Analytical linear perturbation theory for highly eccentric satellite orbits

Kaula's satellite linear perturbation theory has been extended for the case of highly eccentric orbits by using elliptic function expansions.On leave from Institute of Applied Astronomy, St.-Petersburg     

The mapping of Kaula's solution into the orbital reference frame

Kaula's celebrated solution to the problem of satellite motion in the gravitational field of a rigid body is transformed to give the perturbation spectra in both position and velocity in the radial, transverse and normal directions of the orbital reference frame. This work is an extension and a refinement of the theory of orbital perturbations due to the geopotential previously published by Rosborough and Tapley (1987).     

A new method for computing the spectrum of the gravitational perturbations on satellite orbits

Some aspects for efficient computation of the tidal perturbation due to the ellipticity effects of the Earth, the luni-solar potential on an Earth-orbiting satellite and the perturbations of the satellite's radial, transverse and normal position components due to the effects of the Earth's gravitational and ocean tide fields are presented. A straightforward method for computing the spectrum of the geopotential and the tidal-induced perturbations of the orbit elements and the radial, transverse and normal components is described.     

The fifth-order analytical solution of the equations of motion of a satellite in orbit around a non-spherical planet

A high-precise analytical theory of a satellite in orbit around a non-spherical planet has been developed. The Poisson's small parameter method has been used. All secular and short-periodic perturbations proportional up to and including a product of five arbitrary harmonic coefficients of the planetary potential expansion are calculated. Long-periodic perturbations are derived with the accuracy of up to the fourth-order, inclusive. The influence of the high-order perturbations on the motion of ETALON-1 satellite has been investigated. The results of comparison of the numerical and analytical integration of the equations of its motion over a five year interval are as follows:

Spectral decomposition of geopotential,earth and ocean tidal perturbations in linear satellite theory

The spectra of geopotential, Earth and ocean tidal perturbations on a satellite can be obtained using Kaula's linear theory, or an extension thereof, as summations of terms depending on four indices l, m, p, q. In this work algorithms are presented that generate the equivalence classes induced by the composition rule of frequency on the set of all (l, m, p, q) combinations up to a maximum degree L and maximum value Q of the last index. These algorithms eliminate the need to search the set of frequencies when the linear theory is programmed on a computer.     

Long period variations of the motion of a satellite due to non-resonant tesseral harmonics of a gravity potential

The main effects of tesseral harmonics of a gravity potential expansion on the motion of a satellite, are short period variations as well as long period variations due to resonances. However, other smaller long period and secular variations can arise from interactions between tesseral terms of the same order. The analytical integration of these effects is developed, using numerical evaluation of Kaula eccentricity and inclination functions. Examples for some Earth's geodetic satellites show that secular effects can reach a few decameters per year. The secular variations can even reach several hundred of meters per year for the Mars natural satellite Phobos.     

Investigation of equilibria of a satellite subjected to gravitational and aerodynamic torques

Attitude motion of a satellite subjected to gravitational and aerodynamic torques in a circular orbit is investigated. In special case, when the center of pressure of aerodynamic forces is located on one of the principal central axes of inertia of the satellite, all equilibrium orientations are determined. Necessary and (or) sufficient conditions of stability are obtained for each equilibrium orientation. Evolution of domains where stability conditions take place is investigated. All bifurcation values of parameters corresponding to qualitative change of domains of stability are determined.     

Dynamics of a satellite orbiting a planet with an inhomogeneous gravitational field

We study the dynamics of a satellite (artificial or natural) orbiting an Earth-like planet at low altitude from an analytical point of view. The perturbation considered takes into account the gravity attraction of the planet and in particular it is caused by its inhomogeneous potential. We begin by truncating the equations of motion at second order, that is, incorporating the zonal and the tesseral harmonics up to order two. The system is formulated as an autonomous Hamiltonian and has three degrees of freedom. After three successive Lie transformations, the system is normalised with respect to two angular co-ordinates up to order five in a suitable small parameter given by the quotient between the angular velocity of the planet and the mean motion of the satellite. Our treatment is free of power expansions of the eccentricity and of truncated Fourier series in the anomalies. Once these transformations are performed, the truncated Hamiltonian defines a system of one degree of freedom which is rewritten as a function of two variables which generate a phase space which takes into account all of the symmetries of the problem. Next an analysis of the system is achieved obtaining up to six relative equilibria and three types of bifurcations. The connection with the original system is established concluding the existence of various families of invariant 3-tori of it, as well as quasiperiodic and periodic trajectories. This is achieved by using KAM theory techniques.     

