Short-period Perturbations Research Articles

Nonsingular vectorial reformulation of the short-period corrections in Kozai’s oblateness solution

We derive a new analytical solution for the first-order, short-periodic perturbations due to planetary oblateness and systematically compare our results to the classical Brouwer–Lyddane transformation. Our approach is based on the Milankovitch vectorial elements and is free of all the mathematical singularities. Being a non-canonical set, our derivation follows the scheme used by Kozai in his oblateness solution. We adopt the mean longitude as the fast variable and present a compact power-series solution in eccentricity for its short-periodic perturbations that relies on Hansen’s coefficients. We also use a numerical averaging algorithm based on the fast-Fourier transform to further validate our new mean-to-osculating and inverse transformations. This technique constitutes a new approach for deriving short-periodic corrections and exhibits performance that are comparable to other existing and well-established theories, with the advantage that it can be potentially extended to modeling non-conservative orbit perturbations.

The orbital evolution of the Sun–Jupiter–Saturn–Uranus–Neptune system on long time scales

The averaged semi-analytical motion theory of the four-planetary problem is constructed up to the third order in planetary masses and the sixth degree in the orbital eccentricities and inclinations. The second system of Poincare elements and the Jacobi coordinate system are used for the construction of the Hamiltonian expansion. The averaged Hamiltonian is obtained in the third approximation by the Hori–Deprit method. All analytical transformations are performed by using CAS Piranha. The constructed equations of motion in averaged elements are numerically integrated by the different methods for the giant planets of the Solar System over a time interval of up to 10 Gyr. The planetary motion is quasi-periodic, and the short-term perturbations of the orbital elements conserve small values in the modeling process. The comparison of obtained amplitudes and periods of the change of the orbital elements with numerical motion theories shows an excellent agreement with them. The properties of the planetary motion are given. The short-periodic perturbations and the precision of the integration are estimated.

Short-period effects of the planetary perturbations on the Sun–Earth Lagrangian point L3

Context. The Lagrangian point L3 of the Sun–Earth system, and its Lyapunov orbits, have been proposed to perform station-keeping, although L3 is only rigorously defined for the extremely simplified model represented by the reduced Sun–Earth–spacecraft system. As in L3 the planetary perturbations (mainly from Jupiter and Venus) are stronger than Earth’s attraction, it is necessary to understand whether or not the dynamics close to L3 persist under such a strong perturbation, allowing for a definition of dynamical substitutes for models that are more realistic than the circular restricted three-body problem. Aims. In this paper we address the problem of the existence of motions that remain close to L3 for a time-span which is relevant for space missions in a model of the Solar System compatible with the precision of JPL digital ephemerides. Methods. First, we computed analytically the main short-period effects of planetary perturbations in a simplified model of the Solar System with the orbits of all the planets co-planar and circular. We then applied the Fast Lyapunov Indicator method in order to find dynamical substitutes that exist for time-spans of hundreds of years in the model of the Solar System that is used to produce the modern ephemerides. Results. We find that the original system is conjugate by a canonical transformation to an averaged system that has an equilibrium close to L3: even if Venus and Jupiter each move the position of this equilibrium by about 218 and 176 km, respectively, in opposite directions, in the model where both the planets are included, their effects almost perfectly compensate for one another, leaving a displacement of about 40 km only. This equilibrium is then mapped in the original system to a quasi-periodic dynamical substitute; the contributions of each planet to the amplitude of this quasi-periodic libration around L3 do not compensate for one another, and sum to about 10 000 km. The Fast Lyapunov Indicator method allowed us to find orbits of any amplitude bigger than this one (up to 0.03 AU) for time-spans of hundreds of years in the model of the Solar System that is used to produce the modern ephemerides. Conclusions. Using a combination of the Hamiltonian averaging method with a new implementation of the Fast Lyapunov Indicator method we find orbits useful for astrodynamics originating at the Sun–Earth Lagrangian point L3 for a realistic model of the Solar System. In particular, this usage of the chaos indicator provides an innovative application of dynamical systems theory to astrodynamics, where the short-period perturbations represent a relevant part of the model.

