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1.
The Sitnikov problem is one of the most simple cases of the elliptic restricted three body system. A massless body oscillates
along a line (z) perpendicular to a plane (x,y) in which two equally massive bodies, called primary masses, perform Keplerian orbits around their common barycentre with
a given eccentricity e. The crossing point of the line of motion of the third mass with the plane is equal to the centre of gravity of the entire
system. In spite of its simple geometrical structure, the system is nonlinear and explicitly time dependent. It is globally
non integrable and therefore represents an interesting application for advanced perturbative methods. In the present work
a high order perturbation approach to the problem was performed, by using symbolic algorithms written in Mathematica. Floquet
theory was used to derive solutions of the linearized equation up to 17th order in e. In this way precise analytical expressions for the stability of the system were obtained. Then, applying the Courant and
Snyder transformation to the nonlinear equation, algebraic solutions of seventh order in z and e were derived using the method of Poincaré–Lindstedt. The enormous amount of necessary computations were performed by extensive
use of symbolic programming. We developed automated and highly modularized algorithms in order to master the problem of ordering
an increasing number of algebraic terms originating from high order perturbation theory. 相似文献
2.
E. K. El-Shewy 《Astrophysics and Space Science》2011,335(2):389-397
A theoretical investigation is carried out for the nonlinear properties of small amplitude electron acoustic solitary waves
(EAWs) in an unmagnetized collisionless plasma consisting of a cold electron fluid and hot electrons obeying κ velocity distribution, and stationary ions. The Korteweg de Vries (KdV) equation that contains the lowest-order nonlinearity
and dispersion is derived from the lowest order of perturbation and a linear inhomogeneous (KdV-type) equation that accounts
for the higher-order nonlinearity and dispersion is obtained. A stationary solution for equations resulting from higher-order
perturbation theory has been found using the renormalization method. The effects of the spectral index κ and the higher-order corrections are found to significantly change the properties (viz. the amplitude, width, electric field
) of the EASWs. A comparison with the Viking Satellite observations in the dayside auroral zone are also discussed. 相似文献
3.
S. B. Faruque 《Celestial Mechanics and Dynamical Astronomy》2003,87(4):353-369
A new analytic expression for the position of the infinitesimal body in the elliptic Sitnikov problem is presented. This solution is valid for small bounded oscillations in cases of moderate primary eccentricities. We first linearize the problem and obtain solution to this Hill's type equation. After that the lowest order nonlinear force is added to the problem. The final solution to the equation with nonlinear force included is obtained through first the use of a Courant and Snyder transformation followed by the Lindstedt–Poincaré perturbation method and again an application of Courant and Snyder transformation. The solution thus obtained is compared with existing solutions, and satisfactory agreement is found. 相似文献
4.
J. S. Aliyev 《Astrophysics and Space Science》1986,121(2):283-300
The functional analytic method of solution is applied to investigation of the radiative transfer equation in spectral lines. A problem of scattering in the spectral line with the frequency redistribution in anisotropic-scattering infinite and semi-infinite media is considered. Continuum absorption in the line is also taken into account.The solution is presented as the exponential function of the operatorA and the functional calculus is developed. The eigenfunction and the expansion coefficients, in terms of which the explicit solution is expressed, have been found. The nonlinear equation and the explicit expressions for theX-function are derived. The albedo problem with the determined expansion coefficients and the intensity of the emergent radiation is given as an example. 相似文献
5.
Hamid Reza Pakzad 《Astrophysics and Space Science》2011,334(1):55-60
Ion acoustic shock waves (IASWs) are studied in a plasma consisting of nonextensive electrons and ions. The dissipation is
taken into account the kinematic viscosity among the plasma constituents. The Korteweg-de Vries-Burgers (KdV-Burgers) equation
is derived by reductive perturbation method. Shock waves are solutions of KdV-Burgers equation. It is shown that acceptable
values of q-parameter (where q stands for the electron nonextensive parameter) are more than 3 in a weakly nonlinear analysis. We have found that the amplitude
of shock waves decreases by an increasing q-parameter. 相似文献
6.
In this paper, we consider a statistical method for distance determination of stellar groups. The method depends on the assumption
that the members of the group scatter around a mean absolute magnitude in Gaussian distribution. The mean apparent magnitude
of the members is then expressed by frequency function, so as to correct for observational incompleteness at the faint end.
The problem reduces to the solution of a highly transcendental equation for a given magnitude parameter α. For the computational developments of the problem, continued fraction by the Top-Down algorithm was developed and applied
for the evaluation of the error function erf(z). The distance equation Λ(y) = 0 was solved by an iterative method of second order of convergence using homotopy continuation technique. This technique
does not need any prior knowledge of the initial guess, a property which avoids the critical situations between divergent
and very slow convergent solutions, that may exist in the applications of other iterative methods depending on initial guess. 相似文献
7.