A note on lower bounds for relative equilibria in the main problem of artificial satellite theory

In the analytical approach to the main problem in satellite theory, the consideration of the physical parameters imposes a lower bound for normalized Hamiltonian. We show that there is no elliptic frozen orbits, at critical inclination, when we consider small values of H, the third component of the angular momentum. The argument used suggests that it might be applied also to more realistic zonal and tesseral models. Moreover, for almost polar orbits, when H may be taken as another small parameter, a different approach that will simplify the ephemerides generators is proposed.     

Canonical theory of perturbations using eccentric anomaly as independent variable

It is shown in this paper how to build a canonical transformation of variables, so that the eccentric anomaly becomes the new independent variable. In the case of eccentric elliptical orbits it changes the equations of motion so, that they can be integrated analytically to any order of approximation comparatively easy.     

Equilibria of a satellite subjected to gravitational and aerodynamic torques with pressure center in a principal plane of inertia

Attitude motion of a satellite subjected to gravitational and aerodynamic torques in a circular orbit is considered. In special case, when the center of pressure of aerodynamic forces is located in one of the principal central planes of inertia of the satellite, all equilibrium orientations are determined. Existence conditions of all equilibria are obtained and evolution of domains with a fixed number of equilibria is investigated in detail. All bifurcation values of the system’s parameters corresponding to the qualitative change of these domains are determined. Sufficient conditions of stability are obtained for each equilibrium orientation using generalized integral of energy.     

A special perturbation method in orbital dynamics

The special perturbation method considered in this paper combines simplicity of computer implementation, speed and precision, and can propagate the orbit of any material particle. The paper describes the evolution of some orbital elements based in Euler parameters, which are constants in the unperturbed problem, but which evolve in the time scale imposed by the perturbation. The variation of parameters technique is used to develop expressions for the derivatives of seven elements for the general case, which includes any type of perturbation. These basic differential equations are slightly modified by introducing one additional equation for the time, reaching a total order of eight. The method was developed in the Grupo de Dinámica de Tethers (GDT) of the UPM, as a tool for dynamic simulations of tethers. However, it can be used in any other field and with any kind of orbit and perturbation. It is free of singularities related to small inclination and/or eccentricity. The use of Euler parameters makes it robust. The perturbation forces are handled in a very simple way: the method requires their components in the orbital frame or in an inertial frame. A comparison with other schemes is performed in the paper to show the good performance of the method.     

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.     

Study of corrections to the dust model via perturbation theory

This work reports on the application of the Eulerian perturbation theory to a recently proposed model of cosmological structure formation by gravitational instability. Its physical meaning is discussed in detail and put in perspective of previous works. The model incorporates in a systematic fashion corrections to the popular dust model owing to multistreaming and, more generally, the small-scale, virialized degrees of freedom. It features a time-dependent length-scale L ( t ) estimated to be   L / r ₀10^-1  [ r ₀( t ) is the non-linear scale, at which   ²=1]  . The model provides a new angle on the dust model and allows us to overcome some of its limitations. Thus, the scale L ( t ) works as a physically meaningful short-distance cut-off for the divergences appearing in the perturbation expansion of the dust model when there is too much initial power on small scales. The model also incorporates the generation of vorticity by tidal forces; according to the perturbational result, the filtered vorticity for standard cold dark matter initial conditions should be significant today only at scales below 1  h ¹ Mpc.     

On the numerical transformation of variables in perturbation theory

Once the generating function of a Lie-type transformation is known, canonical variables can be transformed numerically by application of a Runge-Kutta type integration method or any other appropriate numerical integration algorithm. The proposed approach avails itself of the fact, that the transformation is defined by a system of differential equations with a small parameter as the independent variable. The integration of such systems arising in the perturbation theories of Hori and Deprit is discussed. The method allows to compute numerical values of periodic perturbations without deriving explicitly the perturbation series. This saving of an algebraic work is achieved at the expense of multiple evaluations of the generator's derivatives.     

The secular effect of gravitational radiation damping on the periastron advance of binary stars in second order perturbation theory

The second order perturbation effect of gravitational radiation damping on the periastron advance of binary stars is studied.The second order analytic solution is obtained based on the first order theory in the 2014 article by Li.Theoretical results show that secular variation exists in the periastron advance of binary stars in the second order theory,but secular variation does not exist in the first order perturbation theory.Numerical results for two compact binary stars(PSR J0737–3039 and M33 X-7)are given,demonstrating the theoretical significance even though the effect is very small.