Earth's shadowing effects by computer algebra

The short-periodic perturbations of orbits of an artificial satellite due to the radiation pressure during one orbital period are influenced by Earth's shadow. For the semi-analytical theories it is necessary to calculate great number of coefficients. Using computer algebra we have computed them for some of applicable semi-analytical theories.

The Implementation of Hori–Deprit Method to the Construction Averaged Planetary Motion Theory by Means of Computer Algebra System Piranha

This article is related to the problem of the construction of planetary motion theory. We have expanded the Hamiltonian of the four-planetary problem into the Poisson series in osculating elements of the second Poincare system. The series expansion is constructed up to the third degree of the small parameter. The averaging procedure of the Hamiltonian is performed by the Hori–Deprit method. It allows to eliminate short-periodic perturbations and sufficiently increase time step of the integration of the equations of motion. This method is based on Lie transformation theory. The equations of motion in averaged elements are constructed as the Poisson brackets of the averaged Hamiltonian and corresponding orbital element. The transformation between averaged and osculating elements is given by the change-variable functions, which are obtained in the second approximation of the Hori–Deprit method. We used computer algebra system Piranha for the implementation of the Hori–Deprit method. Piranha is an echeloned Poisson series processor authored by F. Biscani. The properties of the obtained series are discussed. The numerical integration of the equations of motion is performed by Everhart method for the Solar system’s giant planets.

Perturbation solutions to the nonlinear motion of displaced orbits

This paper analytically describes the nonlinear motion of displaced orbits obtained by low-thrust propulsion, and focuses on the first-order analytical derivation of perturbation solutions of osculating Keplerian element (OKE) time histories. A virtual Earth (VE) model is developed to transform non-Keplerian orbits above the Earth into Keplerian orbits around the VE by treating the thrust and coordinate transformation effects as perturbations. The numerical OKEs computed from the Cartesian position and velocity exhibit secular and periodic variations. The short-period perturbations of the orbital elements are derived by the conventional quasi-mean element method. The differential equations of the secular and long-period perturbations are linearized for an analytical solution. A comparison between the analytical OKEs and numerical OKEs of a specific displaced orbit validates the correctness and good accuracy of the analytical derivation. One contribution of this paper is that the analytical OKEs can be applied for a fast calculation of a displaced orbit in the preliminary mission design phase, which has more significance than accurate but complicated time-consuming calculations by complete variational equations. Another contribution is revealing the connections between the amplitude as well as frequency of perturbation solutions and displaced orbits from a physical perspective, from which the topology and geometrical size of a displaced orbit can be easily inferred from the time history of OKEs.

Detection and analysis of short-period geomagnetic perturbations during increased solar activity and magnetic storms

The dynamics of geomagnetic field variations on the eve and during the periods of magnetic storms of 2015 and 2017 has been studied (data of the horizontal component of the Earth’s magnetic field of the terrestrial station network were used). The method developed by the authors based on wavelet transform and adaptive threshold functions was applied. Weak geomagnetic disturbances synchronously appearing at the stations and preceding the onset of strong magnetic storms was extracted. The correlation of the isolated geomagnetic disturbances with the AE-index is shown, both in the occurrence times and in intensities. On the basis of comparison with interplanetary environment data (interplanetary magnetic field data and the solar wind parameters were analyzed), and also based on the results of other author works [1, 2]. we assume that the isolated effects have solar nature. The research is supported by the grant of the Russian Science Foundation No. 14-11-00194.

