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1.
The differential equations of motion of the elliptic restricted problem of three bodies with decreasing mass are derived. The mass of the infinitesimal body varies with time. We have applied Jeans' law and the space-time transformation of Meshcherskii. In this problem the space-time transformation is applicable only in the special case whenn=1,k=0,q=1/2. We have applied Nechvile's transformation for the elliptic problem. We find that the equations of motion of our problem differ from that of constant mass only by a small perturbing force.  相似文献   

2.
The differential equations of motion of the elliptic restricted problem of three bodies, an infinitesimal spherical body with decreasing mass and two tri-axial bodies are derived. We have applied Jeans's law and the space-time of Meshcherskii in the special case whenn=1,k=0,q=1/2. Also Nechvíle's transformation for the elliptic problem be applied for this case.  相似文献   

3.
This paper investigates the triangular libration points in the photogravitational restricted three-body problem of variable mass, in which both the attracting bodies are radiating as well and the infinitesimal body vary its mass with time according to Jeans’ law. Firstly, applying the space-time transformation of Meshcherskii in the special case when q=1/2, k=0, n=1, the differential equations of motion of the problem are given. Secondly, in analogy to corresponding problem with constant mass, the positions of analogous triangular libration points are obtained, and the fact that these triangular libration points cease to be classical ones when α≠0, but turn to classical L 4 and L 5 naturally when α=0 is pointed out. Lastly, introducing the space-time inverse transformation of Meshcherskii, the linear stability of triangular libration points is tested when α>0. It is seen that the motion around the triangular libration points become unstable in general when the problem with constant mass evolves into the problem with decreasing mass.  相似文献   

4.
The circular restricted three-body problem, where two primaries are taken as heterogeneous oblate spheroid with three layers of different densities and infinitesimal body varies its mass according to the Jeans law, has been studied. The system of equations of motion have been evaluated by using the Jeans law and hence the Jacobi integral has been determined. With the help of system of equations of motion, we have plotted the equilibrium points in different planes (in-plane and out-of planes), zero velocity curves, regions of possible motion, surfaces (zero-velocity surfaces with projections and Poincaré surfaces of section) and the basins of convergence with the variation of mass parameter. Finally, we have examined the stability of the equilibrium points with the help of Meshcherskii space–time inverse transformation of the above said model and revealed that all the equilibrium points are unstable.  相似文献   

5.
A short derivation is given of the regularized equations of motion for the perturbed two-body problem. This method is then applied to the slightly modified time transformation dt/ds=r/ω.  相似文献   

6.
This article deals with the region of motion in the Sitnikov four-body problem where three bodies (called primaries) of equal masses fixed at the vertices of an equilateral triangle. Fourth mass which is finite confined to moves only along a line perpendicular to the instantaneous plane of the motions of the primaries. Contrary to the Sitnikov problem with one massless body the primaries are moving in non-Keplerian orbits about their centre of mass. It is investigated that for very small range of energy h the motion is possible only in small region of phase space. Condition of bounded motions has been derived. We have explored the structure of phase space with the help of properly chosen surfaces of section. Poincarè surfaces of section for the energy range ?0.480≤h≤?0.345 have been computed. We have chosen the plane (q 1,p 1) as surface of section, with q 1 is the distance of a primary from the centre of mass. We plot the respective points when the fourth body crosses the plane q 2=0. For low energy the central fixed point is stable but for higher value of energy splits in to an unstable and two stable fixed points. The central unstable fixed point once again splits for higher energy into a stable and three unstable fixed points. It is found that at h=?0.345 the whole phase space is filled with chaotic orbits.  相似文献   

7.
The problem of stability of the Lagrangian pointL 4 in the circular restricted problem of three bodies is investigated close to the 1 : 2 commensurability of the long and short period libration. By stability we define boundedness of the solution for a given initial finite displacement from the equilibrium point as function of the mass parameter close to the commensurability. A rigorous treatment close to the resonance condition is possible using a transformation that diagonalizes the matrix related to the linear part of the equations of motion. The so obtained equations are further transformed to action angle type variables. Then using an isolated resonance approach, only the slowly varying terms are kept in the equations and two independent isolating first integrals can be found. These integrals finally enable us to solve the stability problem in an exact way. The so obtained results are compared to numeric integration of the equations of motion and are found to be in perfect agreement.  相似文献   

