On the stability of close binaries in hierarchical three-body systems

The Hill-type stability (cf. closure of the zero-velocity curves in the circular restricted three-body problem) of general hierarchical three-body systems is examined analytically in the case where the total mass of the binary is small in comparison to the mass of the external body (e.g. systems of the type Planet-Satellite-Sun, Planet-Planet-Star, etc.). This is compared with results derived by Szebehely, Markellos and Roy in the Planet-Satellite-Sun case of the circular restricted three-body problem. It is demonstrated how the Hill-type stability is affected by the sense of revolution of the binary, i.e. corotational or contrarotational, and the mass ratio within the binary. The effect of the difference in longitudes of the bodies in their orbits is also examined.     

On a four-body problem

In this paper we study a particular four-body problem: three bodies revolve around their center of mass in circular orbits under the influence of their mutual gravitational attraction, while a fourth body moves in the plane defined by the three bodies but non influencing their motion. The linear stability of the eight equilibrium points is studied, and it is found that it depends on the values of the masses.     

Hill stability of the satellite in the elliptic restricted four-body problem

We consider an elliptic restricted four-body system including three primaries and a massless particle. The orbits of the primaries are elliptic, and the massless particle moves under the mutual gravitational attraction. From the dynamic equations, a quasi-integral is obtained, which is similar to the Jacobi integral in the circular restricted three-body problem (CRTBP). The energy constant \(C\) determines the topology of zero velocity surfaces, which bifurcate at the equilibrium point. We define the concept of Hill stability in this problem, and a criterion for stability is deduced. If the actual energy constant \(C_{\mathrm{ac}}\ ( {>} 0 ) \) is bigger than or equal to the critical energy constant \(C_{\mathrm{cr}}\), the particle will be Hill stable. The critical energy constant is determined by the mass and orbits of the primaries. The criterion provides a way to capture an asteroid into the Earth–Moon system.     

The Hill stability of low mass binaries in hierarchical triple systems

The Hill stability of the low mass binary system in the presence of a massive third body moving on a wider inclined orbit is investigated analytically. It is found that, in the case of the third body being on a nearly circular orbit, the region of Hill stability expands as the binary/third body mass ratio increases and the inclination (i) decreases. This i-dependence decreases very quickly with increasing eccentricity (e ₂) of the third body relative to the binary barycentre. In fact, if e ₂ is not extremely small, the Hill stable region can be approximately expressed in a closed form by setting i = 90°, and it contracts with increasing e ₂ as ${e_2^2}$ for sufficiently low mass binary. Our analytic results are then applied to the observed triple star systems and the Kuiper belt binaries.     

On the stability of circumbinary planetary systems

The dynamics of circumbinary planetary systems (the systems in which the planets orbit a central binary) with a small binary mass ratio discovered to date is considered. The domains of chaotic motion have been revealed in the “pericentric distance–eccentricity” plane of initial conditions for the planetary orbits through numerical experiments. Based on an analytical criterion for the chaoticity of planetary orbits in binary star systems, we have constructed theoretical curves that describe the global boundary of the chaotic zone around the central binary for each of the systems. In addition, based on Mardling’s theory describing the separate resonance “teeth” (corresponding to integer resonances between the orbital periods of a planet and the binary), we have constructed the local boundaries of chaos. Both theoretical models are shown to describe adequately the boundaries of chaos on the numerically constructed stability diagrams, suggesting that these theories are efficient in providing analytical criteria for the chaoticity of planetary orbits.     

On the photo-gravitational restricted four-body problem with variable mass

This paper deals with the photo-gravitational restricted four-body problem (PR4BP) with variable mass. Following the procedure given by Gascheau (C. R. 16:393–394, 1843) and Routh (Proc. Lond. Math. Soc. 6:86–97, 1875), the conditions of linear stability of Lagrange triangle solution in the PR4BP are determined. The three radiating primaries having masses \(m_{1}\), \(m_{2}\) and \(m_{3}\) in an equilateral triangle with \(m_{2}=m_{3}\) will be stable as long as they satisfy the linear stability condition of the Lagrangian triangle solution. We have derived the equations of motion of the mentioned problem and observed that there exist eight libration points for a fixed value of parameters \(\gamma (\frac{m \ \text{at time} \ t}{m \ \text{at initial time}}, 0<\gamma\leq1 )\), \(\alpha\) (the proportionality constant in Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), \(0\leq\alpha\leq2.2\)), the mass parameter \(\mu=0.005\) and radiation parameters \(q_{i}, (0< q_{i}\leq1, i=1, 2, 3)\). All the libration points are non-collinear if \(q_{2}\neq q_{3}\). It has been observed that the collinear and out-of-plane libration points also exist for \(q_{2}=q_{3}\). In all the cases, each libration point is found to be unstable. Further, zero velocity curves (ZVCs) and Newton–Raphson basins of attraction are also discussed.     

On the stability of Earth-like planets in multi-planet systems

We present a continuation of our numerical study on planetary systems with similar characteristics to the Solar System. This time we examine the influence of three giant planets on the motion of terrestrial-like planets in the habitable zone (HZ). Using the Jupiter–Saturn–Uranus configuration we create similar fictitious systems by varying Saturn’s semi-major axis from 8 to 11 AU and increasing its mass by factors of 2–30. The analysis of the different systems shows the following interesting results: (i) Using the masses of the Solar System for the three giant planets, our study indicates a maximum eccentricity (max-e) of nearly 0.3 for a test-planet placed at the position of Venus. Such a high eccentricity was already found in our previous study of Jupiter–Saturn systems. Perturbations associated with the secular frequency g ₅ are again responsible for this high eccentricity. (ii) An increase of the Saturn-mass causes stronger perturbations around the position of the Earth and in the outer HZ. The latter is certainly due to gravitational interaction between Saturn and Uranus. (iii) The Saturn-mass increased by a factor 5 or higher indicates high eccentricities for a test-planet placed at the position of Mars. So that a crossing of the Earth’ orbit might occur in some cases. Furthermore, we present the maximum eccentricity of a test-planet placed in the Earth’ orbit for all positions (from 8 to 11 AU) and masses (increased up to a factor of 30) of Saturn. It can be seen that already a double-mass Saturn moving in its actual orbit causes an increase of the eccentricity up to 0.2 of a test-planet placed at Earth’s position. A more massive Saturn orbiting the Sun outside the 5:2 mean motion resonance (a _S  ≥9.7 AU) increases the eccentricity of a test-planet up to 0.4.     

Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem

We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous binary collision orbits in the symmetric collinear four-body problem with masses 1, m, m, 1, and also in a symmetric planar four-body problem with equal masses. In both problems, the assumed symmetries reduce the determination of linear stability to the numerical computation of a single real number. For the collinear problem, this verifies the earlier numerical results of Sweatman for linear stability with respect to collinear and symmetric perturbations.     

On the global stability of single‐planet systems

In this paper, we estimate the global stability properties of single‐planet systems by using a catalogue of stability maps. The data of the catalogue were used to generate probability values on the mass parameter–eccentricity plane for the occurrence of stable orbits. We showed that the probability data can be well approximated by a second order surface. Using the resulted formula the likelihood of finding Earth‐like planets in single‐planet systems can be easily estimated. As an example, we derived estimations for four known exoplanetary systems. Our formula can be useful in selecting target stars for future space missions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the stability of small quasiperiodic motions in the hamiltonian systems

Orbital stability of quasiperiodic motions in the many dimensional autonomic hamiltonian systems is considered. Studied motions are supposed to be not far from equilibrium, the number of their basic frequencies may be not equal to the number of degrees of freedom, and the procedure of their construction is supposed to be converged. The stability problem is solved in the strict nonlinear mode.Obtained results are used in the stability investigation of small plane motions near the lagrangian solutions of the three-dimensional circular restricted three-body problem. The values of parameters for which the plane motions are unstable have been found.

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On the stability of equilibria of Hamiltonian systems near the main resonances

The equilibrium point O of an autonomous Hamiltonian system of two degrees of freedom is considered for small-oscillation frequencies related as ₂=2₁+. If under the precise resonance (=0) the equilibrium is unstable, the inner diameter () of the domain of stability containing the point O is estimated. It is shown that for the normalized variables ()/b where b is the corresponding resonance coefficient. The estimates () for other main resonances are reported.     

On hierarchical cosmology

Progress in laboratory studies of plasmas and in the methods of transferring the results to cosmic conditions, together within situ measurements in the magnetospheres, are now causing a ‘paradigm transition’ in cosmic plasma physics. This involves an introduction ofinhomogeneous models with double layers, filaments, ‘cell walls’, etc. Independently, it has been discovered that the mass distribution in the universe is highly inhomogeneous; indeed,hierarchical. According to de Vaucouleurs, the escape velocity of cosmic structures is 10²–10³ times below the Laplace-Schwarzschild limit, leaving avoid region which is identified as a key problem in cosmology. It is shown that a plasma instability in the dispersed medium of the structures may produce this void and, hence, explain the hierarchical structure. The energy which is necessary may derive either from gravitation or from annihilation caused by a breakdown of cell walls. The latter alternative is discussed in detail. It leads to a ‘Fireworks Model’ of the evolution of the metagalaxy. It is questioned whether the homogeneous four-dimensional big bang model can survive in an universe which is inhomogeneous and three-dimensional.     

Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53.     

Revealing the existence and stability of equilibrium points in the circular autonomous restricted four-body problem with variable mass

We have numerically investigated the circular autonomous restricted four-body problem where the fourth particle of variable mass is moving under the gravitational influence of three bodies known as primaries. Moreover, these primaries move in circular orbit around their common center of mass in such a way that their configuration remains an equilateral triangle configuration. The effect of the parameter α on the existence as well as on the locations of the libration points are investigated. The parametric variation of the positions of the libration points and zero velocity curves are also revealed when the parameter α (which occurs in Jeans’ law) increases. Moreover, the Newton–Raphson basins of convergence corresponding to the libration points are unveiled numerically when the parameter α increases. The obtained results strongly suggest that the study of the evolution of the attracting domains of the proposed dynamical system is worth studying in spite of their complexity.     

Stability of the coplanar planetary four-body system

We consider the coplanar planetary four-body problem, where three planets orbit a large star without the cross of their orbits. The system is stable if there is no exchange or cross of orbits. Starting from the Sundman inequality, the equation of the kinematical boundaries is derived. We discuss a reasonable situation, where two planets with known orbits are more massive than the third one. The boundaries of possible motions are controlled by the parameter c~2E. If the actual value of c~2E is less than or equal to a critical value(c~2 E)cr, then the regions of possible motions are bounded and therefore the system is stable.The criteria obtained in special cases are applied to the Solar System and the currently known extrasolar planetary systems. Our results are checked using N-body integrator.     

Restricted four-body problem. The case of a central configuration: Libration points and their stability

We consider the problem of the motion of a zero-mass body in the vicinity of a system of three gravitating bodies forming a central configuration.We study the case where two gravitating bodies of equal mass lie on the same straight line and rotate around the central body with the same angular velocity. Equations for calculating the equilibrium positions in this system have been derived. The stability of the equilibrium points for a system of three gravitating bodies is investigated. We show that, as in the case of libration points for two bodies, the collinear points are unstable; for the triangular points, there exists a ratio of the mass of the central body to the masses of the extreme bodies, 11.720349, at which stability is observed.