Short-Period and Long-Period Effects of Weak Gravitational Waves

We compute the long-term orbital variation of a test particle orbiting a central body acted upon by normal incident of plane gravitational wave. We use the tools of celestial mechanics to give the first order solution of canonical equations of long-period and short-period terms of the perturbed Hamiltonian of gravitational waves. We consider normal incident of plane gravitational wave and characteristic size of bound—two body system (earth’s satellite or planet) is much smaller than the wavelength of the wave and the wave’s frequency nw is much smaller than the particle’s orbital np. We construct the Hamiltonian of the gravitational waves in terms of the canonical variables (l,g,h,L,G,H) and we solve the canonical equations numerically using Runge-Kutta fourth order method using language MATHEMATICA V10. Taking Jupiter as practical example we found that there are long period perturbations on ω,Ω and i and not changing with revolution and the short period perturbations on a, e and M changing with revolution during the interval of time (t−t0 ) which is changing from 0→4π.

Dynamical modeling and lifetime analysis of geostationary transfer orbits

The dynamics and lifetime reduction of geostationary transfer orbits (GTOs) are of great importance to space debris mitigation. The orbital dynamics, subjected to a complex interplay of multiple perturbations, are complicated and sensitive to the initial conditions and model parameters. In this paper, a simple but effective non-singular orbital dynamics model in terms of Milankovitch elements is derived. The orbital dynamics, which include the Earth oblateness, luni-solar perturbations, and atmospheric drag, are averaged over the orbital motion of the GTO object, or, as needed, also over the orbital motions of the Moon and Sun, to eliminate the short-period terms. After the averaging process, the effect of the atmospheric drag assumes a simple analytical form. The averaged orbital model is verified through a numerical simulation compared with commercial orbit propagators. GTO lifetime reduction by using the luni-solar perturbations is studied. It is shown that the long-period luni-solar perturbation is induced by the precession of the GTO orbital plane and apsidal line, whereas the short-period perturbation is induced by the periodic luni-solar orbital motions. The long- and short-period perturbations are isolated and studied separately, and their global distribution with respect to the orbital geometry is given. The desired initial orbital geometry with a short orbital lifetime is found and verified by a numerical simulation.

Application of Semi-analytical Satellite Theory orbit propagator to orbit determination for space object catalog maintenance

Catalog maintenance for Space Situational Awareness (SSA) demands an accurate and computationally lean orbit propagation and orbit determination technique to cope with the ever increasing number of observed space objects. As an alternative to established numerical and analytical methods, we investigate the accuracy and computational load of the Draper Semi-analytical Satellite Theory (DSST). The standalone version of the DSST was enhanced with additional perturbation models to improve its recovery of short periodic motion. The accuracy of DSST is, for the first time, compared to a numerical propagator with fidelity force models for a comprehensive grid of low, medium, and high altitude orbits with varying eccentricity and different inclinations. Furthermore, the run-time of both propagators is compared as a function of propagation arc, output step size and gravity field order to assess its performance for a full range of relevant use cases. For use in orbit determination, a robust performance of DSST is demonstrated even in the case of sparse observations, which is most sensitive to mismodeled short periodic perturbations. Overall, DSST is shown to exhibit adequate accuracy at favorable computational speed for the full set of orbits that need to be considered in space surveillance. Along with the inherent benefits of a semi-analytical orbit representation, DSST provides an attractive alternative to the more common numerical orbit propagation techniques.

Oscillatory-rotational processes in the Earth motion about the center of mass: Interpolation and forecast

The celestial-mechanics approach (the spatial version of the problem for the Earth-Moon system in the field of gravity of the Sun) is used to construct a mathematical model of the Earth’s rotational-oscillatory motions. The fundamental aspects of the processes of tidal inhomogeneity in the Earth rotation and the Earth’s pole oscillations are studied. It is shown that the presence of the perturbing component of gravitational-tidal forces, which is orthogonal to the Moon’s orbit plane, also allows one to distinguish short-period perturbations in the Moon’s motion. The obtained model of rotational-oscillatory motions of the nonrigid Earth takes into account both the basic perturbations of large amplitudes and the more complicated small-scale properties of the motion due to the Moon short-period perturbations with combination frequencies.

