Families <Emphasis Type="Italic">c</Emphasis> and <Emphasis Type="Italic">i</Emphasis> of periodic solutions of the restricted problem for μ = 5 × 10<Superscript>−5</Superscript>

We consider the circular planar restricted three-body problem with the mass parameter μ = 5 × 10^?5. Two families of periodic solutions are calculated: family c, starting from the collinear fixed point L ₁, and the initial part of familyi, which begins by direct circular orbits of an infinitely small radius around the body of bigger mass. The calculated families are very close to the generating ones, which we described earlier. In particular, the existence of the predicted zigzag structure of characteristics of family iis verified. New properties of the planar and vertical traces are discovered.     

On bridges B(pL, qS) around triangular libration points

When μ is smaller than Routh’s critical value μ ₁ = 0.03852 . . . , two planar Lyapunov families around triangular libration points exist, with the names of long and short period families. There are periodic families which we call bridges connecting these two Lyapunov families. With μ increasing from 0 to 1, how these bridges evolve was studied. The interval (0,1) was divided into six subintervals (0, μ ₅), (μ ₅, μ ₄), (μ ₄, μ ₃), (μ ₃, μ ₂), (μ ₂, μ ₁), (μ ₁, 1), and in each subinterval the families B(pL, qS) were studied, along with the families B(qS, qS′). Especially in the interval (μ ₂, μ ₁), the conclusion that the bridges B(qS, qS′) do not exist was obtained. Connections between the short period family and the bridges B(kS, (k + 1)S) were also studied. With these studies, the structure of the web of periodic families around triangular libration points was enriched.     

Existence and stability of triangular points in the restricted three-body problem with numerical applications

In this paper, we prove that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary in the planar circular restricted three-body problem, considering the effect of oblateness for J ₂ and J ₄. After that, we show that the triangular points are stable for 0<μ<μ _c and unstable when , where μ _c is the critical mass parameter which depends on the coefficients of oblateness. On the other hand, we produce some numerical values for the positions of the triangular points, μ and μ _c using planets systems in our solar system which emphasis that the range of stability will decrease; however this range sometimes is not affected by the existence of J ₄ for some planets systems as in Earth–Moon, Saturn–Phoebe and Uranus–Caliban systems.     

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.     

Bifurcation of central configuration in the 2N+1 body problem

The bifurcation of central configuration in the Newtonian N-body problem for any odd number N ≥ 7 is shown. We study a special case where 2n particles of mass m on the vertices of two different coplanar and concentric regular n-gons (rosette configuration) and an additional particle of mass m₀ at the center are governed by the gravitational law he 2n+1 body problem. This system is of two degrees of freedom and permits only one mass parameter μ =m ₀/m. This parameter μ controls the bifurcation. If n≥ 3, namely any odd N ≥ 7, then the number of central configurations is three when μ ≥ μ _c , and one when μ ≥ μ _c . By combining the results of the preceding studies and our main theorem, explicit examples of bifurcating central configuration are obtained for N ≤ 13, for any odd N ∈ [15,943], and for any N ≥ 945.     

Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies

In this paper, we study the existence of libration points and their linear stability when the three participating bodies are axisymmetric and the primaries are radiating, we found that the collinear points remain unstable, it is further seen that the triangular points are stable for 0<μ<μ _c , and unstable for where , it is also observed that for these points the range of stability will decrease. In addition to this we have studied periodic orbits around these points in the range 0<μ<μ _c , we found that these orbits are elliptical; the frequencies of long and short orbits of the periodic motion are affected by the terms which involve parameters that characterize the oblateness and radiation repulsive forces. The implication is that the period of long periodic orbits adjusts with the change in its frequency while the period of short periodic orbit will decrease.     

