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1.
We study planar central configurations of the five-body problem where three of the bodies are collinear, forming an Euler
central configuration of the three-body problem, and the two other bodies together with the collinear configuration are in
the same plane. The problem considered here assumes certain symmetries. From the three bodies in the collinear configuration,
the two bodies at the extremities have equal masses and the third one is at the middle point between the two. The fourth and
fifth bodies are placed in a symmetric way: either with respect to the line containing the three bodies, or with respect to
the middle body in the collinear configuration, or with respect to the perpendicular bisector of the segment containing the
three bodies. The possible stacked five-body central configurations satisfying these types of symmetries are: a rhombus with
four masses at the vertices and a fifth mass in the center, and a trapezoid with four masses at the vertices and a fifth mass
at the midpoint of one of the parallel sides. 相似文献
2.
In the n-body problem a central configuration is formed if the position vector of each particle with respect to the center
of mass is a common scalar multiple of its acceleration vector. We consider the problem: given a collinear configuration of
four bodies, under what conditions is it possible to choose positive masses which make it central. We know it is always possible
to choose three positive masses such that the given three positions with the masses form a central configuration. However
for an arbitrary configuration of four bodies, it is not always possible to find positive masses forming a central configuration.
In this paper, we establish an expression of four masses depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically we show that there is a compact region
in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of
the compact region, it is always possible to choose positive masses to make the configuration central. 相似文献
3.
Josep M. Cors Jaume Llibre Merce Ollé 《Celestial Mechanics and Dynamical Astronomy》2004,89(4):319-342
We study the planar central configurations of the 1 +n body problem where one mass is large and the other n masses are infinitesimal and equal. We find analytically all these central configurations when 2≤n≤4. Numerically, first we provide evidence that when n9 the only central configuration is the regular n-gon with the large mass in its barycenter, and second we provide also evidence of the existence of an axis of symmetry for
every central configuration.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
4.
New stacked central configurations for the planar 5-body problem 总被引：1，自引：0，他引：1
Jaume Llibre Luis Fernando Mello Ernesto Perez-Chavela 《Celestial Mechanics and Dynamical Astronomy》2011,110(1):43-52
A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m
1, m
2, m
3 (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m
4, m
5 are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by
{1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem. 相似文献
5.
We consider the problem of 4 bodies of equal masses in R
3 for the Newtonian r−1 potential. We address the question of the absolute minima of the action integral among (anti)symmetric loops of class H
1 whose period is fixed. It is the simplest case for which the results of [4] (corrected in [5]) do not apply: the minima cannot
be the relative equilibria whose configuration is an absolute minimum of the potential among the configurations having a given
moment of inertia with respect to their center of mass. This is because the regular tetrahedron cannot have a relative equilibrium
motion in R
3 (see [2]). We show that the absolute minima of the action are not homographic motions. We also show that if we force the
configuration to admit a certain type of symmetry of order 4, the absolute minimum is a collisionless orbit whose configuration
‘hesitates’ between the central configuration of the square and the one of the tetrahedron. We call these orbits ‘hip-hop’.
A similar result holds in case of a symmetry of order 3 where the central configuration of the equilateral triangle with a
body at the center of mass replaces the square.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
Alan Almeida santos 《Celestial Mechanics and Dynamical Astronomy》2004,90(3-4):213-238
We consider some questions on central configurations of five bodies in space. In the first one, we get a general result of
symmetry for the restricted problem of n+1 bodies in dimension n-1. After that, we made the calculation of all c.c. for n=4.
In our second result, we extend a theorem of symmetry due to [Albouy, A. and Libre, I.: 2002, Contemporary Math.
292, 1-16] on non-convex central configurations with 4 unit masses and an infinite central mass. We obtain similar results in
the case of a big, but finite central mass. Finally, we continue the study by [Schmidt, D.S.: 1988, Contemporary Math.