Analytical and Semi‐Analytical Treatment of the Satellite Motion in a Resisting Medium

The orbital dynamics of an artificial satellite in the Earth′s atmosphere is considered. An analytic first‐order atmospheric drag theory is developed using Lagrange′s planetary equations. The short periodic perturbations due to the geopotential of all orbital elements are evaluated. And to construct a second‐order analytical theory, the equations of motion become very complicated to be integrated analytically; thus we are forced to integrate them numerically using the method of Runge‐Kutta of fourth order. The validity of the theory is checked on the already decayed Indian satellite ROHINI where its data are available.

Determination of an instantaneous Laplace plane for Mercury’s rotation

Mercury is the target of two space missions: MESSENGER, which carried out its first and second flybys of Mercury on January 14, 2008 and October 6, 2008, and the ESA/JAXA space mission BepiColombo, scheduled to arrive at Mercury in 2020. The preparation of these missions requires a good knowledge of the rotation of Mercury. This paper presents studies performed by the MORE/ROMEO Team of BepiColombo mission. In particular, we show here an analytical study of the rotation of Mercury, and the determination of a suitable Laplace plane that can be considered as a reference inertial plane for the duration of the mission. Our main results are a prediction of the short-period perturbations that should be detected in the longitudinal rotational motion of Mercury, and an easy method to define an instantaneous Laplace plane taking into account the very small precession of Mercury’s orbit. This plane is afterward chosen as constant reference plane. An estimation of the (small) error induced by this choice is also computed.

Long term evolution of quasi-circular Trojan orbits

Trojan asteroids undergo very large perturbations because of their resonance with Jupiter. Fortunately the secular evolution of quasi circular orbits remains simple—if we neglect the small short period perturbations. That study is done in the approximation of the three dimensional circular restricted three-body problem, with a small mass ratio μ—that is about 0.001 in the Sun Jupiter case. The Trojan asteroids can be defined as celestial bodies that have a “mean longitude”, M + ω + Ω, always different from that of Jupiter. In the vicinity of any circular Trojan orbit exists a set of “quasi-circular orbits” with the following properties: (A) Orbits of that set remain in that set with an eccentricity that remains of the order of the mass ratio μ. (B) The relative variations of the semi-major axis and the inclination remain of the order of \({\sqrt{\mu}}\) . (C) There exist corresponding “quasi integrals” the main terms of which have long-term relative variations of the order of μ only. For instance the product c(1 – cos i) where c is the modulus of the angular momentum and i the inclination. (D) The large perturbations affect essentially the difference “mean longitude of the Trojan asteroid minus mean longitude of Jupiter”. That difference can have very large perturbations that are characteristics of the “horseshoes orbit”. For small inclinations it is well known that this difference has two stable points near ±60° (Lagange equilibrium points L4 and L5) and an unstable point at 180° (L3). The stable longitude differences are function of the inclination and reach 180° for an inclination of 145°41′. Beyond that inclination only one equilibrium remains: a stable difference at 180°.

MASSES, LUMINOSITIES, AND ORBITAL COPLANARITIES OF THE μ ORIONIS QUADRUPLE-STAR SYSTEM FROM PHASES DIFFERENTIAL ASTROMETRY

μ Orionis was identified by spectroscopic studies as a quadruple-star system. Seventeen high-precision differential astrometry measurements of μ Ori have been collected by the Palomar High-precision Astrometric Search for Exoplanet Systems (PHASES). These show both the motion of the long-period binary orbit and short-period perturbations superimposed on that caused by each of the components in the long-period system being themselves binaries. The new measurements enable the orientations of the long-period binary and short-period subsystems to be determined. Recent theoretical work predicts the distribution of relative inclinations between inner and outer orbits of hierarchical systems to peak near 40 and 140 degrees. The degree of coplanarity of this complex system is determined, and the angle between the planes of the A–B and Aa–Ab orbits is found to be 136.7 ± 8.3 degrees, near the predicted distribution peak at 140 degrees; this result is discussed in the context of the handful of systems with established mutual inclinations. The system distance and masses for each component are obtained from a combined fit of the PHASES astrometry and archival radial velocity observations. The component masses have relative precisions of 5% (component Aa), 15% (Ab), and 1.4% (each of Ba and Bb). The median size of the minor axes of the uncertainty ellipses for the new measurements is 20 micro-arcseconds (μas). Updated orbits for δ Equulei, κ Pegasi, and V819 Herculis are also presented.

