Bounded relative motion under zonal harmonics perturbations

The problem of finding natural bounded relative trajectories between the different units of a distributed space system is of great interest to the astrodynamics community. This is because most popular initialization methods still fail to establish long-term bounded relative motion when gravitational perturbations are involved. Recent numerical searches based on dynamical systems theory and ergodic maps have demonstrated that bounded relative trajectories not only exist but may extend up to hundreds of kilometers, i.e., well beyond the reach of currently available techniques. To remedy this, we introduce a novel approach that relies on neither linearized equations nor mean-to-osculating orbit element mappings. The proposed algorithm applies to rotationally symmetric bodies and is based on a numerical method for computing quasi-periodic invariant tori via stroboscopic maps, including extra constraints to fix the average of the nodal period and RAAN drift between two consecutive equatorial plane crossings of the quasi-periodic solutions. In this way, bounded relative trajectories of arbitrary size can be found with great accuracy as long as these are allowed by the natural dynamics and the physical constraints of the system (e.g., the surface of the gravitational attractor). This holds under any number of zonal harmonics perturbations and for arbitrary time intervals as demonstrated by numerical simulations about an Earth-like planet and the highly oblate primary of the binary asteroid (66391) 1999 KW4.     

Relativistic satellite orbits: central body with higher zonal harmonics

Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order \((\mathrm{GM}/c^2R) \times J_k\) with \(k \ge 2\) for the Earth are very small (of order \( 7 \times 10^{-10}\,\times \,J_k\)) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit \(J_k/c^2\)-terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian \(J_k\)-terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node \(<\!{\dot{\Omega }}\!>\) and argument of the pericenter \(<\!{\dot{\omega }}\!>\). Finally orders of magnitude are given and discussed.     

Analysis of 27 satellite orbits to determine odd zonal harmonics in the geopotential

The odd zonal harmonics in the geopotential are the terms independent of longitude and antisymmetric about the Equator: they define the ‘pear-shape’ effect. The coeffecients J₃, J₅, J₇,…of these harmonics have been evaluated by analysing the variations in eccentricity of 27 orbits covering wide range of inclinations. We use again most of the orbits from our previous (1969) evaluations, but we now have the advantage of 3 accurate orbits at inclinations between 60° and 66°, where the variations in eccentricity become very large, and 3 near-equatorial orbits, at inclinations between 3° and 15°, whereas previously there were none at inclinations lower than 28°. The new data lead to much more accurate and reliable values for the coeffecients. Our recommended set, which terminates at J₁₇, is

$\text{10}{}^9\text{J}{}_3\text{= ?2531 ± 7}\text{10}{}^9\text{J}{}_{11}\text{= 159 ± 16}\text{J}{}_5\text{= ?246 ± 9}\text{J}{}_{13}\text{= ?131 ± 22}\text{J}{}_7\text{= ?326 ± 11}\text{J}{}_{15}\text{= ?26 ±24}\text{J}{}_9\text{= ?94 ± 12}\text{J}{}_{17}\text{= ?258 ± 19}$

. With this new set of values the pear-shape tendency of the Earth amounts to 44.7 m at the poles, instead of the previous 40 m, though the new geoid is within 1 m of the old at latitudes away from the poles.     

Effect of 3rd-degree gravity harmonics and Earth perturbations on lunar artificial satellite orbits

In a previous work we studied the effects of (I) the J ₂ and C ₂₂ terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J ₂ + J ₃’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.     

Three-dimensional subsatellite motion under air drag and oblateness perturbations

The three-dimensional relative motion of a subsatellite with respect to a reference station in an elliptical orbit is studied. A general theory based on the variation of the relative elements, i.e. the instantaneous differences between the orbital parameters of the subsatellite and those of the station, is formulated in order to incorporate arbitrary perturbing forces acting on both satellites. The loss of precision inherent in the subtraction of almost identical quantities is avoided by the consistent use of difference variables. In the absence of perturbations exact analytical representations can be obtained for the relative state parameters. The influences of air drag and Earth's oblateness on the relative motion trajectories are investigated and illustrated graphically for a number of cases.     

On the perturbation of the anomalistic period of a highly eccentric orbit satellite due to the zonal harmonics

In this note a simple formula is given for the perturbation of the anomalistic period of a highly eccentric orbit due to the zonal harmonics. This perturbation depends essentially only on the semi-major axisa, the eccentricitye (or pericentre radius r =a(1-e)) and the latitude of the pericentre.     