Paolo Lanzano 《Earth, Moon, and Planets》1986,34(3):283-304
We examine the problem of the thermoelastic deformation of a spherical Earth with constant elastic parameters heated from within by the spontaneous decay of radiogenic elements.The problem consists of the simultaneous solution of the Navier-Stokes equation and the heat conduction equation. We reach an integrodifferential equation which we solve by means of the Laplace transform and the Green's function approach.We obtain analytic solutions for the temperature distribution and radial deformation as infinite series of functions of the radial distance and time, depending also on a sequence of eigenvalues. We provide particular solutions for the case when the two specific heats C
p and C
v are approximately equal. p ]We believe that our analytic results are applicable to the study of the oceanic lithosphere deformations. Our approach could be successfully applied to ascertain the deformation according to other regimes of internal heating. 相似文献
8.
The nonlinear propagation of ion acoustic waves in ideal plasmas consisting of degenerate electrons and positrons, and isothermal ions is investigated. The Korteweg de Vries (K-dV) equation that contains the lowest order nonlinearity and dispersion is derived from the lowest order of perturbation and a linear inhomogeneous (K-dV type) equation that accounts for the higher order nonlinearity and the dispersion relation is obtained. The stationary wave solution for these equations has been found using the renormalization method. Also, the effects of electrons and positrons densities and ion temperature on the amplitude and width of solitary waves are investigated, numerically. It is seen that higher order corrections significantly change the properties of the K-dV solitons. Also, it is found that both compressive and rarefactive solitary waves can be propagated in such plasma system. 相似文献
9.
The resonant absorption of small amplitude surface Alfvén waves is studied in nonlinear incompressible MHD for a viscous and resistive plasma. The reductive perturbation method is used to obtain the equation that governs the spatial and temporal behaviour of small amplitude nonlinear surface Alfvén waves. Numerical solutions to this equation are obtained under the initial condition that att = 0 the spatial variation is purely sinusoidal. The numerical results show that nonlinearity accelerates the wave damping due to resonant absorption. Resonant absorption is a more efficient wave damping mechanism than can be anticipated on the basis of linear theory. 相似文献
10.
Using a multiple time-scale method, the weakly nonlinear waves on a self-gravitating incompressible fluid column are investigated. The analysis reveals that near the wavenumberk=k
c
, the amplitude modulation of a standing wave can be described by the nonlinear Schrödinger equation with the roles of time and space variables interchanged. The nonlinear cutoff wavenumber, which depends sensitively on initial conditions, can then be derived from the nonlinear Schrödinger equation so obtained. The finite amplitude standing wave is stable against modulation. 相似文献
11.
G.V. Groves 《Planetary and Space Science》1982,30(3):219-244
From the equations of classical tidal theory with Newtonian cooling (Chapman and Lindzen, Atmospheric Tides: thermal and gravitational, Reidel, 1970), formulae are obtained for wind, temperature and pressure oscillations generated by thermal, gravitational and lower-boundary excitations of given frequency. The analysis is an extension of that of Butler and Small (Proc. R. Soc. Lond.A274, 91, 1963) who formulated solutions of the vertical structure equation in terms of two independent solutions of the homogeneous equation and derived expressions for surface pressure oscillations. A comprehensive formulation is presented for wind, temperature and pressure oscillations as functions of height with the above-mentioned sources of excitation and an upper-boundary radiation condition. The formulae obtained are applied at the surface leading to evaluations of the surface oscillation weighting function Wp(z) which weights the thermal excitation at height z according to its differential contribution to the surface oscillation. The formulae are shown to simplify at heights above a region of excitation and evaluations are undertaken of the thermal response weighting function Wt(z) which weights the thermal excitation at height z according to its differential contribution to the oscillation at any height above the region of thermal excitation. Computational procedures are described for obtaining two independent solutions of the homogeneous equation and results are presented for an adopted profile of atmospheric scale height. The problem of deriving the surface pressure oscillation due to a tidal potential is briefly reviewed and results are presented as an example of the application of formulae that have been derived. 相似文献
12.
M. S. Wheatland 《Solar physics》2007,245(2):251-262
Improvements to an existing method for calculating nonlinear force-free magnetic fields (Wheatland, Solar Phys. 238, 29, 2006) are described. In particular a solution of the 3-D Poisson equation using 2-D Fourier transforms is presented. The improved
nonlinear force-free method is demonstrated in application to linear force-free test cases with localized nonzero values of
the normal component of the field in the boundary. These fields provide suitable test cases for nonlinear force-free calculations
because the boundary conditions involve localized nonzero values of the normal components of the field and of the current
density, and because (being linear force-free fields) they have more direct numerical solutions. Despite their simplicity,
fields of this kind have not been recognized as test cases for nonlinear methods before. The examples illustrate the treatment
of the boundary conditions on current in the nonlinear force-free method, and in particular the limitations imposed by field
lines that connect outside of the boundary region. 相似文献
13.
B. P. Kondratyev 《Solar System Research》2011,45(5):447-458
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. 相似文献
14.