Simulation of the effect of Galilean satellites in problems on the dynamics of Jupiter’s near and distant satellites

Various simplified representations of the motion of Galilean satellites are considered which are used in simulations of the near and distant satellites of Jupiter. For the near satellites it has been shown that of all simplified models the highest accuracy is attained with a model in which the motion of Galilean satellites is described by formulas of circular motion, and the accuracy is such that the model can well be used for preprocessing of observed data. The problem of short-period perturbations has been stated which arises in a high-accuracy simulation of the motion of distant satellites in view of the motion of fast circumjovial large satellites. It has been shown that this problem is solved most effectively with the use of Gaussian rings.

Special perturbation theory methods in celestial mechanics. I. Principles for the construction and substantiation of the application

The ideas and principles for the construction of methods in special perturbation theory are discussed, and their application to the solution of problems in classical celestial mechanics is substantiated. The problem of shortperiod perturbations and their effect on the numerical integration is investigated.

Analytical second-order drag theory with application for lifetime determination of artificial satellites in low earth orbits

A second-order atmospheric drag theory based on the usage of TD88 model is constructed. It is developed to the second order in terms of TD88 small parameters K n , j . The short periodic perturbations, of all orbital elements, are evaluated. The secular perturbations of the semi-major axis and of the eccentricity are obtained. The theory is applied to determine the lifetime of the satellites in the low Earth orbits, e.g. ROHINI (1980 62A), and to predict the lifetime of the microsatellite MIMOSA. The secular perturbations of the nodal longitude and of the argument of perigee due to the Earth’s gravity are taken into account up to the second order in Earth’s oblateness.

A second order analytical atmospheric drag theory based on the TD88 thermospheric density model

A second order atmospheric drag theory based on the usage of TD88 model is constructed. It is developed to the second order in terms of TD88 small parameters K n,j . The short periodic perturbations, of all orbital elements, are evaluated. The secular perturbations of the semi-major axis and of the eccentricity are obtained. The theory is applied to determine the lifetime of the satellites ROHINI (1980 62A), and to predict the lifetime of the microsatellite MIMOSA. The secular perturbations of the nodal longitude and of the argument of perigee due to the Earth’s gravity are taken into account up to the second order in Earth’s oblateness.

Semianalytic theory of motion for close-Earth spherical satellites including drag and gravitational perturbations

A semianalytic theory for long-term dynamics of a low Earth orbit artificial satellite, or a space debris particle, is presented. The empirical model TD88 of the neutral atmosphere density distribution, simple enough to allow analytic removal of short period perturbations, is the principal part of the theory. In order to compare our results with observations of passive spherical bodies, we also include the main gravitational effects, notably those of the zonal harmonics J 2 , … , J 9 ( J 2 2 terms are also considered), and for low eccentricities and inclinations we use the description in nonsingular elements. Starting with a single set of initial orbital data and giving a certain “confidential interval” on the resulting lifetimes, we find a reasonably good agreement with observations over the timescale of years, the mean computed lifetimes being a few percent apart from the real ones. The strength of our theory is the computational efficiency, since we need only ≃ 5 s (on a PC equipped with Intel Celeron 1.7 GHz) to propagate the 10-year orbital arc, while maintaining a lot of the physics of the motion under drag (solar flux, geomagnetic activity, local time, geographic latitude). The online calculation as well as the code are available on the internet.