8.
This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L 1, L 2 and L 3 correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations.  相似文献   

9.
We have analyzed the motion of an infinitesimal mass in the restricted four-body problem with solar wind drag. It is assumed that the forces which govern the motion are mutual gravitational attractions of the primaries, radiation pressure force and solar wind drag. We have derived the equations of motion and found the Jacobi integral, zero velocity surfaces, and particular solutions of the system. It is found that three collinear points are real when the radiation factor 0<β<0.1 whereas only one real point is obtained when 0.125<β<0.2. The stability property of the system is examined with the help of Poincaré surface of section (PSS) and Lyapunov characteristic exponents (LCEs). It is found that in presence of drag forces LCE is negative for a specific initial condition, hence the corresponding trajectory is regular whereas regular islands in the PSS are expanded.  相似文献   

10.
We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N ? 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z 0 values, where z 0 lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N ? 1 primary masses.  相似文献   

11.
Various families of periodic solutions are shown to exist in the three body problem, in which two of the bodies are close to a commensurability in mean motions about the third body, the primary, which is considerably more massive than the other two. The cases considered are
  1. The non-planar circular restricted problem (in which one of the secondary bodies has zero mass, and the other moves in a fixed circular orbit about the primary).
  2. The planar non-restricted problem (in which the three bodies move in a plane, and both secondaries have finite mass).
  3. The planar elliptical restricted problem (in which the three bodies move in a plane, one of the secondary bodies has zero mass, and the other moves in a fixed elliptical orbit about the primary).
The method used is to eliminate all short period terms from the Hamiltonian of the motion by means of a von Zeipel transformation, leaving only the long period terms which are due to the commensurability. Hence only the long period part of the motion is considered, and the variables used differ from the variables describing the full motion by a series of short-period trigonometric terms of the order of the ratio of the mass of the secondaries to that of the primary body. It is shown that solutions of the long-period problem in which the variables remain constant are equivalent to solutions in the full motion in which the bodies periodically return to the same configuration, and these are the types of periodic solution that are shown to exist. The form of the disturbing function, and hence of the equations of motion, is found up to the fourth powers of the eccentricities and inclination by considering the d'Alembert property. The coefficients of the terms appearing in this expansion are functions of the semi-major axes of the orbits of the secondary bodies. Expressions for these coefficients are not worked out as they are not required. Lete, n, m be the orbital eccentricity, mean motion and mass of one of the secondary bodies, and lete′, n′, m′ be the corresponding quantities for the other. (The mass of the primary is taken as unity). In cases (a) and (c) we will havem=0. In case (a)e′ will be zero, and in case (c) it will be a constant. Leti be the mutual inclination of the orbits of the secondary bodies. Suppose the commensurability is of the form(p+q) n =pn′, wherep andq are relatively prime integers, and put γ=(p+q) n/n′?p. The families of periodic solutions shown to exist are as follows. For q=1 No periodic solutions are found withi≠0 in case (a), and none withe′≠0, in case (c). In case (b) periodic solutions are found in whiche=0 (m′/γ),e′=0 (m/γ) for values of γ away from the exact commensurability. As γ approaches zero thene ande′ become 0 (1). For q≠1 Case (a). Families of periodic solutions bifurcating from the family withe=0, i=0 are shown to exist. Families in whichi=0 ande becomes non-zero exist for all values ofq. Families in whiche=0 andi becomes non-zero exist for even values ofq. Families in whiche andi become non-zero simultaneously exist for odd values ofq. Case (b). No families are found other than those withe=e′=0. Case (c). Families are found bifurcating from the familye=e′=0 in whiche ande′ become non-zero simultaneously. For all these solutions existence is only demonstrated close to the point of bifurcation, where all the variables are small, as the method uses series expansions ine, e′ andi. From the form of the solutions it is clear that the non-zero variables will become large for values of γ away from the bifurcation point.  相似文献   

