Robe’s Circular Restricted Three-Body Problem Under Oblate and Triaxial Primaries

This paper analyzes Robe??s circular restricted three-body problem when the hydrostatic equilibrium figure of the first primary is assumed to be an oblate spheroid, the shape of the second primary is considered as a triaxial rigid body, and the full buoyancy force of the fluid is taken into account. It is found that there is an equilibrium point near the center of the first primary, another equilibrium point exists on the line joining the centers of the primaries and there exist infinite number of equilibrium points on an ellipse in the orbital plane of the second primary. It is also observed that under certain conditions, all these equilibrium points can be stable. The most interesting and distinguishable results of this study are the existence of elliptical points and their stability.     

Studying the Equilibrium Points of the Modified Circular Restricted Three-body Problem: The Case of Sun–Haumea System

We intend to study a modified version of the planar Circular Restricted Three-Body Problem(CRTBP) by incorporating several perturbing parameters. We consider the bigger primary as an oblate spheroid and emitting radiation while the small primary has an elongated body. We also consider the perturbation from a disk-like structure encompassing this three-body system. First, we develop a mathematical model of this modified CRTBP.We have found there exist five equilibrium points in this modified CRTB...     

Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory

This paper deals with the stability analysis of the triangular equilibrium points for the generalized problem of the photogravitational restricted three body where both the primaries are radiating. The problem is generalized in the sense that the eccentricity of the orbits and the oblateness due to both the primaries and infinitesimal are considered. The stability in the case of linear resonance are analyzed based on the Floquet’s theory for finding the characteristic exponent for a system containing periodic coefficients. It was found that the critical value of μ for the stability boundary for parametric excitation is dependent on the oblateness of the primaries as well as infinitesimal.     

Out-of-plane equilibrium points and their stability in the Robe’s problem with oblateness and triaxiality

This paper examines the existence and stability of the out-of-plane equilibrium points of a third body of infinitesimal mass when the equations of motion are written in the three dimensional form under the set up of the Robe’s circular restricted three-body problem, in which the hydrostatic equilibrium figure of the first primary is an oblate spheroid and the second one is a triaxial rigid body under the full buoyancy force of the fluid. The existence of the out of orbital plane equilibrium points lying on the xz-plane is noticed. These points are however unstable in the linear sense.     

The effect of the Earth’s oblateness on the Moon’s physical libration in latitude

The Moon’s physical libration in latitude generated by gravitational forces caused by the Earth’s oblateness has been examined by a vector analytical method. Libration oscillations are described by a close set of five linear inhomogeneous differential equations, the dispersion equation has five roots, one of which is zero. A complete solution is obtained. It is revealed that the Earth’s oblateness: a) has little effect on the instantaneous axis of Moon’s rotation, but causes an oscillatory rotation of the body of the Moon with an amplitude of 0.072″ and pulsation period of 16.88 Julian years; b) causes small nutations of poles of the orbit and of the ecliptic along tight spirals, which occupy a disk with a cut in a center and with radius of 0.072″. Perturbations caused by the spherical Earth generate: a) physical librations in latitude with an amplitude of 34.275″; b) nutational motion for centers of small spiral nutations of orbit (ecliptic) pole over ellipses with semi-major axes of 113.850″ (85.158″) and the first pole rotates round the second one along a circle with radius of 28.691″; c) nutation of the Moon’s celestial pole over an ellipse with a semi-major axis of 45.04″ and with an axes ratio of about 0.004 with a period of T = 27.212 days. The principal ellipse’s axis is directed tangentially with respect to the precession circumference, along which the celestial pole moves nonuniformly nearly in one dimension. In contrast to the accepted concept, the latitude does not change while the Moon’s poles of rotation move. The dynamical reason for the inclination of the Moon’s mean equator with respect to the ecliptic is oblateness of the body of the Moon.     