Analytical Reference Functions <Emphasis Type="Italic">F</Emphasis>(λ) for the Sun's Limb Darkening and Its Absolute Continuum Intensities (λλ 300 to 1100 m)

It is shown (1) that the coefficients A_i of the limb darkening functions I(μ)/I_center = P₅ (μ) = ∑A_i μⁱ (i = 0... 5; μ = cos ϑ), which had been published by Neckel and Labs (Solar Phys. 153, 91, 1994), can well be approximated by analytical functions of wavelength λ, and (2) that at first sight purely formal extrapolation of the functions P₅(μ) to the very limb (μ = 0.0) is not meaningless: in combination with absolute intensities for the disk center these functions yield ‘limb intensities’ which all correspond to almost the same ‘limb temperature’, T_limb≈4746 K. Together these results lead to ‘reference functions’ which can quickly yield rather reliable values of the Sun's continuum intensities, for any values of μ and λ.     

The elliptic restricted problem at the 3 : 1 resonance

Four 3 : 1 resonant families of periodic orbits of the planar elliptic restricted three-body problem, in the Sun-Jupiter-asteroid system, have been computed. These families bifurcate from known families of the circular problem, which are also presented. Two of them, I _c , II _c bifurcate from the unstable region of the family of periodic orbits of the first kind (circular orbits of the asteroid) and are unstable and the other two, I _e , II _e , from the stable resonant 3 : 1 family of periodic orbits of the second kind (elliptic orbits of the asteroid). One of them is stable and the other is unstable. All the families of periodic orbits of the circular and the elliptic problem are compared with the corresponding fixed points of the averaged model used by several authors. The coincidence is good for the fixed points of the circular averaged model and the two families of the fixed points of the elliptic model corresponding to the families I _c , II _c , but is poor for the families I _e , II _e . A simple correction term to the averaged Hamiltonian of the elliptic model is proposed in this latter case, which makes the coincidence good. This, in fact, is equivalent to the construction of a new dynamical system, very close to the original one, which is simple and whose phase space has all the basic features of the elliptic restricted three-body problem.     

Planar electron-acoustic solitary waves and double layers in a two-electron-temperature plasma with nonthermal ions

A rigorous theoretical investigation of nonlinear electron-acoustic (EA) waves in a plasma system (containing cold electrons, hot electrons obeying a Boltzmann distribution, and hot ions obeying a nonthermal distribution) is studied by the reductive perturbation method. The modified Gardner (MG) equation is derived and numerically solved. It has been found that the basic characteristics of the EA Gardner solitons (GSs), which are shown to exist for α around its critical value α _c [where α is the nonthermal parameter, α _c is the value of α corresponding to the vanishing of the nonlinear coefficient of the Korteweg-de Vries (K-dV) equation, e.g. α _c ≃0.31 for μ=n _h0/n _i0=0.5, σ=T _h /T _i =10, n _h0, n _i0 are, respectively, hot electron and nonthermal ion number densities at equilibrium, T _h (T _i ) is the hot electron (ion) temperature], are different from those of the K-dV solitons, which do not exist for α around α _c , and mixed K-dV solitons, which are valid around α∼α _c , but do not have any corresponding double layers (DLs) solution. The parametric regimes for the existence of the DLs, which are found to be associated with positive potential, are obtained. The present investigations can be observed in various space plasma environments (viz. the geomagnetic tail, the auroral regions, the cusp of the terrestrial magnetosphere, etc.).     

Periodic orbits in the generalized perturbed restricted three-body problem

This paper studies the existence and stability of equilibrium points under the influence of small perturbations in the Coriolis and the centrifugal forces, together with the non-sphericity of the primaries. The problem is generalized in the sense that the bigger and smaller primaries are respectively triaxial and oblate spheroidal bodies. It is found that the locations of equilibrium points are affected by the non-sphericity of the bodies and the change in the centrifugal force. It is also seen that the triangular points are stable for 0<μ<μ _c and unstable for m_c ￡ m < \frac12\mu_{c}\le\mu <\frac{1}{2}, where μ _c is the critical mass parameter depending on the above perturbations, triaxiality and oblateness. It is further observed that collinear points remain unstable.     

The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem

This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits.     

Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.     

