On the oscillations in Mercury's obliquity

Mercury's capture into the 3:2 spin–orbit resonance can be explained as a result of its chaotic orbital dynamics. One major objective of MESSENGER and BepiColombo spatial missions is to accurately measure Mercury's rotation and its obliquity in order to obtain constraints on internal structure of the planet. Analytical approaches at the first-order level using the Cassini state assumptions give the obliquity constant or quasi-constant. Which is the obliquity's dynamical behavior deriving from a complete spin–orbit motion of Mercury simultaneously integrated with planetary interactions? We have used our SONYR model (acronym of Spin–Orbit N-bodY Relativistic model) integrating the spin–orbit N-body problem applied to the Solar System (Sun and planets). For lack of current accurate observations or ephemerides of Mercury's rotation, and therefore for lack of valid initial conditions for a numerical integration, we have built an original method for finding the libration center of the spin–orbit system and, as a consequence, for avoiding arbitrary amplitudes in librations of the spin–orbit motion as well as in Mercury's obliquity. The method has been carried out in two cases: (1) the spin–orbit motion of Mercury in the 2-body problem case (Sun–Mercury) where an uniform precession of the Keplerian orbital plane is kinematically added at a fixed inclination (S_2K case), (2) the spin–orbit motion of Mercury in the N-body problem case (Sun and planets) (S_n case). We find that the remaining amplitude of the oscillations in the S_n case is one order of magnitude larger than in the S_2K case, namely 4 versus 0.4 arcseconds (peak-to-peak). The mean obliquity is also larger, namely 1.98 versus 1.80 arcminutes, for a difference of 10.8 arcseconds. These theoretical results are in a good agreement with recent radar observations but it is not excluded that it should be possible to push farther the convergence process by drawing nearer still more precisely to the libration center. We note that the dynamically driven spin precession, which occurs when the planetary interactions are included, is more complex than the purely kinematic case. Nevertheless, in such a N-body problem, we find that the 3:2 spin–orbit resonance is really combined to a synchronism where the spin and orbit poles on average precess at the same rate while the orbit inclination and the spin axis orientation on average decrease at the same rate. As a consequence and whether it would turn out that there exists an irreducible minimum of the oscillation amplitude, quasi-periodic oscillations found in Mercury's obliquity should be to geometrically understood as librations related to these synchronisms that both follow a Cassini state. Whatever the open question on the minimal amplitude in the obliquity's oscillations and in spite of the planetary interactions indirectly acting by the solar torque on Mercury's rotation, Mercury remains therefore in a stable equilibrium state that proceeds from a 2-body Cassini state.     

Dynamical evolution of NEAs: Close encounters, secular perturbations and resonances

We discuss the main mechanisms affecting the dynamical evolution of Near-Earth Asteroids (NEAs) by analyzing the results of three numerical integrations over 1 Myr of the NEA (4179) Toutatis. In the first integration the only perturbing planet is the Earth. So the evolution is dominated by close encounters and looks like a random walk in semimajor axis and a correlated random walk in eccentricity, keeping almost constant the perihelion distance and the Tisserand invariant. In the second integration Jupiter and Saturn are present instead of the Earth, and the 3/1 (mean motion) and v ₆ (secular) resonances substantially change the eccentricity but not the semimajor axis. The third, most realistic, integration including all the three planets together shows a complex interplay of effects, with close encounters switching the orbit between different resonant states and no approximate conservation of the Tisserand invariant. This shows that simplified 3-body or 4-body models cannot be used to predict the typical evolution patterns and time scales of NEAs, and in particular that resonances provide some “fast-track” dynamical routes from low-eccentricity to very eccentric, planet-crossing orbits.     

The stability of vertical motion in the N-body circular Sitnikov problem

We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N ? 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z ₀ values, where z ₀ lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N ? 1 primary masses.     

Integrable approximation of J 2-perturbed relative orbits

Most existing satellite relative motion theories utilize mean elements, and therefore cannot be used for calculating long-term bounded perturbed relative orbits. The goal of the current paper is to find an integrable approximation for the relative motion problem under the J ₂ perturbation, which is adequate for long-term prediction of bounded relative orbits with arbitrary inclinations. To that end, a radial intermediary Hamiltonian is utilized. The intermediary Hamiltonian retains the original structure of the full J ₂ Hamiltonian, excluding the latitude dependence. This formalism provides integrability via separation, a fact that is utilized for finding periodic relative orbits in a local-vertical local-horizontal frame and determine an initialization scheme that yields long-term boundedness of the relative distance. Numerical experiments show that the intermediary-based computation of orbits provides long-term bounded orbits in the full J ₂ problem for various inclinations. In addition, a test case is shown in which the radial intermediary-based initial conditions of the chief and deputy satellites yield bounded relative distance in a high-precision orbit propagator.     

