On the character of evolution of orbits in the vicinity of the 1 : 3 Kirkwood gap

The character of orbital evolution for bodies moving near the if 1 : 3 commensurability with Jupiter was studied by model calculations for the time interval of ～500 years. A comparison of oscillations of the orbital elements a, e, q and q′ is made for ensembles of bodies along three starting orbits in the vicinity of the sharp commensurability with Jupiter. These orbits are eccentric ones of low inclinations having perihelia near the Earth's orbit. Examples of a deceleration of the rate of orbital evolution near the sharp commensurability are revealed. The existence of a group of asteroids connected with the Kirkwood gap, i.e., being in a resonant motion with Jupiter, is suggested. A connection of asteroids 887 Alinda and 1915 Quetzalcoatl with this gap is confirmed.     

Families of asymmetric periodic solutions of the restricted problem of three bodies for the Sun-Jupiter mass ratio and their relationship with the symmetric families

Message derived a method to detect bifurcations of a family of asymmetric periodic solutions from a family of symmetric periodic solutions in the restricted problem of three bodies for the limiting case when the second body has zero mass. This is used to examine several small integer commensurabilities. A total of 21 exterior and 21 interior small integer commensurabilities are examined and bifurcations (two in number) are found to exist only for exterior commensurabilities (q+1):1,q=1, 2,, 7. On investigating other commensurabilities of this form for values ofq up to 50 two bifurcations are still found to exist for each. The eccentricities of the two bifurcation orbits are given for eachq up to 20. For a Sun-Jupiter mass ratio the complete family of asymmetric periodic solutions associated withq=1, 2,..., 5, and the initial segments of the asymmetric family withq=6, 7,..., 12, have been numerically determined. The family associated withq=5 contains some unstable orbits but all orbits in the other four complete families are stable. The five complete families each begin and end on the same symmetric family. The network of asymmetric and symmetric families close to the commensurabilities (q+1):1,q=1, 2,..., 5 is discussed.     

Numerical exploration of the photogravitational restricted five-body problem

We study numerically the restricted five-body problem when some or all the primary bodies are sources of radiation. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points are given. We found that the number of the collinear equilibrium points of the problem depends on the mass parameter β and the radiation factors q _i , i=0,…,3. The stability of the equilibrium points are also studied. Critical masses associated with the number of the equilibrium points and their stability are given. The network of the families of simple symmetric periodic orbits, vertical critical periodic solutions and the corresponding bifurcation three-dimensional families when the mass parameter β and the radiation factors q _i vary are illustrated. Series, with respect to the mass (and to the radiation) parameter, of critical periodic orbits are calculated.     

Expansion of the perturbing function of a satellite restricted four-body problem

A satellite four-body problem is the problem of motion of an artificial satellite of a planet in a region of the space where perturbations due to the gravitational field of the planet are of the same order as perturbations due to influences of two perturbing bodies. In this paper an expansion of the perturbing function into a Fourier series in terms of angular Keplerian elements ( _j , _j ,M _j :j=0,1,2) (designations are standard) is obtained taking into account a sharp commensurability of the typen/ ₀=(p+q)/p (n is the mean motion of the artificial satellite and ₀ is the angular velocity of rotation of the planet,p andq are integers).The coefficients of the Fourier series are the functions of the positional Keplerian elements (a _j ,e _j ,i _j ;j=0, 1, 2) (designations are standard) and, in particular, are series in terms ofe _j that, generally speaking, can be written out to an accuracy ofe _j ¹⁹ .The expansion obtained can be used for the construction of a semianalytical theory of motion of resonant satellites on the basis of conditionally periodic solutions of the restricted four-body problem.     