Evolution of the General Solution of the Restricted Problem Covering Symmetric and Escape Solutions

The work presented in paper I (Papadakis, K.E., Goudas, C.L.: Astrophys. Space Sci. (2006)) is expanded here to cover the evolution of the approximate general solution of the restricted problem covering symmetric and escape solutions for values of μ in the interval [0, 0.5]. The work is purely numerical, although the available rich theoretical background permits the assertions that most of the theoretical issues related to the numerical treatment of the problem are known. The prime objective of this work is to apply the ‘Last Geometric Theorem of Poincaré’ (Birkhoff, G.D.: Trans. Amer. Math. Soc. 14, 14 (1913); Poincaré, H.: Rend. Cir. Mat. Palermo 33, 375 (1912)) and compute dense sets of axisymmetric periodic family curves covering the initial conditions space of bounded motions for a discrete set of values of the basic parameter μ spread along the entire interval of permissible values. The results obtained for each value of μ, tested for completeness, constitute an approximation of the general solution of the problem related to symmetric motions. The approximate general solution of the same problem related to asymmetric solutions, also computable by application of the same theorem (Poincaré-Birkhoff) is left for a future paper. A secondary objective is identification-computation of the compact space of escape motions of the problem also for selected values of the mass parameter μ. We first present the approximate general solution for the integrable case μ = 0 and then the approximate solution for the nonintegrable case μ = 10⁻³. We then proceed to presenting the approximate general solutions for the cases μ = 0.1, 0.2, 0.3, 0.4, and 0.5, in all cases building them in four phases, namely, presenting for each value of μ, first all family curves of symmetric periodic solutions that re-enter after 1 oscillation, then adding to it successively, the family curves that re-enter after 2 to 10 oscillations, after 11 to 30 oscillations, after 31 to 50 oscillations and, finally, after 51 to 100 oscillations. We identify in these solutions, considered as functions of the mass parameter μ, and at μ = 0 two failures of continuity, namely: 1. Integrals of motion, exempting the energy one, cease to exist for any infinitesimal positive value of μ. 2. Appearance of a split into two separate sub-domains in the originally (for μ = 0) unique space of bounded motions. The computed approximations of the general solution for all values of μ appear to fulfill the ‘completeness’ criterion inside properly selected sub-domains of the domain of bounded motions in the (x, C) plane, which means that these sub-domains are filled countably densely by periodic family curves, which form a laminar flow-line pattern. The family curves in this pattern may, or may not, be intersected by a ‘basic’ family curve segment of order from 1 up to 3. The isolated points generating asymptotic solutions resemble ‘sink’ points toward which dense sets of periodic family curves spiral. The points in the compact domain in the (x, C) plane resting outside the domain of bounded motions (μ = 0), including the gap between the two large sub-domains (μ > 0) created by the aforementioned split, generate escape motions. The gap between the two large sub-domains of bounded motions grows wider for growing μ. Also, a number of compact gaps that generate escape motions exist within the body of the two sub-domains of bounded motions. The approximate general solutions computed include symmetric, heteroclinic, asymptotic, collision and escape solutions, thus constituting one component of the full approximate general solution of the problem, the second and final component being that of asymmetric solutions.     

Vertical stability of periodic solutions around the triangular equilibrium points

The vertical stability character of the families of short and long period solutions around the triangular equilibrium points of the restricted three-body problem is examined. For three values of the mass parameter less than equal to the critical value of Routh (μ _R ) i.e. for μ = 0.000953875 (Sun-Jupiter), μ = 0.01215 (Earth-Moon) and μ = μ _R = 0.038521, it is found that all such solutions are vertically stable. For μ > (μ _R ) vertical stability is studied for a number of ‘limiting’ orbits extended to μ = 0.45. The last limiting orbit computed by Deprit for μ = 0.044 is continued to a family of periodic orbits into which the well known families of long and short period solutions merge. The stability characteristics of this family are also studied.     