Artificial Martian frozen orbits and Sun-Synchronous orbits using continuous low-thrust control

This paper analyses three types of artificial orbits around Mars pushed by continuous low-thrust control: artificial frozen orbits, artificial Sun-Synchronous orbits and artificial Sun-Synchronous frozen orbits. These artificial orbits have similar characteristics to natural frozen orbits and Sun-Synchronous orbits, and their orbital parameters can be selected arbitrarily by using continuous low-thrust control. One control strategy to achieve the artificial frozen orbit is using both the transverse and radial continuous low-thrust control, and another to achieve the artificial Sun-Synchronous orbit is using the normal continuous low-thrust control. These continuous low-thrust control strategies consider J ₂, J ₃, and J ₄ perturbations of Mars. It is proved that both control strategies can minimize characteristic velocity. Relevant formulas are derived, and numerical results are presented. Given the same initial orbital parameters, the control acceleration and characteristic velocity taking into account J ₂, J ₃, and J ₄ perturbations are similar to those taking into account J ₂ perturbations for both Mars and the Earth. The control thrust of the orbit around Mars is smaller than that around the Earth. The magnitude of the control acceleration of ASFOM-4 (named as Artificial Sun-Synchronous Frozen Orbit Method 4) is the lowest among these strategies and the characteristic velocity within one orbital period is only 0.5219 m/s for the artificial Sun-Synchronous frozen orbit around Mars. It is evident that the relationship among the control thrusts and the primary orbital parameters of Martian artificial orbits is always similar to that of the Earth. Simulation shows that the control scheme extends the orbital parameters’ selection range of three types of orbits around Mars, compared with the natural frozen orbit and Sun-Synchronous orbit.     

Hamiltonian Stability of Spin–Orbit Resonances in Celestial Mechanics

The behaviour of ‘resonances’ in the spin-orbit coupling in celestial mechanics is investigated in a conservative setting. We consider a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. The continuous system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability character of the periodic orbit. We denote by ε_* (p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε_* (p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities) and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analytical solutions for J 2-perturbed unbounded equatorial orbits

While solutions for bounded orbits about oblate spheroidal planets have been presented before, similar solutions for unbounded motion are scarce. This paper develops solutions for unbounded motion in the equatorial plane of an oblate spheroidal planet, while taking into account only the J ₂ harmonic in the gravitational potential. Two cases are distinguished: A pseudo-parabolic motion, obtained for zero total specific energy, and a pseudo-hyperbolic motion, characterized by positive total specific energy. The solutions to the equations of motion are expressed using elliptic integrals. The pseudo-parabolic motion unveils a new orbit, termed herein the fish orbit, which has not been observed thus far in the perturbed two-body problem. The pseudo-hyperbolic solutions show that significant differences exist between the Keplerian flyby and the flyby performed under the the J ₂ zonal harmonic. Numerical simulations are used to quantify these differences.     

Bounded orbits in the elliptic restricted three-body problem

Special solutions of the planar rectilinear elliptic restricted 3-body problem are investigated for the limiting case e=1. Numerical integration is performed for primaries of equal masses. Starting values which define circular orbit solutions lead to bounded solutions if the initial radius a₀ is larger than 3.74 in units of the primaries' semimajor axis a. A comparison with the Eulerian two-fixedcentre problem is presented in order to understand qualitatively the characteristic features of bounded orbits and the transition to escape orbits.     

On the Stability Regions of the Trojan Asteroids

The orbits of fictitious bodies around Jupiter’s stable equilibrium points L ₄ and L ₅ were integrated for a fine grid of initial conditions up to 100 million years. We checked the validity of three different dynamical models, namely the spatial, restricted three body problem, a model with Sun, Jupiter and Saturn and also the dynamical model with the Outer Solar System (Jupiter to Neptune). We determined the chaoticity of an orbit with the aid of the Lyapunov Characteristic Exponents (=LCE) and used also a method where the maximum eccentricity of an orbit achieved during the dynamical evolution was examined. The goal of this investigation was to determine the size of the regions of motion around the equilibrium points of Jupiter and to find out the dependance on the inclination of the Trojan’s orbit. Whereas for small inclinations (up to i=20°) the stable regions are almost equally large, for moderate inclinations the size shrinks quite rapidly and disappears completely for i>60^°. Additionally, we found a difference in the dynamics of orbits around L ₄ which – according to the LCE – seem to be more stable than the ones around L ₅.     

On Brown's conjecture

We examine the conjecture made by Brown (1911) that in the restricted three body problem, the long period family of periodic orbits aroundL ₄, ends on a homoclinic orbit toL ₃. By numerical integration we establish that for the mass ratio Sun-Jupiter such a homoclinic orbit toL ₃ does not exist but that there exists a family of homoclinic orbits to periodic orbits aroundL ₃.     