Approximate analysis for relative motion of satellite formation flying in elliptical orbits

This paper studies the relative motion of satellite formation flying in arbitrary elliptical orbits with no perturbation. The trajectories of the leader and follower satellites are projected onto the celestial sphere. These two projections and celestial equator intersect each other to form a spherical triangle, in which the vertex angles and arc-distances are used to describe the relative motion equations. This method is entitled the reference orbital element approach. Here the dimensionless distance is defined as the ratio of the maximal distance between the leader and follower satellites to the semi-major axis of the leader satellite. In close formations, this dimensionless distance, as well as some vertex angles and arc-distances of this spherical triangle, and the orbital element differences are small quantities. A series of order-of-magnitude analyses about these quantities are conducted. Consequently, the relative motion equations are approximated by expansions truncated to the second order, i.e. square of the dimensionless distance. In order to study the problem of periodicity of relative motion, the semi-major axis of the follower is expanded as Taylor series around that of the leader, by regarding relative position and velocity as small quantities. Using this expansion, it is proved that the periodicity condition derived from Lawden’s equations is equivalent to the condition that the Taylor series of order one is zero. The first-order relative motion equations, simplified from the second-order ones, possess the same forms as the periodic solutions of Lawden’s equations. It is presented that the latter are further first-order approximations to the former; and moreover, compared with the latter more suitable to research spacecraft rendezvous and docking, the former are more suitable to research relative orbit configurations. The first-order relative motion equations are expanded as trigonometric series with eccentric anomaly as the angle variable. Except the terms of order one, the trigonometric series’ amplitudes are geometric series, and corresponding phases are constant both in the radial and in-track directions. When the trajectory of the in-plane relative motion is similar to an ellipse, a method to seek this ellipse is presented. The advantage of this method is shown by an example.     

Tesseral harmonic perturbations for high order and degree harmonics

A new formula has been derived for geopotential expressed in terms of orbital elements. The summation sequence was changed so that the terms of the same frequencies would be grouped and the generalized lumped coefficients were derived. The proposed formula has the same form for both odd and evenl-m.Applying Hori's perturbation method, new formulae were derived for tesseral harmonic perturbations in nonsingular orbital elements:l+g, h, e cosg,e sing, L, andH. We show the possibility of effective application of the derived formulae to the calculation of orbits of very low satellites taking into account the coefficients of tesseral harmonics of the Earth's gravitational field up to high orders and degrees. As an example the perturbations up to the order and degree of 90 for the orbit of GRM satellites were calculated. The calculations were carried out on an IBM AT personal computer.     

Luni-solar perturbations of an Earth satellite

Luni-solar perturbations of the orbit of an artificial Earth satellite are given by modifying the analytical theory of an artificial lunar satellite derived by the author in recent papers. Expressions for the first-order changes, both secular and periodic, in the elements of the geocentric Keplerian orbit of the earth satellite are given, the moon's geocentric orbit, including solar perturbations in it, being found by using Brown's lunar theory.The effects of Sun and Moon on the satellite orbit are described to a high order of accuracy so that the theory may be used for distant earth satellites.     

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.     

Application of vector spherical harmonics for kinematic analysis of stars from zonal catalogues

We solve the problem on a kinematic analysis of the three-dimensional velocity field of stars from zonal catalogues, i.e., catalogues in which the stars are presented at all right ascensions in some declination zones. We have constructed a system of vector spherical harmonics with the properties of completeness and orthogonality for a chosen declination zone. We suggest a method that allows the Ogorodnikov-Milne model parameters in the Galactic coordinate system to be estimated by analyzing the proper motions and radial velocities of stars in the equatorial coordinate system. The vector spherical harmonics are shown to have the following advantages over the standard approach based on a direct leastsquares estimation of the parameters for a specific model. First, in contrast to the standard approach, the new method can reveal all systematic components of the velocity field irrespective of a particular model. Second, it allows one to get rid of the correlation between the sought-for parameters, which presents a serious problem for the conventional method in the case of zonal catalogues. Third, the method of vector spherical harmonics allows the kinematic parameters to be estimated at least by two techniques. Comparison of these two solutions makes it possible to test the standard kinematic model for compatibility with the observational data. The developed method has been tested on the basis of numerical experiments and applied for a kinematic analysis of the proper motions of Tycho-2 stars in the southern hemisphere for which the parallaxes can be estimated using data from the Tycho-2 Spectral Type Catalogue.     

Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite

Gravity-gradient perturbations of the attitude motion of a tumbling tri-axial satellite are investigated. The satellite center of mass is considered to be in an elliptical orbit about a spherical planet and to be tumbling at a frequency much greater than orbital rate. In determining the unperturbed (free) motion of the satellite, a canonical form for the solution of the torque-free motion of a rigid body is obtained. By casting the gravity-gradient perturbing torque in terms of a perturbing Hamiltonian, the long-term changes in the rotational motion are derived. In particular, far from resonance, there are no long-period changes in the magnitude of the rotational angular momentum and rotational energy, and the rotational angular momentum vector precesses abound the orbital angular momentum vector.At resonance, a low-order commensurability exists between the polhode frequency and tumbling frequency. Near resonance, there may be small long-period fluctuations in the rotational energy and angular momentum magnitude. Moreover, the precession of the rotational angular momentum vector about the orbital angular momentum vector now contains substantial long-period contributions superimposed on the non-resonant precession rate. By averaging certain long-period elliptic functions, the mean value near resonance for the precession of the rotational angular momentum vector is obtained in terms of initial conditions.     

The motion of a lunar satellite

Presented in this theory is a semianalytical solution for the problem of the motion of a satellite in orbit around the moon. The principal perturbations on such a body are due to the nonspherical gravity field of the moon, the attraction of the earth, and, to a lesser degree, the attraction of the sun. The major part of the problem is solved by means of the celebrated von Zeipel Method, first successfully applied to the motion of an artificial earth satellite by Brouwer in 1959. After eliminating from the Hamiltonian all terms with the period of the satellite and those with the period of the moon, it is suggested to solve the remaining problem with the aid of numerical integration of the modified equations of motion.This theory was written in 1964 and presented as a dissertation to Yale University in 1965. Since then a great deal has been learned about the gravity field of the moon. It seems that quite a number of recently determined gravity coefficients would qualify as small quantities of order two. Hence, according to the truncation criteria employed, they should be considered in the present theory. However, the author has not endeavored to update the work accordingly. The final results, therefore, are incomplete in the lunar gravitational perturbations. Nevertheless, the theory does give the largest such variations and it does present the methods by which perturbations may be derived for any gravity terms not actually developed.     

Geopotential harmonics of order 15 and even degree,from changes in orbital eccentricity at resonance

When a satellite orbit decaying slowly under the action of air drag experiences 15th-order resonance with the Earth's gravitational field, so that the ground track repeats after 15 rev, the orbital eccentricity may suffer appreciable changes due to perturbations from the gravitational harmonics of order 15 and even degree (16, 18, 20…). In this paper the changes in eccentricity at resonance for six satellites in near-circular orbits at inclinations between 56 and 90° have been analysed to derive 11 pairs of equations linking the harmonic coefficients of order 15 and (even) degree l, $\text{C}{}_{\mathrm{l,15}}\text{and}\text{S}{}_{\mathrm{l,15}}$ in the usual notation. These equations (together with eight constraint equations) are solved to give:

     

18.

Tidal effects and the motion of a satellite     

Kovalevsky  J. 《Celestial Mechanics and Dynamical Astronomy》1984,34(1-4):243-244

The effect of resonant planetary perturbations on the evolution of the orbit of a satellite driven by tidal forces is studied in this paper. The basic equations that govern it are similar to the equations found in orbit-orbit and in spin-orbit couplings. The general form of these equations is: A general treatment of such equations, proposed earlier (J. Kovalevsky, in Dynamical Trapping and Evolution of the Solar system, IAU Colloquium n^o74, V. V. Markellos and Y. Kozai, eds., 1983) is sketched.In particular, the effects of the large long periodic variations of the excentricity e' of the planet are analysed on an example taken from the lunar theory and the Earth's general theory due to Bretagnon.The argument of the well known planetary term =18 V-16T due to the tidal friction and quasi-periodic variations due to the presence of e' in the expression of the mean motion of the Moon. Their joint effect, has been to produce in the past resonant situations for this argument that repeated more than 100 times. Every such situation can be treated by equation (1).Numerical integration, using conditions that might have occurred while or similar other arguments were quasi resonant, have produced the following results: (a) In some cases, the argument becomes temporarily resonant. Between the capture to and the escape from the resonance, the semi-major axis undergoes oscillations, but the tidal secular evolution is stopped. (b) In other cases, the argument is not trapped into a resonant conditions, but the semi-major axis undergoes a quick change while d/dt is close to zero.A number of arguments that have been quasi resonant in the past history of the Earth-Moon system has been identified from the Chapront and Chapront-Touzé Lunar Theory. It appears that the phenomena described are frequent features in the evolution of the Lunar orbit.     

19.

