Perturbation of a satellite orbiter by the oblateness of the primary planet

The perturbation of an orbiter around a large satellite of a giant planet (Jupiter, Saturn, Uranus or Neptune) produced by the oblateness of the planet is investigated. The perturbing force of theJ ₂-term (general case) and theJ ₄-term (special case of small eccentricity and inclination) is expanded in an appropriate form and the main term and the parallactic term are given explicitly. The variations of the orbital elements are derived using the stroboscopic method. An example shows that the perturbation of the orbit cannot be neglected.     

Canonical Modelling of Relative Spacecraft Motion Via Epicyclic Orbital Elements

This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an example, closed-form solutions for J₂-invariant orbits are obtained.     

Relativistic effects and solar oblateness from radar observations of planets and spacecraft

We used more than 250 000 high-precision American and Russian radar observations of the inner planets and spacecraft obtained in the period 1961–2003 to test the relativistic parameters and to estimate the solar oblateness. Our analysis of the observations was based on the EPM ephemerides of the Institute of Applied Astronomy, Russian Academy of Sciences, constructed by the simultaneous numerical integration of the equations of motion for the nine major planets, the Sun, and the Moon in the post-Newtonian approximation. The gravitational noise introduced by asteroids into the orbits of the inner planets was reduced significantly by including 301 large asteroids and the perturbations from the massive ring of small asteroids in the simultaneous integration of the equations of motion. Since the post-Newtonian parameters and the solar oblateness produce various secular and periodic effects in the orbital elements of all planets, these were estimated from the simultaneous solution: the post-Newtonian parameters are β = 1.0000 ± 0.0001 and γ = 0.9999 ± 0.0002, the gravitational quadrupole moment of the Sun is J₂ = (1.9 ± 0.3) × 10^?7, and the variation of the gravitational constant is ?/G = (?2 ± 5) × 10^?14 yr^?1. The results obtained show a remarkable correspondence of the planetary motions and the propagation of light to General Relativity and narrow significantly the range of possible values for alternative theories of gravitation.     

On the choice of reference orbit,canonical variables,and perturbation method in satellite theory

The author's second-order artificial satellite theory (Aksnes, 1970) is reviewed and compared with that of Kozai (1962). These theories differ in that the former makes use of: (1) an intermediate orbit, being a rotating ellipse instead of a fixed ellipse, (2) Hill variables instead of Delaunay variables, and (3) Hori's perturbation method in Lie series rather than Von Zeipel's method in Taylor series.It is demonstrated that because of these differences, the former theory enjoys a greater simplicity and compactness, it is non-singular at zero eccentricity, and the process of deriving the perturbations is considerably simplified (Aksnes, 1972). For example, the number of second-order short-period terms due to the planet's oblateness (J ₂) is reduced by a factor of about three (Hori, 1970). The intermediate orbit and Hori's perturbation method contribute about equally to this reduction.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Spin-orbit coupling: A unified theory of orbital and rotational resonances

The dynamics of the spin-orbit interaction of a sphereM ₈ and a rotating asymmetrical rigid bodyM _a are examined. No restrictions are imposed on the masses, on the orientation of the rotation axis to the orbit plane, or on the orbit eccentricity. The zonal potential harmonics ofM _a induce a precession of the spin axis as well as a precession of the orbit plane, the net effect being a uniform precession of the node on an invariant plane normal to the constant total angular momentum of the system. In general, the effect of the tesseral harmonics is to induce short-period perturbations of small amplitude in both the orbital and spin motions. Resonances are shown to exist whenever the orbital and rotational periods are commensurable. In any resonant state a single coordinate is found to represent both orbital and spin perturbations; and the system may be described as trapped in a localized potential well. The resultant spin and orbit librations are in phase with a common period. The relative amplitudes of the spin/orbit modes are determined by the characteristic parameter =M _a M _s a ² /3(M _a +M _s )C, wherea is the semimajor axis of the orbit, andC is the moment of inertia ofM _a about the rotation axis. When ga1, the solutions reduce to those for pureorbital resonance, in whichM _s librates in an appropriate reference frame while the rotation rate of the asymmetrical body remains constant. In the opposite extreme of 1, the solutions are appropriate to purerotational resonance, in which the orbital motion is unperturbed but the spin ofM _a librates. In each of these special cases the equations developed herein on the basis of a single theory are in agreement with those previously determined from separate theories of spin and orbital resonances.     