Collinear Equilibrium Solutions of Four-body Problem

We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses. Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two mass ratios as μ ₁ = m ₁/M _T and μ ₂ = m ₂/M _T, where m ₁ and m ₂ are the two unequal masses and M _T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four types of four-body problems.     

Conservation of both current and helicity in a quadrupolar model for solar flares

A model for a solar flare, involving magnetic reconnection transferring flux and current between current-carrying magnetic loops connecting two pairs of footpoints, is generalized to include conservation of magnetic helicity during reconnection, as well as conservation of current at all four footpoints. For a set of force-free loops, with the ith loop having flux F _i and current I _i, the self and mutual helicities are proportional to the self and mutual inductances with the constant of proportionality determined by α_i=F _i/μ₀ I _i. In a constant-α model, the change in magnetic energy is proportional to the change in helicity, and conservation of helicity implies conservation of magnetic energy, so that a flare cannot occur. In a quadrupolar model, with α₁>α₂ initially, α₁ increases and α₂ decreases when flux and current are transferred from loops 1 and 2 to loops 3 and 4. A model that conserves both current and helicity is constructed; it depends on the initial αs, and otherwise is somewhat simpler than when helicity is neglected.     

Effect of perturbation on the location of libration point in the robe restricted problem of three bodies

The effect of small perturbation in the Coriolis and centrifugal forces on the location of libration point in the ‘Robe (1977) restricted problem of three bodies’ has been studied. In this problem one body,m ₁, is a rigid spherical shell filled with an homogeneous incompressible fluid of densityϱ ₁. The second one,m ₂, is a mass point outside the shell andm ₃ is a small solid sphere of densityϱ ₃ supposed to be moving inside the shell subject to the attraction ofm ₂ and buoyancy force due to fluidϱ ₁. Here we assumem ₃ to be an infinitesimal mass and the orbit of the massm ₂ to be circular, and we also suppose the densitiesϱ ₁, andϱ ₃ to be equal. Then there exists an equilibrium point (−μ + (ɛ′μ)/(1 + 2μ), 0, 0).     

Families of Periodic Orbits Emanating From Homoclinic Orbits in the Restricted Problem of Three Bodies

We describe and comment the results of a numerical exploration on the evolution of the families of periodic orbits associated with homoclinic orbits emanating from the equilateral equilibria of the restricted three body problem for values of the mass ratio larger than μ ₁. This exploration is, in some sense, a continuation of the work reported in Henrard [Celes. Mech. Dyn. Astr. 2002, 83, 291]. Indeed it shows how, for values of μ. larger than μ ₁, the Trojan web described there is transformed into families of periodic orbits associated with homoclinic orbits. Also we describe how families of periodic orbits associated with homoclinic orbits can attach (or detach) themselves to (or from) the best known families of symmetric periodic orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability of equilibrium points in a CR3BP with oblate bodies enclosed by a circular cluster of material points

We have investigated an improved version of the classic restricted three-body problem where both primaries are considered oblate and are enclosed by a homogeneous circular planar cluster of material points centered at the mass center of the system. In this dynamical model we have examined the effect on the number and on the linear stability of the equilibrium locations of the small particle due to both, the primaries’ oblateness and the potential created by the circular cluster. We have drawn the zero-velocity surfaces and we have found that in addition to the usual five Lagrangian equilibrium points of the classic restricted three-body problem, there exist two new collinear points L _n1,L _n2 due to the potential from the circular cluster of material points. Numerical investigations reveal that with the increase in the mass of the circular cluster of material points, L _n2 comes nearer to the more massive primary, while L _n1 moves away from it. Owing to oblateness of the bodies, L _n1 comes nearer to the more massive primary, while L _n2 moves towards the less massive primary. The collinear equilibrium points remain unstable, while the triangular points are stable for 0<μ<μ _c and unstable for $\mu_{c} \le \mu \le \frac{1}{2}$ , where μ _c is the critical mass ratio influenced by oblateness of the primaries and the potential from the circular cluster of material points. The oblateness and the circular cluster of material points have destabilizing tendency.