On the unmodeled perturbations in the motion of uranus     

Adrian Brunini 《Celestial Mechanics and Dynamical Astronomy》1992,53(2):129-143

In this paper we apply a numerical method to determine unmodeled perturbations in an attempt to explain the observed discrepancies in the motion of Uranus. We find that the estimated perturbation shows some significant periods that could be attributed to insufficient knowledge of the perturbations from some of the known planets. On the assumption that the gravitational attraction of an unknown planet is the origin of the deviations, the best planar solution of the inverse problem is a planet of 0.6 Earth masses, with true longitude of 133° (1990.5), semi major axis a = 44 AU and eccentricity e = 0.007.     

20.

Direct perturbations of the planets on the Moon's motion     

D. Standaert 《Celestial Mechanics and Dynamical Astronomy》1980,22(4):357-369

The aim of the present paper is to present the theoretical background of a method to compute the planetary perturbations on the Moon's motion. We formulate an algorithm based upon the Lie transform method and well-suited to the particular problem at hand.This algorithm is being implemented using Henrard's Semi-Analytical Lunar Ephemeris (SALE) as solution of the Main Problem and Bretagnon's planetary theory. The accuracy of the solution is intended to be about 0".001 for terms of period up to 2000 years.To illustrate the interest of our approach, we comment on some preliminary results obtained about the direct perturbations due to Venus on the Moon's longitude. The final results will be the subject of another paper.     

$\text{l}$	109$\text{C}{}_{\mathrm{l,15}}$	109$\text{S}{}_{\mathrm{l,15}}$
16	?13.7 ± 1.3	?18.5 ± 2.7
18	?42.3 ± 1.8	?34.7 ± 3.4
20	10.5 ± 3.1	29.8 ± 5.2
22	?8.6 ± 3.8	?20.2 ± 7.4

Tidal effects and the motion of a satellite

The effect of resonant planetary perturbations on the evolution of the orbit of a satellite driven by tidal forces is studied in this paper. The basic equations that govern it are similar to the equations found in orbit-orbit and in spin-orbit couplings. The general form of these equations is: A general treatment of such equations, proposed earlier (J. Kovalevsky, in Dynamical Trapping and Evolution of the Solar system, IAU Colloquium n^o74, V. V. Markellos and Y. Kozai, eds., 1983) is sketched.In particular, the effects of the large long periodic variations of the excentricity e' of the planet are analysed on an example taken from the lunar theory and the Earth's general theory due to Bretagnon.The argument of the well known planetary term =18 V-16T due to the tidal friction and quasi-periodic variations due to the presence of e' in the expression of the mean motion of the Moon. Their joint effect, has been to produce in the past resonant situations for this argument that repeated more than 100 times. Every such situation can be treated by equation (1).Numerical integration, using conditions that might have occurred while or similar other arguments were quasi resonant, have produced the following results: (a) In some cases, the argument becomes temporarily resonant. Between the capture to and the escape from the resonance, the semi-major axis undergoes oscillations, but the tidal secular evolution is stopped. (b) In other cases, the argument is not trapped into a resonant conditions, but the semi-major axis undergoes a quick change while d/dt is close to zero.A number of arguments that have been quasi resonant in the past history of the Earth-Moon system has been identified from the Chapront and Chapront-Touzé Lunar Theory. It appears that the phenomena described are frequent features in the evolution of the Lunar orbit.     

On the unmodeled perturbations in the motion of uranus

In this paper we apply a numerical method to determine unmodeled perturbations in an attempt to explain the observed discrepancies in the motion of Uranus. We find that the estimated perturbation shows some significant periods that could be attributed to insufficient knowledge of the perturbations from some of the known planets. On the assumption that the gravitational attraction of an unknown planet is the origin of the deviations, the best planar solution of the inverse problem is a planet of 0.6 Earth masses, with true longitude of 133° (1990.5), semi major axis a = 44 AU and eccentricity e = 0.007.     

Direct perturbations of the planets on the Moon's motion

The aim of the present paper is to present the theoretical background of a method to compute the planetary perturbations on the Moon's motion. We formulate an algorithm based upon the Lie transform method and well-suited to the particular problem at hand.This algorithm is being implemented using Henrard's Semi-Analytical Lunar Ephemeris (SALE) as solution of the Main Problem and Bretagnon's planetary theory. The accuracy of the solution is intended to be about 0".001 for terms of period up to 2000 years.To illustrate the interest of our approach, we comment on some preliminary results obtained about the direct perturbations due to Venus on the Moon's longitude. The final results will be the subject of another paper.