Nodal and periastron precession of inclined orbits in the field of a rapidly rotating neutron star

We derive a formula for the nodal precession frequency and the Keplerian period of a particle at an arbitrary orbital inclination (with a minimum latitudinal angle reached at the orbit) in the post-Newtonian approximation in the external field of an oblate rotating neutron star (NS). We also derive formulas for the nodal precession and periastron rotation frequencies of slightly inclined low-eccentricity orbits in the field of a rapidly rotating NS in the form of asymptotic expansions whose first terms are given by the Okazaki-Kato formulas. The NS gravitational field is described by the exact solution of the Einstein equation that includes the NS quadrupole moment induced by rapid rotation. Convenient asymptotic formulas are given for the metric coefficients of the corresponding space-time in the form of Kerr metric perturbations in Boyer-Lindquist coordinates.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.     

Sun-synchronous orbits near critical inclination

The long period dynamics of Sun-synchronous orbits near the critical inclination 116.6° are investigated. It is known that, at the critical inclination, the average perigee location is unchanged by Earth oblateness. For certain values of semimajor axis and eccentricity, orbit plane precession caused by Earth oblateness is synchronous with the mean orbital motion of the apparent Sun (a Sun-synchronism). Sun-synchronous orbits have been used extensively in meteorological and remote sensing satellite missions. Gravitational perturbations arising from an aspherical Earth, the Moon, and the Sun cause long period fluctuations in the mean argument of perigee, eccentricity, inclination, and ascending node. Double resonance occurs because slow oscillations in the perigee and Sun-referenced ascending node are coupled through the solar gravity gradient. It is shown that the total number and infinitesimal stability of equilibrium solutions can change abruptly over the Sun-synchronous range of semimajor axis values (1.54 to 1.70 Earth radii). The effect of direct solar radiation pressure upon certain stable equilibria is investigated.     

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.     

A noncanonical analytic solution to theJ 2 perturbed two-body problem

The motion of a satellite subject to an inverse-square gravitational force of attraction and a perturbation due to the Earth's oblateness as theJ ₂ term is analyzed, and a uniform, analytic solution correct to first-order inJ ₂, is obtained using a noncanonical approach. The basis for the solution is the transformation and uncoupling of the differential equations for the model. The resulting solution is expressed in terms of elementary functions of the independent variable (the ‘true anomaly’), and is of a compact and simple form. Numerical results are comparable to existing solutions.     

Analytical solution for the three-dimensional motion of a circumbinary planet around a binary star

By using the method of separating rapid and slow subsystem, we obtain an analytical solution for a stable three-dimensional motion of a circumbinary planet around a binary star. We show that the motion of the planet is more complicated than it was obtained for this situation analytically by Farago and Laskar (2010). Namely, in addition to the precession of the orbital plane of the planet around the angular momentum of the binary (found by Farago and Laskar (2010)), there is simultaneously the precession of the orbital plane of the planet within the orbital plane. We show that the frequency of this additional precession is different from the frequency of the precession of the orbital plane around the angular momentum of the binary. We demonstrate that this problem is mathematically equivalent both to the problem of the motion of a satellite around an oblate planet and to the problem of a hydrogen Rydberg atom in the field of a high-frequency linearly-polarized laser radiation, thus discovering yet another connection between astrophysics and atomic physics. We point out that all of the above physical systems have a higher than geometrical symmetry, which is a counterintuitive result. In particular, it is manifested by the fact that, while the elliptical orbit of the circumbinary planet (around a binary star) or of the satellite (around an oblate planet) or of the Rydberg electron (in the laser field) undergoes simultaneously two types of the precession, the shape of the orbit does not change. The fact that a system, consisting of a circumbinary planet around a binary star, possesses the hidden symmetry should be of a general physical interest. Our analytical results could be used for benchmarking future simulations.     

Analytical approach using KS elements to short-term orbit predictions including J 2

Analytical solutions using KS elements are derived. The perturbation considered is the Earth's zonal harmonic J ₂. The series expansions include terms of fourth power in the eccentricity. Only two of the nine KS element equations are integrated analytically due to the reasons of symmetry. The analytical solution is suitable for short-term orbit computations. Numerical studies show that reasonably good estimates of the orbital elements can be obtained in one step of 10 to 30 degrees of eccentric anomaly for near-Earth orbits of moderate eccentricity. For application purposes, the analytical solution can be effectively used for onboard computation in the navigation and guidance packages, where the modelling of J ₂ effect becomes necessary.     

