Vertical Critical Orbits Research Articles

The phase space structure of retrograde mean motion resonances with Neptune: the 4/5, 7/9, 5/8 and 8/13 cases

We compute planar and three-dimensional retrograde periodic orbits in the vicinity of the restricted three-body problem (RTBP) with the Sun and Neptune as primaries and we concentrate on the dynamics of higher-order exterior mean motion resonances with Neptune. By using the circular planar model as the basic model, families of retrograde symmetric periodic orbits are computed at the 4/5, 7/9, 5/8 and 8/13 resonances. We determine the bifurcation points from the planar circular to the planar elliptic problem and we find all the corresponding families. In order to obtain a global view of the families of periodic orbits, the eccentricity of the primaries takes values in the whole interval 0<e'<1. Then, we find all the possible vertical critical orbits (v.c.o) of the planar circular problem and we proceed to the three-dimensional circular restricted 3-body problem. In this model, retrograde periodic orbits are generated mainly from the retrograde v.c.o. Also, if we continue families of direct orbits for i>90^circ , then we can obtain families of 3D symmetric retrograde periodic orbits. The linear stability is examined too. Stable periodic orbits are associated with phase space domains of resonant motion where TNOs can be captured. In order to study the phase space structure of the above resonances, we construct dynamical stability maps for the whole inclination interval (0<i<180^circ ) by using the well-known “MEGNO Chaos Indicator”. Finally, we discuss about TNOs which are currently located at these resonances.

Spatial resonant periodic orbits in the restricted three-body problem

The quest of exo-Earths has become a prominent field. In this work, we study the stability of non-coplanar planetary configurations consisting of an inclined inner terrestrial planet in mean-motion resonance with an outer giant planet. We examine the families of circular and elliptic symmetric periodic orbits with respect to the vertical stability, and identify the vertical critical orbits from which the spatial families emanate. We showcase that stable spatial periodic orbits can exist for both prograde and retrograde motion in 3/2, 2/1, 5/2, 3/1, 4/1 and 5/1 resonances for broad ranges of inclinations, when the giant evolves on a circular orbit. When the orbit of the giant is elliptic, only the 2/1 resonance has stable periodic orbits up to high inclinations, while the 3/1, 4/1 and 5/1 resonances possess segments of stability for low inclinations. Furthermore, we show that regular motion can also take place in the vicinity of both horizontally and vertically stable planar periodic orbits, even for very high inclinations. Finally, the results are discussed in the context of asteroid dynamics.

Families of three-dimensional periodic solutions in the circular restricted four-body problem

In this paper we study the 3D symmetric periodic orbits of the circular restricted four-body problem, through their bifurcation from plane orbits. To this end, we calculate the special generating planar periodic orbits, i.e. the vertical-critical orbits, from five basic planar families of symmetric periodic orbits of the problem. We study the simplest 3D periodic orbits and so we restrict our calculations to simple vertical bifurcation orbits leading to families of one or two revolutions three-dimensional symmetric periodic orbits. We found 21 such families of simple 3D symmetric periodic orbits and typical orbits of all symmetry type 3D orbits are presented. The stability of each calculated 3D periodic orbit is also studied. Characteristic curves as well as stability diagrams of families of 3D periodic orbits are illustrated.

Vertical instability and inclination excitation during planetary migration

We consider a two-planet system migrating under the influence of dissipative forces that mimic the effects of gas-driven (Type II) migration. It has been shown that, in the planar case, migration leads to resonant capture after an evolution that forces the system to follow families of periodic orbits. Starting with planets that differ slightly from a coplanar configuration, capture can, also, occur and, additionally, excitation of planetary inclinations has been observed in some cases. We show that excitation of inclinations occurs, when the planar families of periodic orbits, which are followed during the initial stages of planetary migration, become vertically unstable. At these points, vertical critical orbits may give rise to generating stable families of \(3D\) periodic orbits, which drive the evolution of the migrating planets to non-coplanar motion. We have computed and present here the vertical critical orbits of the \(2/1\) and \(3/1\) resonances, for various values of the planetary mass ratio. Moreover, we determine the limiting values of eccentricity for which the “inclination resonance” occurs.

Resonant periodic orbits in the exoplanetary systems

The planetary dynamics of $4/3$, $3/2$, $5/2$, $3/1$ and $4/1$ mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to $60^\circ$ of mutual planetary inclination, but in most families, the stability does not exceed $20^\circ$-$30^\circ$, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the $4/3$ and $5/2$ resonance, respectively.

