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.     

Parametric excitation induced by solar pressure torque on the roll-yaw attitude motion of a gravity-gradient stabilized spacecraft

The parametric excitation of a gravity gradient stabilized spacecraft induced by the periodic solar pressure torque is discussed. The solar pressure torque in the linearized equations of motion appears as linear terms with periodic coefficients. The attitude stability is analyzed numerically through the calculation of the Floquet multiplier. The perturbation method is also applied to identify the instability condition analytically. It is made clear that the periodic solar pressure torque can destabilize the coupled roll and yaw attitude motion of the spacecraft. It is also shown that the conditions of parametric resonance are included in the gravity gradient stability condition. Nonlinear simulations are also carried out to verify the effect of the parametric resonance. The numerical simulation using actual parameters shows that the spacecraft inevitably experiences a large amplitude attitude motion due to the periodic solar pressure torque even if the gravity gradient stability condition is satisfied.     

Effects of gravity-gradient torque on the rotational motion of A triaxial satellite in a precessing elliptic orbit

A method of general perturbations, based on the use of Lie series to generate approximate canonical transformations, is applied to study the effects of gravity-gradient torque on the rotational motion of a triaxial, rigid satellite. The center of mass of the satellite is constrained to move in an elliptic orbit about an attracting point mass. The orbit, which has a constant inclination, is free to precess and spin. The method of general perturbations is used to obtain the Hamiltonian for the nonresonant secular and long-period rotational motion of the satellite to second order inn/0, wheren is the orbital mean motion of the center of mass and0 is a reference value of the magnitude of the satellite's rotational angular velocity. The differential equations derivable from the transformed Hamiltonian are integrable and the solution for the long-term motion may be expressed in terms of Jacobian elliptic functions and elliptic integrals. Geometrical aspects of the long-term rotational motion are discussed and a comparison of theoretical results with observations is made.     

Perturbations of a geo-centric synchronous satellite with resonance

We have investigated the resonances due to the perturbations of a geo-centric synchronous satellite under the gravitational forces of the Sun, the Moon and the Earth including it’s equatorial ellipticity. The resonances at the points resulting from (i) the commensurability between \(\dot{\theta}_{0}\) (steady-state orbital angular rate of the satellite) and \(\dot{\theta}_{m}\) (angular velocity of the moon around the earth) and (ii) the commensurability between \(\dot{\theta}_{0}\) and \(\dot{\psi}_{0}\) (steady-state regression rate of the synchronous satellite) are analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure as given in Brown and Shook (Planetary Theory, Cambridge University Press, Cambridge, 1933). We have observed that as θ _m (0^°≤θ _m ≤45^°) and ψ (0^°≤ψ≤135^°) increase, the amplitude decreases and the time period also decreases. We have also shown the effect of ψ on amplitude and time period for 0^°≤Γ≤45^°, where Γ is the angle measured from the minor axis of the earth’s equatorial ellipse to the projection of the satellite on the plane of the equator.     

The motion of a satellite in resonance with the second-degree sectorial harmonic

The solution to the motion of a satellite in an eccentric orbit and in resonance with the second-degree sectorial harmonic of the potential field is developed. The method of solution used parallels the well known von Zeipel method of general perturbations. The solution consists of expressions for the variations of the Delaunay variables. These expressions are composed of the perturbations developed by Brouwer in 1959 for the motion of an artificial satellite plus first-order perturbations due to the second-degree sectorial harmonic (in terms of the Legendre normal elliptic integrals of the first and second kind).This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

On the motion of a satellite in resonance with its rotating planet

The influence of resonance perturbations due to the gravitational field of an oblate planet on its satellite whose motion is commensurable with rotation of the planet has been investigated. It has been shown that in special case of the critical inclination or circular orbit the Lagrange equations can be integrated for all resonance terms simultaneously. The method is applied to the investigation of the motion of the 12-hour communication and navigation satellites of the Molniya and Navstar type. The computations has been performed by the use of four models of the geopotential.     

Theory of satellite orbit-orbit resonance

On the basis of the strong mathematical and physical parallels between orbit-orbit and spin-orbit resonances, the dynamics of mutual orbit perturbations between two satellites about a massive planet are examined, exploiting an approach previously adopted in the study of spin-orbit coupling. The satellites are assumed to have arbitrary mass ratio and to move in non-intersecting orbits of arbitrary size and eccentricity. Resonances are found to exist when the mean orbital periods are commensurable with respect to some rotating axis, which condition also involves the apsidal and nodal motions of both satellites. In any resonant state the satellites are effectively trapped in separate potential wells, and a single variable is found to describe the simultaneous librations of both satellites. The librations in longitude are 180° out-of-phase, with fixed amplitude ratio that depends only on their relative masses and semimajor axes. At the same time the stroboscopic longitude of conjunction also librates about the commensurate axis with the same period. The theory is applicable to Saturn's resonant pairs Titan-Hyperion and Mimas-Tethys, and in these cases our calculated libration periods are in reasonably good agreement with the observed periods.This research supported under a grant from the California Institute of Technology President's Fund and NASA Contract NAS 7-100.     

Analysis of solar radiation pressure induced coupled liberations of a gravity stabilized axisymmetric satellite

A planar, fixed-orbit model of the rotation of the planet Mercury is analyzed. The model includes only the solar torques on the planet's permanent asymmetry and its solar tidal bulge. For this model, it is shown that the zero of the averaged tidal torque corresponds to an asymptotically stable periodic solution of the second kind which, for two tidal torque representations, is close to the asymptotically stable equilibrium point corresponding to an exact 32 spin-orbit resonance. A conjecture that the current rotation state of Mercury is due to transfer from capture by the zero of the averaged tidal torque to 32 resonance capture with changes in the eccentricity of the planet's orbit is discussed briefly.     