Bifurcations of dust acoustic solitary waves and periodic waves in an unmagnetized plasma with nonextensive ions

Bifurcations of dust acoustic solitary waves and periodic waves in an unmagnetized plasma with q-nonextensive velocity distributed ions are studied through non-perturbative approach. Basic equations are reduced to an ordinary differential equation involving electrostatic potential. After that by applying the bifurcation theory of planar dynamical systems to this equation, we have proved the existence of solitary wave solutions and periodic wave solutions. Two exact solutions of the above waves are derived depending on the parameters. From the solitary wave solution and periodic wave solution, the effect of the parameter (q) is studied on characteristics of dust acoustic solitary waves and periodic waves. The parameter (q) significantly influence the characteristics of dust acoustic solitary and periodic structures.     

Periodic orbits around an oblate spheroid

The general properties of certain differential systems are used to prove the existence of periodic orbits for a particle around an oblate spheroid.In a fixed frame, there are periodic orbits only fori=0 andi near /2. Furthermore, the generating orbits are circles.In a rotating frame, there are three families of orbits: first a family of periodic orbits in the vicinity of the critical inclination; secondly a family of periodic orbits in the equatorial plane with 0<e<1; thirdly a family of periodic orbits for any value of the inclination ife=0.     

Effect of oblateness, perturbations, radiation and varying masses on the stability of equilibrium points in the restricted three-body problem

This paper investigates the combined effect of small perturbations ε,ε′ in the Coriolis and centrifugal forces, radiation pressure q _i , and changing oblateness of the primaries A _i (t) (i=1,2) on the stability of equilibrium points in the restricted three body problem in which the primaries is a supergiant eclipsing binary system which consists of a pair of bright oblate stars having the appearance of a giant peanut in space and their masses assumed to vary with time in the absence of reactive forces. The equations of motion are derived and the equilibrium points are obtained. For the autonomized system, it is seen that there are more than a pair of the triangular points as κ→∞; κ being the arbitrary sum of the masses of the primaries. In the case of the collinear points, two additional equilibrium points exist on the line joining the primaries when simultaneously κ+ε′<0 and both primaries are oblate, i.e., 0<α _i ?1. So there are five collinear equilibrium points in this case. Two non-planar equilibrium points exist for κ>1. Hence, there are at least nine equilibrium points of the system. The stability of these points is explored analytically and numerically. It is seen that the collinear and triangular points are stable with respect to certain conditions controlled by κ while the non-planar equilibrium points are unstable.     

Some special restricted four-body problems—II. From Caledonia to Copenhagen

Special analytical solutions are determined for restricted, coplanar, four-body equal mass problems, including the Caledonian problem, where the masses M_i = M for i = 1,2,3,4. Most of these solutions are shown to reduce to the Lagrange solutions of the Copenhagen problem of three bodies by reducing two of the masses (m_i = m for i = 1,2) in the four-body equal mass problem to zero while maintaining their equality of mass. In so doing, families of special solutions to the four-body problem are shown to exist for any value of the mass ratio μ = m/M.     

Intensity and polarization line profiles in a semi-infinite Rayleigh-scattering planetary atmosphere. III. Variation of polarization profiles and the Stokes parameter Q over the disk

The variation of the polarization profiles, the Stokes parameters Q andU, and the angle defining the plane of polarization along the intensity equator and along the mirror meridian, on whichμ = μ ₀, in a Rayleighscattering atmosphere is studied. It is found that these variations are more complex than thought hitherto, particularly at large phase angles.     

Periodic motions around a collinear equilibrium solution of the general three-body problem

The collinear equilibrium position of the circular restricted problem with the two primaries at unit distance and the massless body at the pointL ₃ is extended to the planar three-body problem with respect to the massm ₃ of the third body; the mass ratio μ of the two primaries is considered constant and the constant angular velocity of the straight line on which the three masses stay at rest is taken equal to 1. As regards periodic motions ‘around’ the equilibrium pointL ₃, four possible extensions from the restricted to the general problem are presented each of them starting with a simple or a doubly periodic orbit of the family α of the Copenhagen category (μ=0.50). Form ₃=0.10, μ=0.50 (i.e. for fixed masses of all three bodies) the characteristic curve of the extended family α is found. The qualitative differences of the families corresponding tom ₃=0 andm ₃=0.10 are discussed.     