81 ] of the bifurcations of the configuration with four unit masses located at the vertices of a equilateral tetrahedron and
a variable mass at the barycenter. Using Liapunov-Schmidt reduction and a result on bifurcation equations, which appear in
[Golubitsley, M., Stewart, L. and Schaeffer, D.: 1988, Singularties and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, New York], we show that there exist indeed seven families of central configurations close to a
regular tetrahedron parameterized by the value of central mass. 相似文献
7.
Allyson Oliveira 《Celestial Mechanics and Dynamical Astronomy》2013,116(1):11-20
In this work we are interested in the central configurations of the planar $1+4$ body problem where the satellites have different infinitesimal masses and two of them are diametrically opposite in a circle. We can think of this problem as a stacked central configuration too. We show that the configurations are necessarily symmetric and the other satellites have the same mass. Moreover we prove that the number of central configurations in this case is in general one, two or three and, in the special case where the satellites diametrically opposite have the same mass, we prove that the number of central configurations is one or two and give the exact value of the ratio of the masses that provides this bifurcation. 相似文献
8.
M. Hénon 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):375-388
We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated. 相似文献
9.
The classical problem of the critical inclination in artificial satellite theory has been extended to the case when a satellite may have an arbitrary, significant mass and the rotation momentum vector is tilted with respect to the symmetry axis of the planet. If the planet’s potential is restricted to the second zonal harmonic, according to the assumptions of the main problem of the satellite theory, two various phenomena can be observed: a critical inclination that asymptotically tends to the well known negligible mass limit, and a critical tilt that can be attributed to the effect of transforming the gravity field harmonics to a different reference frame. Stability of this particular solution of the two rigid bodies problem is studied analytically using a simple pendulum approximation. 相似文献
10.
R. Broucke J. D. Anderson L. Blitzer E. Davoust H. Lass 《Celestial Mechanics and Dynamical Astronomy》1981,24(1):63-82
The article describes the solutions near Lagrange's circular collinear configuration in the planar problem of three bodies with three finite masses. The article begins with a detailed review of the properties of Lagrange's collinear solution. Lagrange's quintic equation is derived and several expressions are given for the angular velocity of the rotating frame.The equations of motion are then linearized near the circular collinear solution, and the characteristic equation is also derived in detail. The different types of roots and their corresponding solutions are discussed. The special case of two equal outer masses receives special attention, as well as the special case of two small outer masses.Finally, the fundamental family of periodic solutions is extended by numerical integration all the wap up to and past a binary collision orbit. The stability and the bifurcations of this family are briefly enumerated. 相似文献
11.
C. Beaugé 《Celestial Mechanics and Dynamical Astronomy》2004,88(1):51-68
In this communication we present an analytical model for the restricted three-body problem, in the case where the perturber is in a parabolic orbit with respect to the central mass. The equations of motion are derived explicitly using the so-called Global Expansion of the disturbing function, and are valid for any eccentricity of the massless body, as well as in the case where both secondary masses have crossing orbits. Integrating the equations of motion over the complete passage of the perturber through the system, we are then able to construct a first-order algebraic mapping for the change in semimajor axis, eccentricity and inclination of the perturbed body.Comparisons with numerical solutions of the exact equations show that the map yields precise results, as long as the minimum distance between both bodies is not too small. Finally, we discuss several possible applications of this model, including the evolution of asteroidal satellites due to background bodies, and simulations of passing stars on extra-solar planets. 相似文献
12.
Montserrat Corbera Josep Maria Cors Jaume Llibre 《Celestial Mechanics and Dynamical Astronomy》2011,109(1):27-43
We consider the Newtonian four-body problem in the plane with a dominat mass M. We study the planar central configurations of this problem when the remaining masses are infinitesimal. We obtain two different
classes of central configurations depending on the mutual distances between the infinitesimal masses. Both classes exhibit
symmetric and non-symmetric configurations. And when two infinitesimal masses are equal, with the help of extended precision
arithmetics, we provide evidence that the number of central configurations varies from five to seven. 相似文献
13.
