Kinematical modeling of the Earth rotation,focusing on the Oppolzer terms in a rigid Earth and the Oppolzer-like terms in an elastic Earth

Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes—both for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation (in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the motion of the outer bodies is taken into account.     

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.     

Precession of the Orbit of a Planet around Stars Revolving along: Low-Eccentricity Orbits in a Binary System: Analytical Solution

In our previous paper (hereafter, paper I) we presented analytical results on the non-planar motion of a planet around a binary star for the cases of the circular orbits of the components of the binary. We found that the orbital plane of the planet (the plane containing the “unperturbed” elliptical orbit of the planet), in addition to precessing about the angular momentum of the binary, undergoes simultaneously the precession within the orbital plane. We demonstrated that the analytically calculated frequency of this additional precession is not the same as the frequency of the precession of the orbital plane about the angular momentum of the binary, though the frequencies of both precessions are of the same order of magnitude. In the present paper we extend the analytical results from paper I by relaxing the assumption that the binary is circular – by allowing for a relatively small eccentricity ε of the stars orbits in the binary. We obtain an additional, ε-dependent term in the effective potential for the motion of the planet. By analytical calculations we demonstrate that in the particular case of the planar geometry (where the planetary orbit is in the plane of the stars orbits), it leads to an additional contribution to the frequency of the precession of the planetary orbit. We show that this additional, ε-dependent contribution to the precession frequency of the planetary orbit can reach the same order of magnitude as the primary, ε-independent contribution to the precession frequency. Besides, we also obtain analytical results for another type of the non-planar configuration corresponding to the linear oscillatory motion of the planet along the axis of the symmetry of the circular orbits of the stars. We show that as the absolute value of the energy increases, the period of the oscillations decreases.     

Extension of Cassini's Laws

We investigate the Cassini's laws which describe the rotational motion in a 1:1 spin-orbit resonance. When this rotational motion follows the conventional Cassini's laws, the figure axis coincides with the angular momentum axis. In this case we underline the differences between the rotational Hamiltonian for a 'slow rotating' body like the Moon and for a 'fast rotating' body like Phobos. Then, we study a more realistic rotational Hamiltonian where the angle J between the figure axis and the angular momentum axis could be different from zero. This Hamiltonian has not been studied before. We have found a new particular solution for this Hamiltonian which could be seen as an extension of the Cassini's laws. In this new solution the angle J is constant, which is not zero, and the precession of the angular momentum plane is equal to the mean motion of the argument of pericenter of the rotating body. This type of rotational motion is only possible when the orbital eccentricity of the rotating body is not zero. This new law enables describing in particular, the Moon mean rotational motion for which the mean value of the angle J is found to be equal to 103.9±0.7 s of arc.     

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.     

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.     

Precession of the orbital nodes of Jupiter and Saturn triggered by the mutual perturbation: A model of two rings

The problem of the precession of the orbital planes of Jupiter and Saturn under the influence of mutual gravitational perturbations was formulated and solved using a simple dynamical model. Using the Gauss method, the planetary orbits are modeled by material circular rings, intersecting along the diameter at a small angle α. The planet masses, semimajor axes and inclination angles of orbits correspond to the rings. What is new is that each ring has an angular momentum equal to the orbital angular momentum of the planet. Contrary to popular belief, it was proved that the orbital resonance 5: 2 does not preclude the use of the ring model. Moreover, the period of averaging of the disturbing force (T ≈ 1332 yr) proves to be appreciably greater than a conventionally used period (≈900 yr). The mutual potential energy of rings and the torque of gravitational forces between the rings were calculated. We compiled and solved the system of differential equations for the spatial motion of rings. It was established that a perturbing torque causes the precession and simultaneous rotation of the orbital planes of Jupiter and Saturn. Moreover, the opposite orbit nodes on the Laplace plane coincide and perform a secular movement in retrograde direction with the same velocity of 25.6″/yr and the period T _J = T _S ≈ 50687 yr. These results are close to those obtained in the general theory (25.93″/yr), which confirms the adequacy of the developed model. It was found that the vectors of the angular velocity of orbital rings move counterclockwise over circular cones and describe circles on the celestial sphere with radii β₁ ≈ 0.8403504° (Saturn) and β₂ ≈ 0.3409296° (Jupiter) around the point which is located at an angular distance of 1.647607° from the ecliptic pole.     

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.     