Numerical Exploration of Chermnykh’s Problem

We study the equilibrium points and the zero-velocity curves of Chermnykh’s problem when the angular velocity ω varies continuously and the value of the mass parameter is fixed. The planar symmetric simple-periodic orbits are determined numerically and they are presented for three values of the parameter ω. The stability of the periodic orbits of all the families is computed. Particularly, we explore the network of the families when the angular velocity has the critical value ω = 2√2 at which the triangular equilibria disappear by coalescing with the collinear equilibrium point L₁. The analytic determination of the initial conditions of the family which emanate from the Lagrangian libration point L₁ in this case, is given. Non-periodic orbits, as points on a surface of section, providing an outlook of the stability regions, chaotic and escape motions as well as multiple-periodic orbits, are also computed. Non-linear stability zones of the triangular Lagrangian points are computed numerically for the Earth–Moon and Sun–Jupiter mass distribution when the angular velocity varies.     

Gamma-ray bursts and the production of cosmogenic radionuclides in the Earth’s atmosphere

An explanation is offered for the impulsive increase in the concentration of cosmogenic radiocarbon in annual tree rings (Δ¹⁴C ～ 12‰) from AD ?775. A possible cause of such an increase could be the high-energy emission from a Galactic gamma-ray burst. It is shown that such an event should not lead to an increase in the total production of ¹⁰Be in the atmosphere, as distinct from the effect of cosmic-ray fluxes on the atmosphere. At the same time, the production of an appreciable amount of ³⁶Cl, which can be detected in Greenland and Antarctica ice samples of the corresponding age, should be expected. This allows the effects caused by a gamma-ray burst and anomalously powerful proton events to be distinguished.     

Robe’s restricted problem of 2+2 bodies with one of the primaries an oblate body

In this problem, one of the primaries of mass \(m^{*}_{1}\) is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ ₁. The smaller primary of mass m ₂ is an oblate body outside the shell. The third and the fourth bodies (of mass m ₃ and m ₄ respectively) are small solid spheres of density ρ ₃ and ρ ₄ respectively inside the shell, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m ₂ is describing a circle around \(m^{*}_{1}\) . The masses m ₃ and m ₄ mutually attract each other, do not influence the motions of \(m^{*}_{1}\) and m ₂ but are influenced by them. We also assume that masses m ₃ and m ₄ are moving in the plane of motion of mass m ₂. In the paper, equilibrium solutions of m ₃ and m ₄ and their linear stability are analyzed. There are two collinear equilibrium solutions for the system. The non collinear equilibrium solutions exist only when ρ ₃=ρ ₄. There exist an infinite number of non collinear equilibrium solutions of the system, provided they lie inside the spherical shell. In a system where the primaries are considered as earth-moon and m ₃,m ₄ as submarines, the collinear equilibrium solutions thus obtained are unstable for the mass parameters μ,μ ₃,μ ₄ and oblateness factor A. In this particular case there are no non-collinear equilibrium solutions of the system.     

Peculiarities of the variations in the coordinates of the Earth’s pole

The appearance of features with cusp points on the diagrams of changes in the coordinates of the Earth’s instantaneous pole (polhodes) is considered as the result of mapping onto the plane of its displacement over the surface during the Earth’s rotational-translational motion. The results of qualitative and quantitative analyses of the data on the coordinates of the Earth’s instantaneous pole are discussed. The basic principles of the theory of Whitney singularities and their application for explaining the bifurcations of the equilibrium positions for the Zeeman catastrophe machine (Arnold 1990) are used in the analyses.     

The Restricted Hill Four-Body Problem with Applications to the Earth–Moon–Sun System

Starting from the four-body problem a generalization of both the restricted three-body problem and the Hill three-body problem is derived. The model is time periodic and contains two parameters: the mass ratio ν of the restricted three-body problem and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the restricted three-body problem is recovered as m → 0 and the classical Hill three-body problem is recovered as ν → 0. This model also predicts motions described by earlier researchers using specific models of the Earth–Moon–Sun system. An application of the current model to the motion of a spacecraft in the Sun perturbed Earth–Moon system is made using Hill's Variation orbit for the motion of the Earth–Moon system. The model is general enough to apply to the motion of an infinitesimal mass under the influence of any two primaries which orbit a larger mass. Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange points are performed. Values of the parameter ν range in the neighborhood of the Earth–Moon value as the parameter m increases from 0 to 0.195 at which point the Hill Variation orbit becomes unstable. Two families of planar periodic orbits are studied in detail as the parameters m and ν vary. These families contain stable and unstable members in the plane and all have the out-of-plane stability. The stable and unstable manifolds of the unstable periodic orbits are computed and found to be trapped in a geometric area of phase space over long periods of time for ranges of the parameter values including the Earth–Moon–Sun system. This model is derived from the general four-body problem by rigorous application of the Hill and restricted approximations. The validity of the Hill approximation is discussed in light of the actual geometry of the Earth–Moon–Sun system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Robe’s circular restricted three-body problem under oblate primaries with perturbations in Coriolis and centrifugal forces