David A. Minton 《Icarus》2008,195(2):698-704
Rubble pile asteroids can attain shapes that are dramatically different from those of rotating, self-gravitating equilibrium fluids. A new numerical technique, called “seed growth,” is demonstrated for calculating three-dimensional bodies that are self-gravitating and rotating, and whose every surface is approximately at a constant angle, ?, with respect to the local horizontal. By altering the configuration of cusps, which are points along a constant longitude path where the surface angle changes sign but not magnitude, multiple solution shapes that satisfy the condition that all surface slopes are at a constant angle are possible. Five different cusp configurations are explored here, three of which yield solutions for 20°???30°. Rotational effects are explored, and it is found that for some solution shapes, the ratios of their shortest to longest dimensions, c/a, can fall outside the limits published in the literature for rotating, cohesionless, spheroidal bodies. Solution shapes show some similarities to observed small bodies, such as the saturnian satellite Atlas, the near-Earth Asteroid 1999 KW4, and some contact binary asteroids. 相似文献
14.
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. 相似文献
15.
In the following paper we argue that each wind-driving star in relative motion with respect to the ambient interstellar medium experiences a force exerted on its central wind-generating body. The exact magnitude of this force depends on the actual geometry of the counterflow configuration of stellar and interstellar winds for a particular kinematic situation which is especially sensitive to whether the interstellar flow is subsonic or supersonic. It will, however, be demonstrated here that this force is of an accelerating nature, i.e., it operates like a rocket-motor, as long as the peculiar motion of the wind-driving star with respect to the ambient interstellar medium remains subsonic.Here we use a specific analytical model to describe theoretically the specific counterflow configuration for the case of the solar system in a subsonic peculiar motion with respect to the local interstellar medium assuming irrotational and incompressible flows. We can work out a quantitative number for the accelerating force governing the Sun's motion at present. The net reaction force exerted on the solar body is then mediated by the asymmetric boundary conditions to which the distant solar wind field has to adapt.Next we study the indirect action of such a force on orbiting Keplerian objects like planets, planetesimals and comets. Since this force only influences the central solar body, but not the planets themselves, the problem is different from the treatment of a constant perturbation force perturbing the Keplerian orbits. We present a perturbation analysis treating the action of a corresponding position-dependent perturbation force resulting in secular changes of the orbital elements of Keplerian objects. It is found that changes are accumulating more rapidly in time the closer to the sun the orbiting bodies are. Main axis and perihelion distances are systematically increasing. Especially pronounced are changes in the perihelion position angle of the objects. For solar wind mass losses larger than the Sun's present value by a factor of 1000 (T-Tauri phase of the Sun,) the migration periods calculated for the planet Mercury are of the same order of magnitude as that for corresponding general relativistic migration. 相似文献
16.
The problem triaxial satellite having a plane of dynamical symmetry in the restricted problem of three bodies has been studied. The first integrals are established and the general solution of the problem can be written in quadratures. The results show that the semi-major axis of the satellite orbit and its rotational angular momentum remain unchanged. The singular solution of this problem has been considered and the elements of satellite orbit can be determined. 相似文献
17.
Stability of interplay motions 总被引：2，自引：2，他引：0
M. Hénon 《Celestial Mechanics and Dynamical Astronomy》1977,15(2):243-261
A family of rectilinear periodic solutions of the three-body problem, in which the central body collides alternately with each of the two other bodies, is investigated numerically for all values of the three masses. It is found that for every mass combination there exists just one solution of this kind. The linear stability of the orbits with respect to arbitrary three-dimensional perturbations is also investigated. Domains of stability and instability are displayed in a triangular mass diagram. Their boundaries form one-parameter families of critical orbits, which are tabulated. Limiting cases where one or two masses vanish are studied in detail. The domains of stability cover nearly one half of the total area in the mass diagram: this reinforces the conclusion that real triple stars might have motions of a kind entirely different from the usual hierarchical arrangement. 相似文献
18.
