This paper reviews recent results on the dynamics of multiple-planet extra-solar systems, including main sequence stars and the pulsar PSR B1257+12 and, comparatively, our own Solar System. Taking into account the degree of gravitational interaction of the planets, the known planetary systems may be separated into four main groups: (Ia) Planets in Mean-motion resonance (Ib) Low-eccentricity near-resonant pairs; (II) Non-resonant planets with a significant secular dynamics; and (III) Weakly interacting planet pairs. Different analytical and numerical tools can help to understand the structure of the phase space, to identify stability mechanisms and to categorize different types of motions in the cases of more significant dynamical interaction. The origin of resonant configurations is discussed in the light of the hypothesis of planetary migration.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

This paper presents recent results concerning the planet formation, planet migration, and the long term stability of planetary systems. Most stars are found in binary systems and binary companions can disrupt both planet formation and stability. We first consider the effects of outer binary companions on the late stages of terrestrial planet formation and show how planet formation depends on the binary periastron. We then consider migration mechanisms for giant planets. In this case, planet scattering produces the full range of orbital eccentricities, but is less effective in moving planets inward (decreasing their semi-major axes). Disk torques are effective at moving planets inward, but not at increasing the eccentricities. We explore a scenario in which disk torques act in concert with planet scattering to provide the full range of orbital elements observed in extrasolar planetary systems. Finally, we consider the longer term stability of Earth-like planets in binary systems; we find that nearly 50 percent of binaries allow for Earth-like planets to remain stable over the current (4.6 Gyr) age of our solar system.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

To date, more than 130 extrasolar planets around main sequence stars are revealed mainly by the Doppler radial velocity measurements. Due to the observational biases, most of the detected planets are moving in orbits close to the host stars, with some in highly eccentric orbits. Dynamical processes during the late stage of planet formation are important to account for the present orbital properties. These processes include: planet migrations and resonance trappings caused by gravitational interactions between protostellar disk and planets, dynamical scattering due to interactions between planets, etc. In this paper, we review the major effects of these dynamical processes on the orbital characteristics of the planet systems.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

In the standard scenario of planet formation, solid planets are formed through accretion of small bodies called planetesimals. The dynamics of planetesimals is important since it controls their growth mode and timescale. Here, I briefly explain the basic dynamics of planetesimals due to the two-body gravitational relaxation process. The important roles of two-body relaxation in a planetesimal system are viscous stirring and dynamical friction. Due to viscous stirring, the random velocities (eccentricities and inclinations) of planetesimals increase, while dynamical friction realizes the energy equipartition of the random energy. I also explain the orbital repulsion of protoplanets which is the coupling effect of two-body scattering and dynamical friction.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Nowadays, more than one hundred extra-solar planets are known, and about a dozen of multi-planetary systems have been discovered. Most of them have been detected by the radial velocity (RV) method. The recovery of orbital parameters from RV data leads to several problems. Usually RV data cover irregularly a short time interval which is frequently shorter than the orbital period of the most distant planet. Moreover, observations contain a noise due to the instabilities of the star. The distribution of this noise is unknown. A precise determination of the dynamical state of a multi-planetary system is important for understanding its stability and evolution. In most cases observers determine the orbital parameters for multi-planetary systems just fitting a sum of Keplerian orbits. The parameters obtained in such a way are in most cases the only accessible data about an extra-solar system because the observes very rarely publish their observations. However, the parameters from a multi-Keplerian fit as it has already been observed by many authors, cannot be interpreted as the osculating elements for actual planetary orbits. Moreover, these parameters can be considered as Keplerian elements of: relative, barycentric or Jacobi orbits. One can find arguments that the interpretation of parameters from a multi-Keplerian fit as elements of Keplerian orbits in the Jacobi coordinates is the most proper one, see [Lee and Peale, 2002; Goździewski et al. 2003].To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

We analyze the secular interactions of two coplanar planets based on a high order (order 12) expansion of the perturbative potential in powers of the eccentricities. The model depends on only two parameters (the ratio of semi-major axis and the mass ratio of the planets) and can be reduced to a one degree of freedom system, allowing for an exhaustive parametric analysis. Following Pauwels (1984) we map the phase space on a sphere, avoiding in this way the artificial singularities introduced by other mappings. We show that the twelve order expansion is able to describe correctly most of the exosolar planetary systems discovered so far, even if the eccentricities of these planets are considerably larger than the eccentricities of our own solar system. The expansion is even able to reproduce, at moderate eccentricities, the secular resonances discovered numerically by Michtchenko and Malhotra (2004) at moderate to large eccentricities.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