12.
C.D. Murray  K. Fox 《Icarus》1984,59(2):221-233
The motion of asteroids near the 3:1 Jovian resonance in the restricted planar case is studied using three numerical methods: (a) integrating the full equations of motion, (b) integrating the averaged equations of motion, and (c) using an algebraic mapping recently developed by Wisdom (1982, Astron. J.87, 577–593). The relative merits of each method are investigated. It is concluded that in the regular regions of the phase space, methods b and c give excellent agreement with each other and that provided the maximum eccentricity emax < 0.4 differences with the exact solution (method a) are <6% in emax and <27% in the period of the oscillations. The additions of higher order terms in the expansion of the averaged Hamiltonian provides marginally better agreement with the full integration. This is probably due to the slow convergence of the expansion of the disturbing function at large eccentricities (e > 0.3). In chaotic regions of the phase there is little agreement between the orbital elements at any given time calculated by each method. However, all methods reflect the qualitative behavior of the chaotic trajectories and give good agreement on the bounds of the motion. Since the map is at least 200 times faster than solving the full equations of motion it is an efficient method of rapidly exploring accessible regions of the chaotic phase space.  相似文献   

13.
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14.
This paper deals with the existence and stability of the non-collinear libration points in the restricted three-body problem when both the primaries are ellipsoid with equal mass and identical in shape. We have determined the equations of motion of the infinitesimal mass which involves elliptic integrals and then we have investigated the existence and stability of the non-collinear libration points. This is observed that the non-collinear libration points exist only in the interval 52°<φ<90° and form an isosceles triangle with the primaries. Further we observed that the non collinear libration points are unstable in 52°<φ<90°.  相似文献   

15.
The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.Originally presented at the AAS/AIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973  相似文献   

16.
The delimitations of the librational motion around the Lagrangian triangular pointsL 4,L 5 are investigated within the framework of the restricted circular three body problem according to Brown's and Thüring's theory. The isotropic mass variation of the primaries does not exceed the order of the small primary and the derivatives of the masses with respect to the time are negligible second order quantities. The amplitude of the maximum elongations with respect to the small primary remains unchanged. The expression for the maximum variation of the distance of the particle from the large mass has the same form as in the classical problem with constant masses.  相似文献   

17.
This paper investigates Robe’s circular restricted three-body problem for two cases: with a Roche ellipsoid-triaxial system and with a Roche ellipsoid-oblate system. Without ignoring any component in both problems, a full treatment is given of the buoyancy force. The relevant equations of motion are established, and the special case where the density of the fluid and that of the infinitesimal mass are equal (D=0) is discussed. The location of the libration point and its stability when the infinitesimal mass is denser than the medium (D>0) are studied and it is found that the point (0,0,0) is the only libration point and this point is stable.  相似文献   

18.
The restricted (equilateral) four-body problem consists of three bodies of masses m 1, m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.  相似文献   

19.
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.  相似文献   

20.
The circular restricted problem of three bodies is investigated analytically with respect to the problem of deriving a second integral of motion besides the well known Jacobian Integral. The second integral is searched for as a correction the angular momentum integral valid in the two body case. A partial differential equation equivalent to the problem is derived and solved approximately by an asymptotic Fourier method assuming either sufficiently small values for the dimensionless mass parameter or sufficiently large distances from the barycentre. The solution of the partial equation then leads to a function of the coordinates, velocities and time being nearly constant, which means that its variation with time is about 40–300 times less than that of the pure angular momentum. By averaging over the remaining fluctuating part of the quasi-integral we are able to integrate the first order equations using a renormalization transformation. This leads to an explicit expression for the approximate solution of the circular problem which describes the motion of the third body orbiting both primaries with nonvanishing initial eccentricity (eccentric planetary type orbits). One of the main results is an explicit formula for the frequency of the perihelion motion of the third body which depends on the mass parameter, the initial distance of the third body from the barycentre and the initial eccentricity. Finally we study orbits of the P-Type, being defined as solutions of the restricted problem with circular initial conditions (vanishing initial eccentricity).  相似文献   

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