A new solution to the lagrange's case of the three‐body problem

The motion of three particles, interacting by gravitational forces, is studied in a new coordinate system given by the principal axes of inertia, as determined by Euler angles, and using the inertia principal moments and an auxiliar angle as coordinates. The solution to the particular Lagrange case of the three‐body problem is reviewed and solved in these new coordinates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Nondestructive determination of the physical properties of Antarctic meteorites: Importance for the meteorite—parent body connection

Photogrammetry is a low-cost, nondestructive approach for producing 3-D models of meteorites for the purpose of determining sample bulk density. Coupled with the use of a nondestructive magnetic susceptibility/electrical conductivity field probe, we present measurements for the interrogation of several physical properties, on a set of Antarctic meteorites. Photogrammetry is an effective technique over a range of sample sizes, with meteorite bulk density results that are closely comparable with literature values, determined using Archimedean glass bead or laser scanning techniques. The technique is completely noncontaminating and suitable for the analysis of rare or fragile samples, although there are limitations for analyzing reflective samples. It is also flexible, and, with variations in equipment setup, may be appropriate for samples of a wide range of sizes. X-ray computed tomography analyses of the same meteorite samples yielded slightly different bulk density results, predominantly for samples below 10 g, although the reason for this is unclear. Such analyses are expensive and potentially damaging to certain features of the sample (e.g., organic compounds), but may be useful in expanding the measurements to accommodate an understanding of internal voids within the sample, lending itself to measurement of grain density. Measurements of bulk density are valuable for comparisons with estimates of the bulk densities of asteroids that are suggested as meteorite parent bodies.     

Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem

A new fully numerical method is presented which employs multiple Poincaré sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory required to store these orbits. This method reduces the calculations required for searching two-dimensional invariant tori to a search for closed orbits, which are the intersection of the invariant tori with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant tori is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme utilizes the invariance of the circles of the maps on these Poincaré sections in order to find the Fourier coefficients that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasiperiodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-halo and Lissajous families of the Sun–Earth RTBP around the L2 libration point are obtained via this method. Results are compared with the existing literature. A numerical method to transform these orbits from the RTBP model to the real ephemeris model of the Solar System is introduced and applied.     

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.     

The effect of radiation pressure on the equilibrium points in the generalized photogravitational restricted three body problem

The existence of equilibrium points and the effect of radiation pressure have been discussed numerically. The problem is generalized by considering bigger primary as a source of radiation and small primary as an oblate spheroid. We have also discussed the Poynting-Robertson (P-R) effect which is caused due to radiation pressure. It is found that the collinear points L ₁,L ₂,L ₃ deviate from the axis joining the two primaries, while the triangular points L ₄,L ₅ are not symmetrical due to radiation pressure. We have seen that L ₁,L ₂,L ₃ are linearly unstable while L ₄,L ₅ are conditionally stable in the sense of Lyapunov when P-R effect is not considered. We have found that the effect of radiation pressure reduces the linear stability zones while P-R effect induces an instability in the sense of Lyapunov.     

Periodic motions of a small body in the Newtonian field of a regular polygonal configuration of ν+1 bodies

We study the simple periodic orbits of a particle that is subject to the gravitational action of the much bigger primary bodies which form a regular polygonal configuration of (ν+1) bodies when ν=8. We investigate the distribution of the characteristic curves of the families and their evolution in the phase space of the initial conditions, we describe various types of simple periodic orbits and we study their linear stability. Plots and tables illustrate the obtained material and reveal many interesting aspects regarding particle dynamics in such a multi-body system.     

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.     

Instabilities in the Sun–Jupiter–Asteroid three body problem

We consider dynamics of a Sun–Jupiter–Asteroid system, and, under some simplifying assumptions, show the existence of instabilities in the motions of an asteroid. In particular, we show that an asteroid whose initial orbit is far from the orbit of Mars can be gradually perturbed into one that crosses Mars’ orbit. Properly formulated, the motion of the asteroid can be described as a Hamiltonian system with two degrees of freedom, with the dynamics restricted to a “large” open region of the phase space reduced to an exact area preserving map. Instabilities arise in regions where the map has no invariant curves. The method of MacKay and Percival is used to explicitly rule out the existence of these curves, and results of Mather abstractly guarantee the existence of diffusing orbits. We emphasize that finding such diffusing orbits numerically is quite difficult, and is outside the scope of this paper.     

