John D. Hadjidemetriou

John D. Hadjidemetriou

Abstract We have proved that the isotropic loss of mass from a binary system is equivalent to a fore proportional to the rate of loss of mass and to the velocity vector, in the direction of motion. Using this force as a perturbation, we have derived the equations of the variation of the elements of the orbit; and from these equations approximate relations for the secular variation of the semi-major axis and the eccentricity have been found. The eccentricity is proved to remain secularly constant, while the semi-major axis increases secularly. On the other hand, the radiation pressure does not alter the results obtained for the isotropic loss of mass, if the total mass of the system is augmented by an additional term depending on the luminosities of the two components.

John D. Hadjidemetriou

A method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section. The method is linear and involves the computation of a 4×4 variational matrix by integrating numerically the differential equations for time intervals of the order of a period. Several properties of this matrix are proved and also it is shown that for a symmetric periodic motion it can be computed by integrating for half the period only.This linear stability analysis is used to study the stability of a family of periodic motions of three bodies with equal masses, in a rotating frame of reference. This family represents motion such that two bodies revolve around each other and the third body revolves around this binary system in the same direction to a distance which varies along the members of the family. It was found that a large part of the family, corresponding to the case where the distance of the third body from the binary system is larger than the dimensions of the binary system, represents stable motion. The nonlinear effects to the linear stability analysis are studied by computing the intersections of several perturbed orbits with the surface of sectiony3=0. In some cases more than 1000 intersections are computed. These numerical results indicate that linear stability implies stability to all orders, and this is true for quite large perturbations.

G. Bozis; John D. Hadjidemetriou

It is proved that a symmetric periodic orbit of the circular planar restricted three-body problem can be continued analytically, when the mass of the third body is small but not negligible, to a periodic motion of the general three-body problem in a rotating frame of reference whose origin coincides with the center of mass of the two bodies with large masses and itsx axis always contains these bodies. The two bodies with the large masses describe periodic motion on thex axis of the rotating frame while the third body, with the small mass, describes a symmetric periodic orbit in this frame. The motion of the two bodies lying on thex axis is always stable, whereas the periodic orbit of the third body in the rotating frame is stable or unstable depending on whether or not the nonzero characteristic exponents of the original periodic orbit of the restricted problem are of the stable or unstable type, respectively. It is also shown that for a fixed value of the small mass of the third body, a family of symmetric periodic orbits of the restricted problem can be continued analytically to a family of periodic motions of the general three-body problem.

John D. Hadjidemetriou

A mapping model is constructed to describe asteroid motion near the 3 : 1 mean motion resonance with Jupiter, in the plane. The topology of the phase space of this mapping coincides with that of the real system, which is considered to be the elliptic restricted three body problem with the Sun and Jupiter as primaries. This model is valid for all values of the eccentricity. This is achieved by the introduction of a correcting term to the averaged Hamiltonian which is valid for small values of the ecentricity.We start with a two dimensional mapping which represents the circular restricted three body problem. This provides the basic framework for the complete model, but cannot explain the generation of a gap in the distribution of the asteroids at this resonance. The next approximation is a four dimensional mapping, corresponding to the elliptic restricted problem. It is found that chaotic regions exist near the 3 : 1 resonance, due to the interaction between the two degrees of freedom, for initial conditions close to a critical curve of the circular model. As a consequence of the chaotic motion, the eccentricity of the asteroid jumps to high values and close encounters with Mars and even Earth may occur, thus generating a gap. It is found that the generation of chaos depends also on the phase (i.e. the angles σ andv) and as a consequence, there exist islands of ordered motion inside the sea of chaotic motion near the 3 : 1 resonance. Thus, the model of the elliptic restricted three body problem cannot explain completely the generation of a gap, although the density in the distribution of the asteroids will be much less than far from the resonance. Finally, we take into account the effect of the gravitational attraction of Saturn on Jupiters orbit, and in particular the variation of the eccentricity and the argument of perihelion. This generates a ‘mixing’ of the phases and as a consequence the whole phase space near the 3 : 1 resonance becomes chaotic. This chaotic zone is in good agreement with the observations.