Extended Schubart Averaging

The purpose of this paper is the presentation of an integrator for the average motion of an asteroid in mean motion commensurability with Jupiter. The program is valid for any (p+q)/p mean motion commensurability (except whenq=0) and uses a double precision version of DE (Shampine and Gordon 1975) as propagator. The averaged equations of motion of the asteroid are evaluated in a non-singular way for any value of the eccentricities and the inclinations and the orbit of Jupiter is described by the most important terms in Longstop 1B (Nobiliet al. 1989). This integrator can be considered as an extension of the well known Schubart Averaging (Schubart 1978) in which Jupiter is moving on a fixed ellipse.     

Partial separability of the Laplace equation in Roche coordinates

The Laplace equation in the coordinatesu, v, w is calledu-separable if there are solutions of the formF(u)G(v, w). If the surfacesv = constant andw = constant are orthogonal tou = constant the coordinate system is called semi-orthogonal. The Laplace equation is notu-separable for the rotation problem semi-orthogonal Roche coordinate system (n0, q=0) or the general problem (n0, q0) ifv andw are analytic functions ofn andq and the coordinate system is proper in some region of then, q plane including the origin,n=q=0 (u is the Roche potential).     

Sur les solutions Lagrangiennes du problème des trois corps solides avec la loi de Weber

In this work we consider the problem of translational-rotational motion of three solid bodies, for which the elementary particles attract each other according to different Weber's laws for each pair of bodies. This problem represents a special case of the generalized problem of three solids considered in a previous work, (Dubochin, 1974) and it gives an example of the verification of the existence conditions for the Lagrangian solutions. In these solutions, the centers of mass always for m an equilateral triangle. Each body has axial symmetry with the plane of symmetry perpendicular to the axis of symmetry rotates uniformly around this axis, which at any instant stays perpendicular to the plane of the triangle formed by the centers of mass. According to Weber's law (Tisserand, 1896) the elementary particles of two bodiesT _i andT _j (i, j=0, 1, 2) are attracted by forces which are proportional to the function $$F_{ij} (W) = \frac{{f_{ij} }}{{\Delta _{ij^2 } }}\left[ {1 - a_{ij} \dot \Delta _{ij^2 } + 2a_{ij} \Delta _{ij} \ddot \Delta _{ij} } \right]$$ wheref _ij anda _ij (in generalf _ji ≠f _ij anda _ji ≠a _ij ) are functions of the timet, and where the real quantities Δ_ij are the mutual distances between the particles of the bodiesT _i andT _j , and where \(\dot \Delta _{ij} \) and \(\ddot \Delta _{ij} \) are their derivatives with respect to the time. The analysis of the general conditions for the Lagrangian solutions gives the following results for the case of Weber's laws.

1. Only the invariant Lagrangian solutions, (the traingle of the centres of mass does not change in time) are possible in this problem.
2. Besides the conditions (NL) obtained in the case of the Newton-Coulomb law, (all thea _ij are zero), the complementary conditions (WL) must be satisfied.

In particular, if all the bodies are spheres or homogeneous ellipsoids, they must necessarily have the same dimensions, but they can have different masses.     

On stationary solutions of some canonical systems of differential equations

In our article (Zhuravlev, 1979) a formal method of constructing conditionally periodic solutions of canonical systems of differential equations with a quick-rotating phase in the case of sharp commensurability was presented. The existence of stationary (or periodic) solutions of an averaged system of differential equations corresponding to the initial system of differential equations is necessary for an effective application of the method for different problems.Evidently, the stationary solutions do not always exist but in numerous papers on stationary solutions (oscillations or motions), the conditions of existence of such solutions are very often not considered at all. Usually a simple assumption is used that the stationary solutions do exist.Otherwise it is well known that Poincaré's theory of periodic solutions (Poincaré, 1892) let one set up conditions of existence of periodic solutions in different systems of differential equations. Particularly, in papers,Mah (1949, 1956), see alsoexmah (1971), the necessary and sufficient conditions of the existence of periodic solutions of (non-canonical) systems of differential equations which are close to arbitrary non-linear systems are given. For canonical autonomous systems of differential equations the conditions of existence of periodic solutions and a method of calculation are presented in the paperMepmah (1952).In our paper another approach is given and the conditions of existence of stationary solutions of canonical systems of differential equations with a quick-rotating phase are proved. For this purpose Delaunay-Zeipel's transformation and Poincaré's small parameter method are used.     