T. A. Heppenheimer 《Celestial Mechanics and Dynamical Astronomy》1973,7(2):177-194
Out-of-plane motion about libration points is studied within the framework of the elliptic restricted three-body problem. Nonlinear motion in the circular restricted problem is given to third order in the out-of-plane amplitudeA
z by Jacobi elliptic functions. Linear motion in the elliptic problem is studied using Mathieu's and Hill's equations. Additional terms needed for a complete third-order theory are found using Lindsted's method. This theory is constructed for the case of collinear libration points; for the case of triangular points, a third-order nonlinear solution is given separately in terms of Jacobi elliptic functions. 相似文献
15.
To illustrate his theory of coronal heating, Parker initially considers the problem of disturbing a homogeneous vertical magnetic
field that is line-tied across two infinite horizontal surfaces. It is argued that, in the absence of resistive effects, any
perturbed equilibrium must be independent of z. As a result random footpoint perturbations give rise to magnetic singularities, which generate strong Ohmic heating in the
case of resistive plasmas. More recently these ideas have been formalized in terms of a magneto-static theorem but no formal
proof has been provided. In this paper we investigate the Parker hypothesis by formulating the problem in terms of the fluid
displacement. We find that, contrary to Parker's assertion, well-defined solutions for arbitrary compressibility can be constructed
which possess non-trivial z-dependence. In particular, an analytic treatment shows that small-amplitude Fourier disturbances violate the symmetry ∂z = 0 for both compact and non-compact regions of the (x, y) plane. Magnetic relaxation experiments at various levels of gas pressure confirm the existence and stability of the Fourier
mode solutions. More general footpoint displacements that include appreciable shear and twist are also shown to relax to well-defined
non-singular equilibria. The implications for Parker's theory of coronal heating are discussed. 相似文献
16.
An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem
J. Hagel 《Celestial Mechanics and Dynamical Astronomy》2009,103(3):251-266
The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the
primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a
coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and
therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original
method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations
at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence
of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically
obtained solutions. 相似文献
17.
M. K. Das 《Astrophysics and Space Science》1985,114(2):295-302
The nonlinear pseudo-radial mode of oscillation of a rotating magnetic star is studied. It is shown that for a general rotational field, the coupling between magnetic field and rotation tends to reduce the average rotational energy parameterT. This result in a lowering of the maximum pulsation amplitudeq
max, which depends on strength of rotation and magnetic field. The configuration tends, therefore, to a new equilibrium state at lower value ofq
max. The analytic solution of the pulsation equation for the case ofy=5/3 in the presence of rotation and magnetic field has also been derived in the Appendix. 相似文献
18.
The nonlinear propagation of ion acoustic waves in an ideal plasmas containing degenerate electrons is investigated. The Korteweg-de-Vries
(K-dV) equation is derived for ion acoustic waves by using reductive perturbation method. The analytical traveling wave solutions
of the K-dV equation investigated, through the (G′/G)-expansion method. These traveling wave solutions are expressed by hyperbolic function, trigonometric functions are rational
functions. When the parameters are taken special values, the solitary waves are derived from the traveling waves. Also, numerically
the effect different parameters on these solitary waves investigated and it is seen that exist only the compressive solitary
waves in Thomas-Fermi plasmas. 相似文献
19.
The redshift (z) dependence of the dispersion relations for free particles is analyzed by taking into account the Lorentz invariance violation. A nonlinear algebraic equation is derived for the momenta of the particles involved in the annihilation reaction of a hard photon from a γ-ray source with a soft cosmic microwave background (CMB) photon near the threshold of this reaction. The solutions of this threshold equation are constructed and analyzed as a function of the redshift. We show that the threshold of the reaction under consideration tends to decrease with increasing z; the energy spectra of γ-ray sources at energies of ~10 TeV must be cut off in accordance with the calculated z dependence. We also calculate the time delay of the light signals from γ-ray sources that corresponds to the Lorentz invariance violation for photons. We discuss the possibility of improving the standard constraints on the Lorentz invariance violation parameters for fields of various physical natures. 相似文献
20.
Nonlinear cylindrical fast magnetoacoustic waves are investigated in a dissipative magnetoplasma comprising of electrons,
positrons, and ions. In this regard, cylindrical Kadomtsev-Petviashvili-Burgers (CKPB) equation is derived using the small
amplitude perturbation expansion method. Furthermore, cylindrical Burgers-Kadomtsev-Petviashvili (Cyl Burgers-KP) for a fast
magnetoacoustic wave is derived, for the first time, for spatial scales larger than the electron/positron skin depths, c/ω
p(e,p). Using the tangent hyperbolic method, the solutions of both planar KPB and Burgers-KP equations are obtained and then subsequently
used as an initial profile to solve their respective counterparts in the cylindrical geometry. The effect of positron concentration,
kinematic viscosity, and plasma β are explored both for the KPB and the Burgers-KP shock waves and the differences between the two are highlighted. The temporal
evolution of the cylindrical fast magnetoacoustic wave is also numerically investigated. The present study may be beneficial
to study the propagation characteristics of nonlinear electromagnetic shock waves in planetary magnetospheres. 相似文献