Generalized YORP evolution: Onset of tumbling and new asymptotic states

Asteroids have a wide range of rotation states. While the majority spin a few times to several times each day in principal axis rotation, a small number spin so slowly that they have somehow managed to enter into a tumbling rotation state. Here we investigate whether the Yarkovsky-Radzievskii-O'Keefe-Paddack (YORP) thermal radiation effect could have produced these unusual spin states. To do this, we developed a Lie-Poisson integrator of the orbital and rotational motion of a model asteroid. Solar torques, YORP, and internal energy dissipation were included in our model. Using this code, we found that YORP can no longer drive the spin rates of bodies toward values infinitely close to zero. Instead, bodies losing too much rotation angular momentum fall into chaotic tumbling rotation states where the spin axis wanders randomly for some interval of time. Eventually, our model asteroids reach rotation states that approach regular motion of the spin axis in the body frame. An analytical model designed to describe this behavior does a good job of predicting how and when the onset of tumbling motion should take place. The question of whether a given asteroid will fall into a tumbling rotation state depends on the efficiency of its internal energy dissipation and on the precise way YORP modifies the spin rates of small bodies.     

Satellite resonance with longitude-dependent gravity—II Effects involving the eccentricity

For a satellite in a nominally circular orbit at arbitrary inclination whose mean motion is commensurable with the Earth's rotation, the dependence of gravity on longitude leads to a resonant variation in eccentricity as well as the long-period oscillation in longitude. Provided forces capable of processing perigee are present, it is shown that the change in eccentricity for a satellite captured in librational resonance is not secular but periodic.

There are corresponding resonance effects for a satellite in a nominally equatorial but eccentric orbit. Here the commensurability condition is that the longitudes of the apses shall be nearly repetitive relative to the rotating Earth. There will be a long-period oscillation in longitude which can take the form of either a libration (trapped) or a circulation (free), and there will also be an oscillation of the orbital plane having the same period as the precession of perigee relative to inertial space.     

Nodal and periastron precession of inclined orbits in the field of a rotating black hole

The inclination of low-eccentricity orbits is shown to significantly affect orbital parameters, in particular, the Keplerian, nodal precession, and periastron rotation frequencies, which are interpreted in terms of observable quantities. For the nodal precession and periastron rotation frequencies of low-eccentricity orbits in a Kerr field, we derive a Taylor expansion in terms of the Kerr parameter at arbitrary orbital inclinations to the black-hole spin axis and at arbitrary radial coordinates. The particle radius, energy, and angular momentum in the marginally stable circular orbits are calculated as functions of the Kerr parameter j and parameter s in the form of Taylor expansions in terms of j to within O[j ⁶]. By analyzing our numerical results, we give compact approximation formulas for the nodal precession frequency of the marginally stable circular orbits at various s in the entire range of the Kerr parameter.     

The LAGEOS satellites orbital residuals determination and the Lense-Thirring effect measurement

The method applied since 1996 for the analysis of the orbital residuals of the LAGEOS satellites in order to measure the Lense-Thirring effect has been the subject of the present work. This method, based on the difference between the orbital elements of consecutive arcs, is explained and analysed also from the analytical point of view. It is proved that this “difference method” works well for the determination of the secular effects, as in the case of the relativistic precession induced by the Earth's angular momentum, but also very useful for the determination and study of the long-term periodic effects. Indeed, the only limitation in the determination of the periodic effects is the possibility of the reduction of their amplitude by a factor which depends from the periodicity of the given perturbation and from the orbital arc length chosen for the satellite during the data analysis. In the case of the Yarkovsky-Schach effect, the main non-gravitational perturbation seen in the LAGEOS satellites orbital residuals, in particular in its perigee rate and eccentricity vector excitation residuals, we show that the “difference method” is quite good also for the determination of the long-period perturbations induced by this subtle non-conservative force.     

LAGEOS II perigee rate and eccentricity vector excitations residuals and the Yarkovsky-Schach effect