Paolo Lanzano 《Astrophysics and Space Science》1968,1(1):92-111
The motion of two rotating spheroidal bodies, constituting the components of a binary system in a weak gravitational field, has been considered up to terms of the second order in the small parameterV/c, whereV denotes the velocity of the bodies andc is the velocity of light.The following simplifying assumptions, consistent with a problem of astronomical interest, have been made: (1) the dimensions of the bodies are small compared with their mutual distance; (2) the bodies consist of matter in the fluid state with internal hydrostatic pressure and their oblateness is due to their own rotation; (3) there exist axial symmetry about the axis of rotation and symmetry with respect to the equatorial plane, the same symmetry properties apply to mass densities and stress tensors.The Fock-Papapetrou method was used to ascertain those terms in the equations of motion which are due to the rotation and to the oblateness of each component. Approximate solutions to the Poisson and wave equations were obtained to express the potential and retarded potential at large distances from the bodies generating them. The explicit evaluation of certain integrals has necessitated the use of the Laplace-Clairaut theory for the equibrium configuration of rotating bodies. The final expressions require the knowledge of the mass density as a function of the mean radius of the equipotential surfaces.As an interpretation of the results, the Lagrangian perturbation equations were employed to evaluate the secular motion of the nodal line for the relative orbit of the two components. The results constitute a generalization of Fock's work and furnish the contribution of the mass distribution to the rotation effect of general relativity. 相似文献
19.
The photo-gravitational problems of two or more bodies have attracted much attention during the last decades. In this paper, radiation is considered as an additional factor influencing the particle motion in a regular polygon formation of N big bodies where the ν = Ν ? 1 primary bodies have equal masses and are located at the vertices of a regular polygon and the Nth primary has different mass and is located at the mass center of the system. We assume that some or all the primary bodies are radiation sources and we numerically explore various cases where symmetry of the resultant force field with respect to the same axis is preserved. For the purposes of our investigation we adopt Radzievski’s theory and assumptions. The material gathered helps us to estimate the radiation effect on the evolution of periodic orbits and their characteristics, such as their periods and their stability. Figures and diagrams illustrate these alterations and document our conclusions. 相似文献
20.
It is well-known that the optical pulsations in DQ Her are due to emission from the magnetic poles of the white dwarf. As
the white dwarf spins on its axis, the magnetic poles sweep into and out of the line of sight due to the fact that the magnetic
axis and the spin axis are not aligned, that is, the DQ Her white dwarf is an `oblique rotator'. So, a central question is
if an initially axisymmetric model simulating the DQ Her white dwarf before its `turn-over' (where the term `turn-over' describes
the process by which the magnetic axis gets inclining relative to the spin axis at a progressively increasing angle, the so-called
`turn-over angle') is indeed susceptible to turn-over. For the puprose of resolving this problem, we compute several axisymmetric
models of the DQ Her white dwarf. Our results show that, for both the rotation periods proposed on the basis of the observational
evidence regarding the optical pulsations of DQ Her (i.e.,71 s or 142 s), the moment of inertia along the rotation axis is
less than the corresponding moment of inertia along the remaining two principal axes of the axisymmetric configuration, I
33 > I
11(=I
22). This is because toroidal magnetic field (tending to derive prolate equidensity surfaces) dominates over rotation (tending,
in turn, to derive oblate equidensity surfaces), mainly in the interior of the star. The situation I
11 < I
33 is known as `dynamical asymmetry', and can cause a turn-over of the magnetic symmetry axis with respect to the rotation axis,
eventually deriving a nonaxisymmetric configuration corresponding to the so-called `perpendicular rotator' with turn-over
angle almost equal to 90°. In this view, our results explain why the DQ Her white dwarf is now an oblique rotator.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献