The 3D restricted three-body problem under angular velocity variation

Nonlinear approximation of periodic motions around the collinear equilibrium points in the case of the restricted three-body problem when the angular velocity of the primaries is not equal to the value of the classical problem (which is unit in the usual units of mass, length and time), is studied. The stability of the equilibrium points and the analytical solutions in their neighborhood constructing series approximations of the periodic orbits in the planar and in the spatial problem, are given. Families which emanate from L1, L2 and L3 both in the plane and in three dimensions as well as their stability for the Earth-Moon mass distribution, are computed. Special generating plane orbits, the vertical-critical orbits, of the families a, b and c of the problem, are determined and presented. We have also computed series of vertical-critical periodic orbits with the angular velocity as parameter. Three-dimensional families which generate from the bifurcation orbits for ω = 2, are given.

Stable Orbits of Planets of a Binary Star System in the Three-Dimensional Restricted Problem

The present research was motivated by the recent discovery of planets around binary stars. Our initial intention was thus to investigate the 3-dimensional nearly circular periodic orbits of the circular restricted problem of three bodies; more precisely Stromgren's class L, (direct) and class m, (retrograde). We started by extending several of Henon's vertical critical orbits of these 2 classes to three dimensions, looking especially for orbits which are near circular and have stable characteristic exponents. We discovered early on that the periodic orbits with the above two qualifications are fairly rare and we decided thus to undertake a systematic exploration, limiting ourselves to symmetric periodic orbits. However, we examined all 16 possible symmetry cases, trying 10 000 sets of initial values for periodicity in each case, thus 160 000 integrations, all with z o or zo equal to 0.1 This gave us a preliminary collection of 171 periodic orbits, all fairly near the xy-plane, thus with rather low inclinations. Next, we integrated a second similar set of 160 000 cases with z o or zo equal to 0.5, in order to get a better representation of the large inclinations. This time, we found 167 periodic orbits, but it was later discovered that at least 152 of them belong to the same families as the first set with 0.1 Our paper quickly describes the definition of the problem, with special emphasis on the symmetry properties, especially for the case of masses with equal primaries. We also allow a section to describe our approach to stability and characteristic exponents, following our paper on this subject, (Broucke, 1969). Then we describe our numerical results, as much as space permits in the present paper. We found basically only about a dozen families with sizeable segments of simple stable periodic orbits. Some of them are around one of the two stars only but we do not describe them here because of a lack of space. We extended about 170 periodic orbits to families of up to 500 members, (by steps of 0.005 in the parameter), although, in many cases, we do not know the real end of the families. We also give an overview of the different types of periodic orbits that are most often encountered. We describe some of the rather strange orbits, (some of which are actually stable).

BIFURCATIONS OF PLANE TO 3D PERIODIC ORBITS IN THE RESTRICTED THREE-BODY PROBLEM WITH OBLATENESS

We consider the bifurcation of 3D periodic orbits from the plane of motion of the primaries in the restricted three-body problem with oblateness. The simplest 3D periodic orbits branch-off at the plane periodic orbits of indifferent vertical stability. We describe briefly suitable numerical techniques and apply them to produce the first few such vertical-critical orbits of the basic families of periodic orbits of the problem, for varying mass parameter μ and fixed oblateness coefficent A1 = 0.005, as well as for varying A1 and fixed μ = 1/2. The horizontal stability of these orbits is also determined leading to predictions about the stability of the branching 3D orbits.

VERTICAL-CRITICAL PERIODIC ORBITS IN THE GENERAL THREE-BODY PROBLEM

We present special generating plane orbits, the vertical-critical orbits, of the coplanar general three-body problem. These are determined numerically for various values of m3, for the entire range of the mass ratio of the two primaries. The vertical-critical orbits are necessary in order to specify the vertically stable segments of the families of plane periodic orbits, and they are also the starting points of the families of the simplest possible three-dimensional periodic orbits, namely the simple and double periodic. The initial conditions of the vertical-critical periodic orbits of the basic families l, m, i, h, b and c and their stability parameters are determined.

A NOTE ON PARAMETER VALUES FOR STABLE LOW-INCLINATION PERIODIC MOTION IN THE RESTRICTED THREE-BODY PROBLEM WITH RADIATION

The intervals of the mass parameter (µ) values for possible stability of the basic families of 3D periodic orbits in the restricted three-body problem determined elsewhere are now extended into regions of theµ - q1 parameter space of the photogravitational restricted three-body problem, where q1 is the radiation factor of m1 and it is assumed that m2 does not radiate. Several 3D periodic orbits corresponding to these regions are computed and tested for stability and seven regions, corresponding to the vertical-critical orbits l1v, l'1v, l6v, m1v, m2v and i1v, survive this stability test, emerging as the regions allowing the simplest types of stable low inclination 3D motion of the infinitesimal particle.

