subject also to a drag and a radial modulated force. The drag is linearly proportional to the velocity and inversely proportional to the square of the distance from the center of attraction. The problem is tackled by exploiting the properties of a suitably devised linearizing map that transforms the nonlinear dynamics into an inhomogeneous linear system of differential equations supplemented by a quadratic objective function. The generating function method is then applied to this new system, and the solution is back transformed in the old variables. The proposed technique, in contrast to the classical optimal control problem, allows us to derive analytic closed-loop solutions without solving any two-point boundary value problem. Applications are discussed.     

High order transfer maps for perturbed Keplerian motion

The paper presents a new semi-analytical technique for the propagation of near-Earth satellite motion. The approach uses differential algebra techniques to compute the high order expansion of the solution of the system’s ordinary differential equation for one orbital revolution, referred to as the transfer map. Once computed, a single high order transfer map (HOTM) can be reused to map an initial condition, or a set of initial conditions, forward in time for many revolutions. The only limiting factor is that the mapped objects must stay close to the reference orbit such that they remain within the region of validity of the HOTM. The performance of the method is assessed through a set of test cases in which both autonomous and non-autonomous perturbations are considered, including the case of continuously propelled trajectories.     

Existence of the solution of Szebehely's equation in three dimensions using a two-parametric family of orbits

The three-dimensional inverse problem is investigated. A quasi-linear system of partial differential equations is derived for the determination of the potential. The solution of this system is studied by a method of differential geometry. A necessary condition for the solution is derived and the determination of the potential is reduced to algebraic equations written in vectorial form. A few examples are also given.