Tides in a body librating about a spin–orbit resonance: generalisation of the Darwin–Kaula theory

The Darwin-Kaula theory of bodily tides is intended for celestial bodies rotating without libration. We demonstrate that this theory, in its customary form, is inapplicable to a librating body. Specifically, in the presence of libration in longitude, the actual spectrum of Fourier tidal modes differs from the conventional spectrum rendered by the Darwin–Kaula theory for a nonlibrating celestial object. This necessitates derivation of formulae for the tidal torque and the tidal heating rate, that are applicable under libration. We derive the tidal spectrum for longitudinal forced libration with one and two main frequencies, generalisation to more main frequencies being straightforward. (By main frequencies we understand those emerging due to the triaxiality of the librating body.) Separately, we consider a case of free libration at one frequency (once again, generalisation to more frequencies being straightforward). We also calculate the tidal torque. This torque provides correction to the triaxiality-caused physical libration. Our theory is not self-consistent: we assume that the tidal torque is much smaller than the permanent-triaxiality-caused torque, so the additional libration due to tides is much weaker than the main libration due to the permanent triaxiality. Finally, we calculate the tidal dissipation rate in a body experiencing forced libration at the main mode, or free libration at one frequency, or superimposed forced and free librations.     

Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body

In this paper, we have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when more massive body is a source of radiation and the smaller primary is an oblate body. We have determined periodic orbits for fixed values of μ, σ and different values of p and h (μ mass ratio of the two primaries, σ oblate parameter, p radiation parameter and h energy constant). These orbits have been determined by giving displacements along the tangent and normal to the mobile co-ordinates as defined by Karimov and Sokolsky (in Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of radiation pressure on the periodic orbits by taking some fixed values of μ and σ.     

Oscillatory orbits in the planar three‐body problem with equal masses

In the free‐fall three‐body problem, distributions of escape, binary, and triple collision orbits are obtained. Interpretation of the results leads us to the existence of oscillatory orbits in the planar three‐body problem with equal masses. A scenario to prove their existence is described. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Spectrophotometric determination of the temperature of the cepheid ξ Geminorum

The absolute energy distributions of the cepheid Gem at several phases of the light cycle have been given. By matching these with suitable model atmospheres, the effective temperatures of the star at these phases have been determined. The radius and effective gravity variations as well as the mass have been derived.     

Determination of the height of the “meteoric explosion”

When cosmic bodies of asteroidal and cometary origin, with a size from 20 to approximately 100 m, enter dense atmospheric layers, they are destroyed with a large probability under the action of aerodynamic forces and decelerated with the transfer of their energy to the air at heights from 20–30 to several kilometers. The forming shock wave reaches the Earth’s surface and can cause considerable damage at great distances from the entry path similar to the action of a high-altitude explosion. We have performed a numerical simulation of the disruption (with allowance for evaporation of fragments) and deceleration of meteoroids having the aforesaid dimensions and entering the Earth’s atmosphere at different angles and determined the height of the equivalent explosion point generating the same shock wave as the fall of a cosmic body with the given parameters. It turns out that this height does not depend on the velocity of the body and is approximately equal to the height at which this velocity is reduced by half. The obtained results were successfully approximated by a simple analytical formula allowing one to easily determine the height of an equivalent explosion depending on the dimensions of the body, its density, and angle of entry into the atmosphere. A comparison of the obtained results with well-known approximate analytical (pancake) models is presented and an application of the obtained formula to specific events, in particular, to the fall of the Chelyabinsk meteorite on February 15, 2013, and Tunguska event of 1908, is discussed.     

Characteristic actionsħ (s) in the structure of the universe and the problem of the origin of galaxies

From the characteristic actions ^(s), we can derive various relations between the basic characteristic quantities of objects and the fundamental constants in known physical laws. The main physical processes which lead to the formation of objects should be included in some such reletions through the fundamental constants. The problem of the origin of galaxies has been considered on the basis of the theory of actions ^(s). It has been shown that in addition to gravitational effect, the dissipation process of the adiabatic density perturbations arising from the Thomson scattering in the early universe is a crucial process in forming galaxies; and if the Hubble constant has a valueH ₀ 50 km s^–1 Mpc^–1, the protogalaxies might be formed just before recombination.     