Tesseral resonance effects on satellite orbits

Resonance effects on satellite orbits due to tesseral harmonics in the potential field have been studied by many authors. Most of these studies have been restricted to nearly circular 24-hour orbits and to the deep resonance regime, where there is exact commensurability between earth rotation and orbit period. Resonance effects have also been noted, however, on eccentric synchronous and subsynchronous orbits and on orbits with far from commensurate periods. These have received much less attention; the object of this paper is to study the whole spectrum of orbits with respect to resonance effects.     

Irregular satellite capture during planetary resonance passage

The passage of Jupiter and Saturn through mutual 1:2 mean-motion resonance has recently been put forward as explanation for their relatively high eccentricities [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461] and the origin of Jupiter's Trojans [Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R., 2005. Nature 435, 462-465]. Additional constraints on this event based on other small-body populations would be highly desirable. Since some outer satellite orbits are known to be strongly affected by the near-resonance of Jupiter and Saturn (“the Great Inequality”; ?uk, M., Burns, J.A., 2004b. Astron. J. 128, 2518-2541), the irregular satellites are natural candidates for such a connection. In order to explore this scenario, we have integrated 9200 test particles around both Jupiter and Saturn while they went through a resonance-crossing event similar to that described by Tsiganis et al. [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461]. The test particles were positioned on a grid in semimajor axes and inclinations, while their initial pericenters were put at just 0.01 AU from their parent planets. The goal of the experiment was to find out if short-lived bodies, spiraling into the planet due to gas drag (or alternatively on orbits crossing those of the regular satellites), could have their pericenters raised by the resonant perturbations. We found that about 3% of the particles had their pericenters raised above 0.03 AU (i.e. beyond Iapetus) at Saturn, but the same happened for only 0.1% of the particles at Jupiter. The distribution of surviving particles at Saturn has strong similarities to that of the known irregular satellites. If saturnian irregular satellites had their origin during the 1:2 resonance crossing, they present an excellent probe into the early Solar System's evolution. We also explore the applicability of this mechanism for Uranus, and find that only some of the uranian irregular satellites have orbits consistent with resonant pericenter lifting. In particular, the more distant and eccentric satellites like Sycorax could be stabilized by this process, while closer-in moons with lower eccentricity orbits like Caliban probably did not evolve by this process alone.     

Influence of a second satellite on the rotational dynamics of an oblate moon

The gravitational influence of a second satellite on the rotation of an oblate moon is numerically examined. A simplified model, assuming the axis of rotation perpendicular to the (Keplerian) orbit plane, is derived. The differences between the two models, i.e. in the absence and presence of the second satellite, are investigated via bifurcation diagrams and by evolving compact sets of initial conditions in the phase space. It turns out that the presence of another satellite causes some trajectories, that were regular in its absence, to become chaotic. Moreover, the highly structured picture revealed by the bifurcation diagrams in dependence on the eccentricity of the oblate body’s orbit is destroyed when the gravitational influence is included, and the periodicities and critical curves are destroyed as well. For demonstrative purposes, focus is laid on parameters of the Saturn–Titan–Hyperion system, and on oblate satellites on low-eccentric orbits, i.e. \(e\approx 0.005\).     

Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass

We investigate the stability of the periodic motion of a satellite, a rigid body, relative to the center of mass in a central Newtonian gravitational field in an elliptical orbit. The orbital eccentricity is assumed to be low. In a circular orbit, this periodic motion transforms into the well-known motion called hyperboloidal precession (the symmetry axis of the satellite occupies a fixed position in the plane perpendicular to the radius vector of the center of mass relative to the attractive center and describes a hyperboloidal surface in absolute space, with the satellite rotating around the symmetry axis at a constant angular velocity). We consider the case where the parameters of the problem are close to their values at which a multiple parametric resonance takes place (the frequencies of the small oscillations of the satellite’s symmetry axis are related by several second-order resonance relations). We have found the instability and stability regions in the first (linear) approximation at low eccentricities.     

The problem of critical inclination combined with a resonance in mean motion in artificial satellite theory

In this paper the two-degree of freedom problem of a geosynchronous artificial satellite orbiting near the critical inclination is studied. First a local approach of this problem is considered. A semi-numerical method, well suited to describe the perturbations of a non-trivial separable system, is then applied such that surfaces of section illustrating the global secular dynamics are obtained. The results are confirmed by numerical integrations of the full Hamiltonian.Research Assistant for the Belgian National Fund for Scientific Research     

An analytical treatment of resonance effects on satellite orbits

A first order analytical approximation of the tesseral harmonic resonance perturbations of the Keplerian elements is presented, and the mean elements (the Keplerian elements with the long period portions averaged out) will also be given in closed form. Finally the results of a numerical test, which compares the analytical solution against a numerical integration of the Lagrange equations of motion, will be summarized.This work was sponsored with the support of the Department of the Air Force under contract F19628-85-C-0002.The views expressed are those of the author and do not reflect the official policy or position of the U.S. Government.     

Comments on ‘the motion of a satellite in resonance with the second-degree sectorial harmonic’ by Dallas and Diehl

A critical appraisal is given of a recently published paper concerning the orbital motion of an artificial satellite in resonance with its rotating primary.     

Oscillating gyroscopes in a satellite

It is shown that a set of three gyroscopes in a satellite can test vital aspects of general relativity in a period of a few days.     

Motion of a satellite in the equatorial plane of a spheroid

An exact, closed-form solution of the problem of the motion of a satellite in the equatorial plane of an oblate body is obtained. It is shown that the classic formula for the motion of the perihelion is a first order approximation to the exact formula.