Periodic orbits in three-dimensional planetary systems

Three-dimensional planetary systems are studied, using the model of the restricted three-body problem for Μ =.001. Families of three-dimensional periodic orbits of relatively low multiplicity are numerically computed at the resonances 3/1, 5/3, 3/5 and 1/3 and their stability is determined. The three-dimensional orbits are found by continuation to the third dimension of the vertical critical orbits of the corresponding planar problem

Three-dimensional periodic orbits in barred galaxies

We study the properties of the families of three-dimensional periodic orbits which bifurcate from the vertical critical orbits of the retrograde family of quasi-circular plane periodic orbits which extend from the galactic center up to infinity. We consider the case of a barred galaxy with a strong central bulge. The values of the parameters are chosen in such a way as to cover the cases of a strong or weak bar with a fast or slow rotation.

The Mechanism of Branching of Three-Dimensional Periodic Orbits from the Plane

AbstractA linearised treatment is presented of vertical bifurcations of symmetric periodic orbits(bifurcations of plane with three-dimensional orbits) in the circular restricted problem. Recent work on bifurcations from vertical-critical orbits (av = ±1) is extended to deal with the v more general situation of bifurcations from vertical self-resonant orbits (av = cos(2Πn/m) for integer m,n) and it is shown that in this more general case bifurcating families of three-dimensional orbits always occur in pairs, the orbital symmetry properties being governed by the evenness or oddness of the integer m. The applicability of the theory to the elliptic restricted problem is discussed.

Three-dimensional periodic motion of three finite masses around collinear equilibrium configurations

Three-dimensional periodic motions of three bodies are shown to exist in the infinitesimal neighbourhood of their collinear equilibrium configurations. These configurations and some characteristic quantities of the emanating three-dimensional periodic orbits are given for many values of the two mass parameters, μ=m2/(m1+m2) andm3, of the general three-body problem, under the assumption that the straight line containing the bodies at equilibrium rotates with unit angular velocity. The analysis of the small periodic orbits near the equilibrium configurations is carried out to second-order terms in the small quantities describing the deviation from plane motion but the analytical solution obtained for the ‘horizontal’ components of the state vector is valid to third-order terms in those quantities. The families of three-dimensional periodic orbits emanating from two of the collinear equilibrium configurations are continued numerically to large orbits. These families are found to terminate at large vertical-critical orbits of the familym of retrograde periodic orbits ofm3 around the ‘primaries’m1 andm2. The series of these termination orbits, formed when the value ofm3 varies, are also given. The three-dimensional orbits are computed form3=0.1.

Bifurcations of planar to three-dimensional periodic orbits in the general three-body problem

We study the generation of three-dimensional periodic orbits of the general three-body problem from special generating plane orbits, the vertical-critical orbits. The bifurcation process is examined analytically and geometrically. A method of obtaining numerically continuous sets of vertical-critical orbits is outlined, and applied for the determination of 16 monoparametric sets including all possible types of such orbits corresponding to all possible types of symmetry of the bifurcating three-dimensional orbits. The stability of all bifurcation orbits is assessed. Examples of three-dimensional periodic orbits generated from the bifurcation orbits are given.

Bifurcations of plane with three-dimensional periodic orbits in the elliptic restricted problem

We show that the procedure employed in the circular restricted problem, of tracing families of three-dimensional periodic orbits from vertical self-resonant orbits belonging to plane families, can also be applied in the elliptic problem. A method of determining series of vertical bifurcation orbits in the planar elliptic restricted problem is described, and one such series consisting of vertical-critical orbits (a v=+1) is given for the entire range (0,1/2) of the mass parameter μ. The initial segments of the families of three-dimensional orbits which bifurcate from two of the orbits belonging to this series are also given.

On the continuation of periodic orbits from the planar to the three-dimensional general three-body problem

It is shown that the vertical critical orbits of the general planar problem of three bodies can be used as starting points for finding monoparametric families of three-dimensional periodic orbits. Several numerical examples are given.

The bifurcation of periodic orbits of the planar circular restrictedN-body problem to the three-dimensional generalN-body problem

It is proved that the vertical critical orbits of the planar circular restricted three-body problem can be used as starting points for finding periodic orbits of the three-dimensional generalN-body problem. A numerical example is given.