A unified theory of the formation of galactic bars

We give arguments for a basically unified formation mechanism of slow (Lynden-Bell) and fast (common) galactic bars. This mechanism is based on an instability that is akin to the well-known instability of radial orbits and is produced by the mutual attraction and alignment of precessing stellar orbits (so far, only the formation of slow bars has been explained in this way). We present a general theory of the low-frequency modes in a disk that consists of orbits precessing at different angular velocities. The problem of determining these modes is reduced to integral equations of moderately complex structure. The characteristic pattern angular velocities Ω_p of the low-frequency modes are of the order of the mean orbital precession angular velocity \(\bar \Omega _{pr}\). Bar modes are also among the low-frequency modes; while \(\Omega _p \approx \bar \Omega _{pr}\) for slow bars, Ω_p for fast bars can appreciably exceed even the maximum orbital precession angular velocity in the disk Ω _pr ^max (however, it remains of the order of these precession angular velocities). The possibility of such an excess of Ω_p over Ω _pr ^max is associated with the effect of “repelling” orbits. The latter tend to move in a direction opposite to the direction in which they are pushed. We analyze the pattern of orbital precession in potentials typical of galactic disks. We note that the maximum radius of an “attracting” circular orbit r_c can serve as a reasonable estimate of the bar length l_b. Such an estimate is in good agreement with the available results of N-body simulations.     

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus’ equilateral equilibrium points is investigated with respect to the inclination of the asteroid’s orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L ₄ and L ₅, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 × 10⁹ years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 ≤ a ≤ 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids’ orbit was varied in the range 0° < i < 60° and the semimajor axes were fixed. It turned out that only four ‘windows’ of stable orbits survive: these are the orbits for the initial inclinations 0° < i < 7°, 9° < i < 13°, 31° < i < 36° and 38° < i < 50°. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.     

The 1/1 resonance in extrasolar planetary systems

We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits.     

Relativistic Precession of Elliptical Orbits of a Circumbinary Planet can be Cancelled

In publications presenting analytical results on the non-coplanar motion of a circumbinary planet it was shown that the unperturbed elliptical orbit of the planet undergoes simultaneously two kinds of the precession: the precession of the orbital plane and the precession of the orbit in its own plane. It is also well-known that there is also the relativistic precession of the planetary orbit in its own plane. In the present paper we study a combined effect of the all of the above precessions. For the general case, where the planetary orbit is not coplanar with the stars orbits, we analyzed the dependence of the critical inclination angle i_c, at which the precession of the planetary orbit in its own plane vanishes, on the angular momentum L of the planet. We showed that the larger the angular momentum, the smaller the critical inclination angle becomes. We presented the analytical result for i_c(L) and calculated the value of L, for which the critical inclination value becomes zero. For the particular case, where the planetary orbit is not coplanar with the stars orbits, we demonstrated analytically that at a certain value of the angular momentum of the planet, the elliptical orbit of the planet would become stationary: no precession. In other words, at this value of the angular momentum, the relativistic precession of the planetary orbit and its precession, caused by the fact that the planet revolves around a binary (rather than single) star, cancel each other out. This is a counterintuitive result.     

Orbital dynamics of the gravitational field in Bardeen space-time

In this paper, the orbital dynamics of the gravitational field in Bardeen space-time are investigated. The equations of the particle’s orbital motion are given by solving the Lagrangian equation, and the stability and types of orbits are studied by means of analysing the effective potential of particles. Also, with the help of phase-plane method, the closed and non-closed orbits of test particle are analysed. We find that the stability and types of orbits in the Bardeen space-time are determined not only by the particles’ energy but also by the angular momentum. And for q=0.5M and b<3.3731M, absorbed by the black hole is the only fate of the test particle. We also find that the position of the innermost stable circular orbit of Bardeen black hole occurs at r _min =5.5722M.     

Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury

In this paper, we consider the elliptic collinear solutions of the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler–Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler–Moulton collinear solution of n-bodies splits into \((n-1)\) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other \((n-2)\) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martínez et al. on 3-body Euler solutions in 2004–2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler–Moulton solution of the 4-body problem with two small masses in the middle.     

The existence of families of periodic orbits in theN-body problem

It is proved that monoparametric families of periodic orbits of theN-body problem in the plane, for fixed values of all masses, exist in a rotating frame of reference whosex axis contains always two of the bodiesP ₁ andP ₂. A periodic motion of theN-body problem is obtained as a continuation ofN–2 symmetric periodic orbits of the circular restricted three-body problem whose periods are in integer dependence, by increasing the masses of the originallyN–2 massless bodiesP ₃, ...,P _k. The analytic continuation, for sufficiently small values of theN–2 bodiesP ₃ ...P _k and finite values for the masses ofP ₁ andP ₂ has been proved by the continuation method and the solution itself has been found explicitly to a linear approximation in the small masses. Also, the possible application of the above periodic orbits to the study of the Solar system and of stellar systems is mentioned.     

Families of periodic orbits in the planar three-body problem

A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm ₃ of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom ₁:m ₂:m ₃?5:5:3. From that point on the family continues for decreasing massesm ₃ until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem.