Canonical planetary perturbation equations for velocity-dependent forces,and the Lense-Thirring precession

A form of planetary perturbation theory based on canonical equations of motion, rather than on the use of osculating orbital elements, is developed and applied to several problems of interest. It is proved that, with appropriately selected initial conditions on the orbital elements, the two forms of perturbation theory give rise to identical predictions for the observable coordinates and velocities, while the orbital elements themselves may be strikingly different. Differences between the canonical form of perturbation theory and the classical Lagrange planetary perturbation equations are discussed. The canonical form of perturbation theory in some cases has advantages when the perturbing forces are velocity-dependent, but the two forms of perturbation theory are equivalent if the perturbing forces are dependent only on position and not on velocity. The canonical form of the planetary perturbation equations are derived and applied to the Lense Thirring precession of a test body in a Keplerian orbit around a rotating mass source.     

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 gravitational perturbation spectrum in linear satellite theory

A new method for calculating the perturbation spectrum in the framework of Kaula's linear satellite theory (LST) is introduced. The novelty of this approach consists in using recent results on the spectral decomposition of the perturbation frequencies in LST to provide a closed formulation for the amplitude and the phase of each line in the perturbation spectrum. The theory presented here can be applied to perturbations in the elements or in the radial and transverse directions due to the geopotential or to the tides. Separate algorithms are developed for application to orbits with circulating or frozen perigee.     

Structure-distortion of polytropic stars by differential rotation: Third-order perturbation theory

In order to specify the structure of a rapidly and differentially rotating gaseous polytrope, we extend Chandrasekhar's perturbation theory to include third-order terms in the perturbation parameter. In the present paper, the theory developed is required for a subsequent numerical treatment of the structure-determination.This research was supported by the Research Development Project of the University of Patras, Greece.     

The oscillation frequencies of magnetic stellar models

A first-order perturbation theory method developed by Goossens to determine the perturbation to the eigenfrequencies of stellar models caused by the presence of a magnetic field is modified slightly, and applied to models with toroidal and poloidal fields. Some limitations of the analysis are pointed out.     

三点相关统计的理论与测量

本文分别从数值模拟、理论和观测的角度研究和分析了三点相关函数以及双谱对尺度、形状和光度等的依赖性,并比较了数值模拟与二阶扰动理论以及暗晕模型的差别。我们发现二阶扰动理论即使在线性尺度上也是与数值模拟存在明显偏差的,它并不足以很好地描述暗物质双谱在大尺度上的行为。如果要与数值模拟更好地吻合,我们需要引入更高阶的修正。我们用其他人的半解析模型构建了对应于我们数值模拟的模拟星系样本,并计算了不同光度星系样本的双谱从而得到了相应的星系偏袒值。我们发现,对星系密度场的泰勒二阶展开是可以适用到准线性尺度k(?)0.15h/Mpc上的,但是要通过三点相关来获得准确的星系偏袒值,我们需要拥有对暗物质密度场的准确估计,这却是二阶非线性扰动理论所无法做到的。暗晕模型在定性描述双谱行为方面是十分有效的,但是它与数值模拟的定量比较还是存在很多不同。要用暗晕模型来精确描述双谱,我们需要对模型的各种设定做出精细的调节和改进。最后,我们还利用了SDSS的最新观测数据测量了红移空间的三点相关函数以及投影三点相关函数,并研究它们对于星系光度、颜色和恒星质量等性质的依赖关系。我们发现,不同于之前的许多工作,星系的归约三点相关函数对光度存在明显的依赖性,而这种依赖性却是与三角形的形状和尺度耦合在一起。三点相关对恒星质量的依赖与光度依赖性十分类似。而颜色的依赖性在小尺度上则比光度和恒星质量更显著一些。     

On a perturbation theory using Lie transforms

Kamel has recently extended to non-Hamiltonian equations a perturbation theory using Lie transforms. We show here how Kamel's extension can be approached from an intrinsic viewpoint, which reformulation leads to a simpler algorithm. Then we complete Kamel's contribution by establishing the rules for inverting the transformation generated by the perturbation theory, and for composing two such transformations.     