This paper examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid, and the shape of the second primary is also an oblate spheroid. The problem is perturbed in the sense that small perturbations given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. The linear stability of this configuration is examined; it is observed that the first axial point is unstable while the second one is conditionally stable and the circular points are unstable.     

The cyclical reversible transitions of the universe’s evolution

In this work we propose cyclical reversible transitions as the scenario in which the universe evolves, through a series consisting of reversible expansion, temporary stability, and contraction. Our model is based on the comparison between local and global time-dependent densities {ρ ₀(τ ₀),ρ(τ)} instead of the critical density ρ _c, local and global time-dependent Hubble parameters {H ₀(τ ₀),H(τ)}, and the variations {Δρ(τ),ΔH(τ)} due to cosmological chaotic fluctuations, which are generally ignored in certain oscillating models. By taking into account all these factors, a rate equation in the form of (H ₀/H)² ∝(ρ ₀/ρ) has been established, and from it we derive some others, to provide a mechanism that is responsible for the cyclical reversible transitions. Also, the problems of singularities, black hole overproduction, and the second law of thermodynamics arising in oscillating universe models are conceptually resolved.     

Changes in the activity of Jupiter’s hemispheres

Solar-induced changes in the reflective properties of the visible disk of Jupiter mostly depend on variations in the Earth’s jovimagnetic latitude. Since the orbit of Jupiter is eccentric (the eccentricity is e = 0.04845) and the planet passes perihelion at the time close to the summer solstice, the atmosphere receives 21% more solar energy in the northern hemisphere than in the southern one. According to the results of our studies, the ratio of the brightness values for the northern and southern tropical and temperate zones is a clear indicator of photometric activity of the processes in the atmosphere of Jupiter. From the analysis of the observational data for the period from 1962 to 2017, the cyclicity in changes of the activity factor of the hemispheres of the planet with a period of 11.87 years was found. This suggests that the atmosphere of Jupiter experiences seasonal restructuring.     

The growth of wind-waves in Titan’s hydrocarbon seas

Ocean wave growth on Titan is considered. The classic Sverdrup–Munk theory for terrestrial wave growth is applied to Titan, and is compared with a simple energy balance model that exposes the effect of Titan’s environmental parameters (air density, gravity, and fluid density). These approaches are compared with the only previously-published (semi-empirical) model (Ghafoor, N.A.-L., Zarnecki, J.C., Challenor, P., Srokosz, M.A. [2000] J. Geophys. Res. 105, 12,077–12,091, hereafter G2k), and allow the impact of various parameters such as atmospheric density to be transparently explored.Our model, like G2k, suggests fully-developed significant wave heights on Titan H_s = 0.2 U², where U is the windspeed (SI units): in dimensionless terms this is rather close to H_s = 0.2 U²/g, a rule of thumb previously noted for terrestrial waves (we find various datasets where the prefactor varies by ～2). It is noted that liquid and air densities affect the growth rate of waves, but not their fully-developed height: for 1 m/s winds wave amplitude reaches 0.15 m (75% of fully-developed) with a fetch of only 1 km, rather faster than predicted by G2k. Liquid viscosity has no major effect on gravity wave growth, but does influence the threshold windspeed at which gravity–capillary waves form in the first place.The model is used to develop predicted ranges for wave height to guide the design of the Titan Mare Explorer (TiME), a proposed Discovery-class mission to float a capsule on Ligeia Mare in 2023. For the expected maximum 1 m/s winds, a significant wave height of 0.2 m and wavelength of ～4 m can be expected. Assuming that wave heights follow Rayleigh statistics as they do on Earth, then given the wave period of ～4 s, individual waves of ～0.6 m might be encountered over a 3 month period.For predicted Titan winds at Kraken Mare, significant wave heights may reach ～0.6 m in the peak of summer but do not exceed the tidal amplitude at its northern end, consistent with the area around Mayda Insula being a tidal flat, while elsewhere on Kraken and Ligeia and at Ontario Lacus, shorelines may be wave- or tidally-dominated, depending on the specific location.     