John D. Hadjidemetriou

Families of nearly circular periodic orbits of the planetary type are studied, close to the 3/1 mean motion resonance of the two planets, considered both with finite masses. Large regions of instability appear, depending on the total mass of the planets and on the ratio of their masses.Also, families of resonant periodic orbits at the 2/1 resonance have been studied, for a planetary system where the total mass of the planets is the 4% of the mass of the sun. In particular, the effect of the ratio of the masses on the stability is studied. It is found that a planetary system at this resonance is unstable if the mass of the outer planet is smaller than the mass of the inner planet.Finally, an application has been made for the stability of the observed extrasolar planetary systems HD82943 and Gliese 876, trapped at the 2/1 resonance.

George Voyatzis; John D. Hadjidemetriou; Dimitri Veras; Harry Varvoglis

Discoveries of exoplanets orbiting evolved stars motivate critical examinations of the dynamics of N-body systems with mass loss. Multi-planet evolved systems are particularly complex because of the mutual interactions between the planets. Here, we study the underlying dynamical mechanisms which can incite planetary escape in two-planet post-main sequence systems. Stellar mass loss alone is unlikely to be rapid and high enough to eject planets at typically-observed separations. However, the combination of mass loss and planetplanet interactions can prompt a shift from stable to chaotic regions of phase space. Consequently, when mass loss ceases, the unstable configuration may cause escape. By assuming a constant stellar mass loss rate, we utilize maps of dynamical stability to illustrate the distribution of regular and chaotic trajectories in phase space. We show that chaos can drive the planets to undergo close encounters, leading to the ejection of one planet. Stellar mass loss can trigger the transition of a planetary system from a stable to chaotic configuration, subsequently causing escape. We find that mass loss non-adiabatically affects planet-planet interaction for the most massive progenitor stars which avoid the supernova stage. For these cases, we present specific examples of planetary escape.

Dimitri Veras; John D. Hadjidemetriou; Christopher A. Tout

The birth and death of planets may be affected by mass outflows from their parent stars during the T-Tauri or post-main-sequence phases of stellar evolution. These outflows are often modelled to be isotropic, but this assumption is not realistic for fast rotators, bipolar jets and supernovae. Here we derive the general equations of motion for the time evolution of a single planet, brown dwarf, comet or asteroid perturbed by anisotropic mass loss in terms of a complete set of planetary orbital elements, the ejecta velocity, and the parent stars co-latitude and longitude. We restrict our application of these equations to 1) rapidly rotating giant stars, and 2) arbitrarily-directed jet outflows. We conclude that the isotropic mass-loss assumption can safely be used to model planetary motion during giant branch phases of stellar evolution within distances of hundreds of au. In fact, latitudinal mass loss variations anisotropically affect planetary motion only if the mass loss is asymmetric about the stellar equator. Also, we demonstrate how constant-velocity, asymmetric bipolar outflows in young systems incite orbital inclination changes. Consequently, this phenomenon readily tilts exoplanetary orbits external to a nascent disc on the order of degrees.

John D. Hadjidemetriou

Abstract The system of equations defining the variation of the orbital elements in the two-body problem with variable mass has been solved analytically for various laws of mass loss. There are two cases in which the eccentricity is a periodic function of the eccentric anomaly with a period of 2π. In all cases the secular variation of the eccentricity is very small, but there are oscillations with variable amplitude. It is also shown that several problems in which the perturbing force depends on the radial distance only, can be transformed into an equivalent two-body problem with variable mass, and be treated as such.

John D. Hadjidemetriou

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, Ic, IIc 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, Ie, IIe, 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 Ic, IIc, but is poor for the families Ie, IIe. 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.

John D. Hadjidemetriou; George Voyatzis

We present families of symmetric and asymmetric periodic orbits at the 1/1 resonance, for a planetary system consisting of a star and two small bodies, in comparison to the star, moving in the same plane under their mutual gravitational attraction. The stable 1/1 resonant periodic orbits belong to a family which has a planetary branch, with the two planets moving in nearly Keplerian orbits with non zero eccentricities and a satellite branch, where the gravitational interaction between the two planets dominates the attraction from the star and the two planets form a close binary which revolves around the star. The stability regions around periodic orbits along the family are studied. Next, we study the dynamical evolution in time of a planetary system with two planets which is initially trapped in a stable 1/1 resonant periodic motion, when a drag force is included in the system. We prove that if we start with a 1/1 resonant planetary system with large eccentricities, the system migrates, due to the drag force, along the family of periodic orbits and is finally trapped in a satellite orbit. This, in principle, provides a mechanism for the generation of a satellite system: we start with a planetary system and the final stage is a system where the two small bodies form a close binary whose center of mass revolves around the star.