Nonlinear perturbation theory for cosmic ray propagation in random magnetic fields

A nonlinear perturbation theory is applied to the problem of pitch angle diffusion of energetic particles in random magnetic fields. To keep the analysis simple, the discussion is restricted to fluctuation fields, consisting of Alfvén waves. It is shown that the failure of quasilinear theory at small particle velocities parallel to the average field can be overcome by a statistically exact treatment of the particle orbits in the first order fields. In fact, for spherical power spectra which, in addition, do not fall off too steeply with increasing frequency, the conventional perturbation theory also leads to formally convergent expressions for the scattering mean free path. These results are shown to be quite satisfactory, even in a quantitative sense. For more general physically realistic power spectra, however, a divergence-free diffusion theory is indispensible. A simple representation for the resulting pitch-angle diffusion coefficient is suggested.     

Modulation of ion acoustic waves

The multiple scales perturbation theory is applied to a system of dispersive waves including plasma waves (ion-acoustic waves). Assuming the amplitudes of waves are slowly varying function of space and time, we find that long-time slow modulation of the complex amplitude can be described by the non-linear Schrödinger equation. This result agrees with that obtained by Shimizu and Ichikawa using the reductive perturbation theory, and agrees exactly with the nonrelativistic limit of the nonlinear Schrödinger equation obtained by Nejoh using the stretching method.     

Construction of a consistent semianalytic theory of a planetary or moon orbiter perturbed by a third body

The third body perturbation of an orbiter of a planet or moon is considered. A very convenient form of the Lagrange equations is given allowing an easy derivation of the various terms of the expansion of the perturbed elements. A careful analysis of the order of magnitude of these terms indicates which ones are required for a consistent theory. It follows that in many practical cases the main term of the disturbing function has to be carried to the second order of the perturbation theory.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.Dedicated to V. Szebehely on the occasion of his 60th birthday     

Account of additional perturbations in canonical system solutions obtained by Lie transforms

The analytical or semi-analytical solution of some canonical system (basic theory) constructed by the perturbation technique based on the Lie transforms is supposed to be known. A method allowing one to obtain the solution of perturbed canonical or non-canonical system using the known basic theory is discussed.     

Restricted quantum-mechanical three-body problems

A new method is presented in a general form to solve the Schrödinger equation of helium-like ions. The wave function is expanded in terms of the eigenfunctions of a moving electron in the field of two Coulombic ions which are fixed in space. This makes the method similar to the Dirac perturbation theory (perturbation theory for time-dependent problems). In the present method an infinitely coupled system of infinitely many second-order ordinary differential equations must be solved instead of one second-order partial differential equation of three variables. The nature of the singular points and boundary conditions are discussed and some general relations are given which are useful for the numerical treatment.     

Asymptotic expansions in the perturbed two-body problem with application to systems with variable mass

The main theorems of the theory of averaging are formulated for slowly varying standard systems and we show that it is possible to extend the class of perturbation problems where averaging might be used. The application of the averaging method to the perturbed two-body problem is possible but involves many technical difficulties which in the case of the two-body problem with variable mass are avoided by deriving new and more suitable equations for these perturbation problems. Application of the averaging method to these perturbation problems yields asymptotic approximations which are valid on a long time-scale. It is shown by comparison with results obtained earlier that in the case of the two-body problem with slow decrease of mass the averaging method cannot be applied if the initial conditions are nearly parabolic. In studying the two-body problem with quick decrease of mass it is shown that the new formulation of the perturbation problem can be used to obtain matched asymptotic approximations.     