Chaotic Scattering in the Restricted Three‐Body Problem II. Small mass parameters

We study the scattering motion of the planar restricted three‐body problem for small mass parameters μ. We consider the symmetric periodic orbits of this system with μ = 0 that collide with the singularity together with the circular and parabolic solutions of the Kepler problem. These divide the parameter space in a natural way and characterize the main features of the scattering problem for small non‐vanishing μ. Indeed, continuation of these orbits yields the primitive periodic orbits of the system for small μ. For different regions of the parameter space, we present scattering functions and discuss the structure of the chaotic saddle. We show that for μ < μc and any Jacobi integral there exist departures from hyperbolicity due to regions of stable motion in phase space. Numerical bounds for μc are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Minicomets in the Solar system and in the Earth’s atmosphere and related phenomena

We consider the history of discovery and justify the existence in the Solar system of a new class of bodies—minicomets, i.e., bodies of cometary nature and composition but of low mass. Two classes of minicomets are distinguished: icy ones similar to the Tunguska meteorite, and snow ones, which break up at high altitudes.     

The origin of the ‘FUN’ anomalies and the high temperature inclusions in the allende meteorite

The discovery of isotopic anomalies in white inclusions of the meteorite Allende has led to fundamental questions concerning the origin of these anomalies and of the white inclusions themselves. An analysis of the FUN anomalies in the inclusions C1 and EK1-4-1 demonstrates that these isotopic anomalies may be decomposed into individual nucleosynthetic components, which have been subjected to separate mass and component fractionations. There is no evidence that any freshlysynthesized material injected into the primitive solar nebula was of abnormal isotopic composition, or that the FUN anomalies were due to an injection of unusual material. Rather, they show the effects of form of interstellar grains whose size or chemistry served as a memory for the nucleosynthetic origins of their constituent atoms. Giant gaseous protoplanets, as described for the early solar nebula by Cameron (1978), are a potential site for achieving both mass and component fractionations, and for producing white inclusions in general.     

A Planar Case of the n + 1 Body Problem: the 'Ring’ Problem

Our intention in this article is to present a new model for the investigation of the motion of a particle of negligible mass in a multibody surrounding. The proposed general planar configuration consists of ν = n-1 primaries arranged in equal arcs on an ideal ring and a central body of different mass located at the centre of mass of the system. We formulate the general equations of motion and we study the stationary solutions and the zero-velocity contours for various values of ν. This revised version was published online in July 2006 with corrections to the Cover Date.     

Chandrasekhar’s Integral Equilibrium Theorems Modified in the Context of Non-Gaussian Kappa Statistics for a Spherically Symmetric Protostar Cloud

Solar System Research - A generalization of Chandrasekhar’s integral theorems on the equilibrium for matter and blackbody radiation in a protostellar gravitating spherically symmetric cloud...     

On the probable change of the radius and nature of aerosol particles in the deep layers of Jupiter’s atmosphere

When analyzing the pressure dependences of the aerosol volume scattering coefficient calculated from the measurement data on the geometric albedo of Jupiter obtained in 1993 in the methane absorption bands at 619, 727, and 842 nm, the signs of probable changes in the parameters of aerosol particles in the deep atmospheric layers were detected and the first estimates of the magnitude of these changes were obtained. It has been found that, in the pressure interval from 4 to 14 bar, the effective radius of particles may increase twofold and more (larger than 0.73 μm) and the real part of the refractive index may grow by 10% (from 1.44 and higher) relative to the values of these parameters in the upper atmosphere. If we take into account these changes, we find no signs of aerosol deep in the atmosphere of Jupiter.