Ocean tidal perturbations on the orbit of the satellite “STARLETTE”

A continuous 300-day, world-wide laser-ranging data of the satellite STARLETTE between October 1976 and July 1977 was used to analyse the perturbation of the orbit by ocean tide. The long-period (longer than 20d) perturbations in the orbital inclination by the four main tide components K₁, P₁, K₂, S₂ were clearly identifiable in the maximum entropy power spectrum and periodogram. By comparing orbits including and excluding the tidal terms, the effect of the tide on the position of the satellite over 5 days was found to be of the order of one meter.     

Orbital history of the Martian satellites with inferences on their origin

Recent Viking results indicate the Martian satellites are composed of carbonaceous chondritic material, suggesting that Phobos and Deimos were once asteroids captured by Mars. On the other hand, the low eccentricities and inclinations of their orbits on the equator of Mars argue against that hypothesis. This paper presents detailed calculations of the tidal evolution of Phobos and Deimos, considering dissipation in both Mars and its satellites simultaneously and using a new method applicable for any value of the eccentricity. In particular, including precession of the satellites' orbits indicates that they have always remained close to their Laplacian plane, so that the orbital planes of Phobos and Deimos switched from near the Martian orbital plane to the Martian equator once the perturbations due to the planetary oblateness dominated the solar perturbations, as they do presently. The results show that Deimos has been little affected by tides, but several billion (10⁹) years ago, Phobos was in a highly eccentric orbit lying near the common plane of the solar system. This outcome is obtained for very reasonable values of dissipation inside Mars and inside Phobos. Implications for the origin of the Martian satellites are discussed.     

Third-order secular Solution of the variational equations of motion of a satellite in orbit around a non-spherical planet

We constructed an analytical theory of satellite motion up to the third order relative to the oblateness parameter of the Earth (J ₂). Equations of secular variations was developed for the first three orbital elements (a, e, i) of an artificial satellite. The secular variations are solved in a closed form.     

Yarkovsky detection opportunities: II. Binary systems

We consider the possibility of detecting the Yarkovsky orbital perturbation acting on binary systems among the near-Earth asteroids. This task is significantly more difficult than for solitary asteroids because the Yarkovsky force affects both the heliocentric orbit of the system's center of mass and the relative orbit of the two components. Nevertheless, we argue these are sufficiently well decoupled so that the major Yarkovsky perturbation is in the simpler heliocentric motion and is observable with the current means of radar astrometry. Over the long term, the Yarkovsky perturbation in the relative motion of the two components is also detectable for the best observed systems. However, here we consider a simplified version of the problem by ignoring mutual non-spherical gravitational perturbations between the two asteroids. With the orbital plane constant in space and the components' rotation poles fixed (and assumed perpendicular to the orbital plane), we do not examine the coupling between Yarkovsky and gravitational effects. While radar observations remain an essential element of Yarkovsky detections, lightcurve observations, with their ability to track occultation and eclipse phenomena, are also very important in the case of binaries. The nearest possible future detection of the Yarkovsky effect for a binary system occurs for (66063) 1998 RO₁ in September 2006. Farther out, even more statistically significant detections are possible for several other systems including 2000 DP₁₀₇, (66391) 1999 KW₄ and 1996 FG₃.     

The variations in eccentricity and apse precession rate of a narrow ring perturbed by a close satellite

We derive a Hamiltonian which describes the first-order perturbations of orbital eccentricity and apse precession rate of a narrow ring due to a close satellite whose orbit is also eccentric. Our treatment covers cases in which the satellite crosses the ring. The level curves of the Hamiltonian are displayed for several values of the parameters. We apply our results to the interaction of Saturn's F ring with its inner shepherd satellite.     

Qualitative features of the evolution of some polar satellite orbits

Two special cases of the problem of the secular perturbations in the orbital elements of a satellite with a negligible mass produced by the joint influence of the oblateness of the central planet and the attraction by its most massive (or main) satellites and the Sun are considered. These cases are among the integrable ones in the general nonintegrable evolution problem. The first case is realized when the plane of the satellite orbit and the rotation axis of the planet lie in its orbital plane. The second case is realized when the plane of the satellite orbit is orthogonal to the line of intersection between the equatorial and orbital planes of the planet. The corresponding particular solutions correspond to those polar satellite orbits for which the main qualitative features of the evolution of the eccentricity and pericenter argument are described here. Families of integral curves have been constructed in the phase plane of these elements for the satellite systems of Jupiter, Saturn, and Uranus.