Robe’s restricted problem of 2+2 bodies when the bigger primary is a Roche ellipsoid and the smaller primary is an oblate body

In this problem, one of the primaries of mass m ₁ is a Roche ellipsoid filled with a homogeneous incompressible fluid of density ρ ₁. The smaller primary of mass m ₂ is an oblate body outside the Ellipsoid. 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 Ellipsoid, 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 have extended the Robe’s restricted three-body problem to 2+2 body problem under the assumption that the fluid body assumes the shape of the Roche ellipsoid (Chandrashekhar in Ellipsoidal figures of equilibrium, Chap. 8, Dover, New York, 1987). We have taken into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid (ii) that originating in the attraction of m ₂ (iii) that arising from the centrifugal force. In this paper, equilibrium solutions of m ₃ and m ₄ and their linear stability are analyzed. We have proved that there exist only six equilibrium solutions of the system, provided they lie within the Roche ellipsoid. In a system where the primaries are considered as Earth-Moon and m ₃,m ₄ as submarines, the equilibrium solutions of m ₃ and m ₄ respectively when the displacement is given in the direction of x ₁-axis or x ₂-axis are unstable.     

A theoretical and empirical study of the response of the high latitude thermosphere to the sense of the “Y” component of the interplanetary magnetic field

The strength and direction of the Interplanetary Magnetic Field (IMF) controls the transfer of solar wind momentum and energy to the high latitude thermosphere in a direct fashion. The sense of “ Y” component of the IMF (BY) creates a significant asymmetry of the magnetospheric convection pattern as mapped onto the high latitude thermosphere and ionosphere. The resulting response of the polar thermospheric winds during periods when BY is either positive or negative is quite distinct, with pronounced changes in the relative strength of thermospheric winds in the dusk-dawn parts of the polar cap and in the dawn part of the auroral oval. In a study of four periods when there was a clear signature of BY, observed by the ISEE-3 satellite, with observations of polar winds and electric fields from the Dynamics Explorer-2 satellite and with wind observations by a ground-based Fabry-Perot interferometer located in Kiruna, Northern Sweden, it is possible to explain features of the high latitude thermospheric circulation using three dimensional global models including BY dependent, asymmetric, polar convection fields. Ground-based Fabry-Perot interferometers often observe anomalously low zonal wind velocities in the (Northern) dawn auroral oval during periods of extremely high geomagnetic activity when BY is positive. Conversely, for BY negative, there is an early transition from westward to southward and eastward winds in the evening auroral oval (excluding the effects of auroral substorms), and extremely large eastward (sunward) winds may be driven in the auroral oval after magnetic midnight. These observations are matched by the observation of strong anti-sunward polar-cap wind jets from the DE-2 satellite, on the dusk side with BY negative, and on the dawn side with BY positive.     

A Monte Carlo Study of the Evolution of the Scale Height of Normal Pulsars in the Galaxy

Based on the undisturbed, finite thickness disk gravitational potential, we carried out 3-D Monte Carlo simulations of normal pulsars. We find that their scale height evolves in a similar way for different velocity dispersions (σv): it first increases linearly with time, reaches a peak, then gradually decreases, and finally approaches a stable asymptotic value. The initial velocity dispersion has a very large influence on the scale height. The time evolution of the scale height is studied. When the magnetic decay age is used as the time variable, the observed scale height has a similar trend as the simulated results in the linear stage, from which we derive velocity dispersions in the range 70- 178km s-1, which are near the statistical result of 90 - 270km s-1 for 92 pulsars with known transverse velocities. If the characteristic age is used as the time variable, then the observed and theoretical curves roughly agree for t > 108 yr only if σv < 25km s-1.     

Effect of the Earth’s compression on the physical libration of the Moon

The effect of the Earth??s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 10⁶ than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon??s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon??s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth??s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth??s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L _? = 90° or L _? = 270°. The famous Cassini??s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth??s compression on the Moon??s libration in longitude, a harmonic with an amplitude of 0.03?? and period of T ₈ = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth??s compression on the variation in spin angular velocity of the Moon proves to be negligible.     

On the relationship of 6300 Å airglow to parameters of the F-region of the ionosphere

It is shown that variations in 6300 Å airglow intensities can, under certain assumptions, be simply related to $\text{?}{}_0\text{F2}$ and its time derivative. In deriving the relationship it is not necessary to assume that the concentration of the neutral atmosphere remains constant and so the relationship is useful on occasions when changes in the neutral atmosphere do occur making it difficult to obtain agreement between observed and calculated 6300 Å intensities; An example is given of a night in which a post-midnight enhancement occurred in the airglow and for which the observations could not be reproduced using a neutral atmosphere constant with time. It is shown that the airglow variations can be explained in terms of the variations of f₀F2, implying that the airglow is due to recombination and that, during the night, changes occurred in the concentrations of the constituents of the neutral atmosphere.