Exospheric perturbations by radiation pressure: II. Solution for orbits in the ecliptic plane

An earlier paper gave solutions for the mean time rates of change of orbital elements of satellite atoms in an exosphere influenced by solar radiation pressure. Each element was assumet to beahve independently. Here the instantaneous rates of change for three elements $\text{(e, ω,}\text{and}\text{θ = ω + Ω)}$ are integrated simultaneously for the case of the inclination i = 0. The results (a) confirm the validity of using mean rates when the orbits are tightly bound to the planet and (b) serve as examples to be reproduced by the complicated numerical solutions required for arbitrary inclination. Strongly bound hydrogen atoms perturbed in Earth orbit by radiation pressure do not seem a likely cause of the geotail extending in the anti-Sun direction. Instead, radiation pressure wil cause those particles' orbits to form a broad fan-shaped tail and to deteriorate into the Earth's atmosphere. Whether loosely bound H atoms are plentiful enough to create the geotail depends on their source function versusr; that question is beyond the scope of this paper.     

Families of three-dimensional axisymmetric periodic orbits in the restricted three-body problem: Sun-Jupiter case

Families of three-dimensional axisymmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits (_v = 1,b _v = 0) of the basic plane familiesi andI. Further the predictor-corrector procedure employed to reveal these families has been described and interesting numerical results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.     

The photometric solution of W UMa-type star BW draconis

The photometric solutions of W UMa-type binary BW Dra have been determined by applying the Wilson and Devinney Code toUBV observations of Rucinski and Kaluzny.It is shown that BW Dra is corresponding to a system with an overcontact configuration and smaller mass ratioq=0.392 andUBV light curves give the converging solutions with non-zero third light.It is proved that the components of BW Dra are older stars (the spectral types are G0 and G3, respectively). They could come into contact later stage of evolution. The photometric solution is similar to the results of Kaluzny and Rucinski. According to the photometric solution and spectroscopic results of Batten and Lu, the absolute parameters are presented too.     

NSVS 7051868: A system in a key evolutionary stage. First multi-color photometric study

The first CCD photometric complete light curves of the eclipsing binary NSVS 7051868 were obtained during six nights in January 2016 in the B, V and I_c bands using the 0.25 m telescope of the Stazione Astronomica Betelgeuse in Magnago, Italy.These observations confirm the short period (P = 0.517 days) variation found by Shaw and collaborators in their online list (http://www.physast.uga.edu/~jss/nsvs/) of periodic variable stars found in the Northern Sky Variability Survey.The light curves were modelled using the Wilson–Devinney code and the elements obtained from this analysis are used to compute the physical parameters of the system in order to study its evolutionary status.A grid of solutions for several fixed values of mass ratio was calculated.A reasonable fit of the synthetic light curves of the data indicate that NSVS 7051868 is an A-subtype W Ursae Majoris contact binary system, with a low mass ratio of q = 0.22, a degree of contact factor f = 35.5% and inclination i = 85°. Our light curves shows a time of constant light in the secondary eclipse of approximately 0.1 in phase. The light curve solution reveals a component temperature difference of about 700 K. Both the value of the fill-out factor and the temperature difference suggests that NSVS 7051868 is a system in a key evolutionary stage of the Thermal Relaxation Oscillation theory.The distance to NSVS 7051868 was calculated as 180 pc from this analysis, taking into account interstellar extinction.