Large-scale bias in the Universe — II. Redshift-space bispectrum

The determination of the density parameter Ω₀ from the large-scale distribution of galaxies is one of the major goals of modern cosmology. However, if galaxies are biased tracers of the underlying mass distribution, linear perturbation theory leads to a degeneracy between Ω₀ and the linear bias parameter b , and the density parameter cannot be estimated. In Matarrese, Verde &38; Heavens we developed a method based on second-order perturbation theory to use the bispectrum to lift this degeneracy by measuring the bias parameter in an Ω₀-independent way. The formalism was developed assuming that one has perfect information on the positions of galaxies in three dimensions. In galaxy redshift surveys, the three-dimensional information is imperfect, because of the contaminating effects of peculiar velocities, and the resulting clustering pattern in redshift space is distorted. In this paper we combine second-order perturbation theory with a model for collapsed, virialized structures, to extend the method to redshift space, and demonstrate that the method should be successful in determining with reasonable accuracy the bias parameter from state-of-the-art surveys such as the Anglo-Australian 2 degree Field Survey and the Sloan Digital Sky Survey.     

Planetary theories

The present condition of planetary theory, i.e., the computation and comprehension of the motion of the planets, is reviewed. Account is taken of the new and anticipated demands of observational accuracy, new mathematical methods in perturbation theory, and the use of the high-speed computer to perform algebraic as well as numerical work.     

Two-stream instability in the presence of longitudinal magnetic field

A multiple sclaes perturbation theory has been applied to investigate the nonlinear behaviour of beam-plasma system near a marginally stable state in the presence of longitudinal magnetic field. The perturbation method leads to a nonlinear Schrödinger equation for the finite amplitude. The coefficients of this equation show that only if the beam is compressed isothermally can there exist a range of wavenumbers for which stabilization might occur. The stable region increases with the applied magnetic field.     

Tidal Evolution of the Kepler Candidate Two-planet Systems

The tidal evolution of ten Kepler candidate two-planet systems is investigated by using the general secular perturbation theory, and then a general picture of tidal evolution for these systems is described. Taking the KOI 1239 system as an example, the tidal effect of the system is studied in detail, the results indicate that the dissipative term of the tidal effect causes the attenuation of planets’ orbital eccentricities, and it plays a dominant role in the process of orbital evolution, however, the conservative term of the tidal effect and the relativistic effect may damp the excitation of the eccentricity of the inner planet under the secular perturbation of the outer planet. In addition, the process of tidal evolution is also affected by both the initial eccentricity of the outer planet and the planet's tidal dissipation coeffcient. At the same time, the numerical simulation on the tidal evolution of the KOI 1239 system is also made, and the numerical results are consistent with those of general secular perturbation theory.     

Gravitational lensing by stars in a galaxy halo: Theory of combined weak and strong scattering

The theory of gravitational lensing of background quasars by stars in the halo of a galaxy is considered. In the limiting case of small ‘optical depth’, only one star is close enough to the beam to cause strong scattering, and the effect of all the other stars is treated as a perturbation with both systematic and random components. The perturbation coming from weak scattering can increase the number of images and the amplification in those cases where the amplification is already high; such events are preferentially selected in flux limited observations. The theory is applicable to the apparent association of background quasars with foreground galaxies. A comparison with earlier work on the same problem is given. The relevance of these results to gravitational lensing by galaxies as perturbed by random inhomogeneities surrounding the ray path is also briefly discussed.     

Algebraic aspects of perturbation theories

In this paper the following problems are considered: Hori's perturbation equations, the composition of two Lie series, the elimination of geometrical (virtual) singularities in perturbation theory, the connection between the methods of Hori and Deprit. The analysis is based on an isomorphism between the Lie algebra of the non-associative algebra of vector fields and a Lie algebra of linear operators. All linear operators, however, form an associative algebra.     

An extended canonical perturbation method

In this investigation, a procedure is described for extending the application of canonical perturbation theories, which have been applied previously to the study of conservative systems only, to the study of non-conservative dynamical systems. The extension is obtained by imbedding then-dimensional non-conservative motion in a 2n-dimensional space can always be specified in canonical form, and, consequently, the motion can be studied by direct application of any canonical perturbation method. The disadvantage of determining a solution to the 2n-dimensional problem instead of the originaln-dimensional problem is minimized if the canonical transformation theory is used to develop the perturbation solution. As examples to illustrate the application of the method, Duffing's equation, the equation for a linear oscillator with cubic damping and the van der Pol equation are solved using the Lie-Hori perturbation algorithm.This research was supported by the Office of Naval Research under Contract N00014-67-a-0126-0013.