We have analysed LAGEOS II perigee rate and eccentricity vector excitation residuals over a period of about 7.8 years, adjusting and computing the satellite orbit with the full set of dynamical models included in the GEODYN II software code. The long-term behaviour of these orbital residuals appears to be characterised by several distinct frequencies which are a clear signature of the Yarkovsky-Schach perturbing effect. This non-gravitational perturbation is not included in the GEODYN II models for the orbit determination and analysis. Through an independent numerical analysis, and using the new LOSSAM model to represent the spin-axis behaviour of the satellite, we propagated the Yarkovsky-Schach effect on LAGEOS II perigee rate and compared the results obtained with the orbital residuals. We have thus been able to satisfactorily fit the amplitude of the Yarkovsky-Schach effect to the observed residuals. Our approach here has proven very successful with very positive results. We have been able to obtain a fractional reduction of about 40% of the post-fit rms with respect to the pre-fit value. When analysing the eccentricity vector residuals, we have been able to obtain a better result in the case of the real component, with a fractional reduction of the post-fit rms of about 49% of the initial value. The analysis of the effect's imaginary component in the eccentricity vector rate is more complicated and deserves additional scrutiny. In this case we need a deeper study which includes the analysis of other unmodelled and mismodelled effects acting on the imaginary component. The study performed in this paper will be of significant relevance not only for the geophysical applications involving LAGEOS II orbit analysis, but also for a refined re-analysis of the general relativistic precession produced by the Earth angular momentum, i.e., the Lense-Thirring effect.     

Relative motion of satellites exploiting the super-integrability of Kepler’s problem

This paper builds upon the work of Palmer and Imre exploring the relative motion of satellites on neighbouring Keplerian orbits. We make use of a general geometrical setting from Hamiltonian systems theory to obtain analytical solutions of the variational Kepler equations in an Earth centred inertial coordinate frame in terms of the relevant conserved quantities: relative energy, relative angular momentum and the relative eccentricity vector. The paper extends the work on relative satellite motion by providing solutions about any elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum case. The geometrical framework assists the design of complex formation flying trajectories. This is demonstrated by the construction of a tetrahedral formation, described through the relevant conserved quantities, for which the satellites are on highly eccentric orbits around the Sun to visit the Kuiper belt.     

A method for averaging the perturbing function in Hill’s problem

A modified method for averaging the perturbing function in Hill’s problem is suggested. The averaging is performed in the revolution period of the satellite over the mean anomaly of its motion with a full allowance for a variation in the position of the perturbing body. At its fixed position, the semimajor axis of the satellite orbit during the revolution of the satellite is constant in view of the evolution equations, while the remaining orbital elements undergo secular and long-period perturbations. Therefore, when the motion of the perturbing body is taken into account, the semimajor axis of the satellite orbit undergoes the strongest perturbations. The suggested approach generalizes the averaging method in which only the linear (in time) term is included in the perturbing function. This method requires no expansion in powers of time. The described method is illustrated by calculating the perturbations of the semimajor axes for two distant satellites of Saturn, S/2000 S 1 and S/2000 S5. An approximate analytic solution is compared with the results of numerical integration of the averaged system of equations of motion for these satellites.     

Orbits in an anisotropic radiation field

The elliptic-type motion around a source whose luminosity is anisotropic is being perturbatively treated. Using a Fourier expansion for the perturbing force, exact expressions describing the evolution of each orbital element are determined from Newton-Euler equations. For near-unit frequency the resonant solutions are pointed out. An approximate expression for the variation of the nodal period is determined, too. The behaviour of the radius vector of the unstable orbit (resonance case) is pointed out, showing periodic variations of constantly increasing amplitude. The (both angular and physical) time scale for escape is estimated. Concrete astronomical situations modellable in this way are mentioned.     

The equations of relative motion in the orbital reference frame

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced.     

Using wavelet approximation to study the influence of Jupiter on the orbital evolution of a distant satellite of Saturn

We suggest a nonstandard methodology for studying the influence of Jupiter on the secular orbital evolution of a distant satellite of Saturn. This influence is tangible only in short time spans near the times of the smallest separation between Jupiter and Saturn, i.e., when the heliocentric longitudes of the two planets coincide. These times are spaced about 20 years apart. To describe the jumplike behavior of perturbations, we suggest approximating the principal part of the perturbing function averaged over the satellite’s motion by a two-parameter exponential wavelet-type (burst) function. The subsequent averaging (smoothing) of the perturbing function allows us to eliminate the 20-year-period terms and obtain an approximate analytical solution in a special case of the problem. The results are illustrated by plots of the variations in the averaged perturbing function and the orbital eccentricity of Saturn’s outer satellite S/2000 S1, which is most strongly perturbed by Jupiter.     

On the true circular orbit of a satellite

The osculating orbit of a planetary satellite moving in the equatorial plane of the central body under the influence of a rotational symmetric perturbation force is elliptical in first order approximation even if the true orbit is always circular. The satellite motion is influenced by a resonance effect due to this perturbing force. An inclined true satellite orbit cannot be circular.