A complete study is made of the 5/2 resonant motion of two planets revolving around a star, in the model of the general planar three body problem. Families of 5/2 resonant symmetric periodic orbits are computed numerically, for the masses of the extrasolar system 47 UMa. The phase of the two planets (alignment or antialignment of perihelia and position of each planet at perihelion or aphelion) plays an important role, and the change of the phase, other things being the same, may destabilize the system. Stable motion exists even in the case where the two planetary orbits intersect. A small value of the eccentricities, for the same phase, stabilizes the system. The above results are applied to the study of 47 UMa, which according to some observations is close to the 5/2 resonance.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

We investigate the dynamical behaviour of a simplified model of our planetary system (Mercury and the planets Uranus and Neptune were excluded) when we change the mass of the Earth via a mass factor $\kappa_E \in [1,300]$. This is done to study the motions in this “model planetary system” as an example for extrasolar systems. It is evident that the new systems under consideration can only serve as a model for a limited number of exosystems because they have massive planets sometimes with large orbital eccentricities. We did these numerical experiments using an already well tested numerical integration method (LIE-integration) in the framework of the Newtonian equations of motions. We can show that these planetary systems are very stable up to several hundred earth masses, but for some specific values of $\kappa_E$ they show a typical chaotic behaviour already in the semi-major axis. It is know from the inner Solar System that the planets move in a small region of weak chaos, but this behaviour (close to $\kappa_E=5$) was quite unexpected. We then use a $1^{st}$ order secular theory to explain the appearance of chaos. The results may serve for a better understanding of the dynamics of some extrasolar planetary systems.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Among more than 120 discovered exo-planets, less than 20 were found in double star systems. Out of this sample we studied the planetary motion in those systems that can be regarded as close binaries, i.e. HD41004 AB, $\gamma$ Cephei and Gliese 86. In this study we concentrate on the first two systems, where the secondaries are M4 V dwarfs at about 20 AU from the host-star. A comparison of previous studies – where the dynamical behavior was studied in the (semi-major axis, inclination) plane (see Dvorak et al. 2003a) for $\gamma$ Cephei and Pilat-Lohinger & Funk (2004) for HD41004 A) – shows significant differences in the stability maps. Our numerical investigation examines the region between 0.5 and 1.2 AU, which is influenced mainly by mean motion resonances when the initial position of the detected planet $a_{gp} < 1.5$ AU. If we move the planet farther away from the host-star (to distances $> 1.5$ AU) we observe an arc-shaped chaotic structure in the dynamical map.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Regularity and chaos of “quasi-stable” (i.e. appearing stable during a finite interval of time) planetary orbits around one component of binary stars is investigated for different values of the binary's mass ratio and orbital eccentricity $e$. The behavior of fictitious planetary orbits around 16 Cyg B-like stars is presented. Among the quasi-stable orbits we found that there exists a (“stability zone”) for every values of $e$, but that the existence of nearly-circular planetary orbits is restricted to values of $e$ less than 0.8. However, not all the quasi-stable orbits are regular: emergence of chaos when $e$ increases is shown in two sets of quasi-stable orbits (each set has fixed initial conditions for the planet, but varying values for $e$ from 0 to 0.99). In the first set, which lies in the “heart” of the stability zone, chaos appears only when $e$ approaches 1. The second one is near the border of the stability zone, and chaos appears as soon as $e$ reaches 0.7. The influence of the binary's orbital eccentricity on the limit between regularity and chaos is therefore stronger in the latter case (wider planetary orbits) than for the former one (closer planetary orbits).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

The photographic observations of 51 Peg with the known planetary companion which has been discovered on the basis of radial velocities (Mayor and Quelos 1995), are being performed at Pulkovo since 1995 by means of 65 cm refractor. So far 46 plates with 170 individual positions have been obtained. The mean error of one exposure is $0.^{\prime\prime}031$ and the error of one plate is $0.^{\prime\prime}020$. The external error of one plate or the error of unit weight is $0.^{\prime\prime}033$, while the error of one yearly position equals to $0.^{\prime\prime}010$. The aim of our observations is to investigate whether this star has some satellites of low mass (stellar or substellar) with periods of rotation from some years to a decade, or more. At present we have the possibility to study its motion over a time span of 8 years and to learn of probable perturbations due to the presence of possible satellites with the periods 0.5–6.0 years and with masses more than $0.2M_\odot$. The absence of satellites with such periods and masses has been shown by means of our observations. The results are compared with the Pulkovo series of the star Gliese 623, whose satellite with the period of $3.76$ yr and mass of $0.09M_\odot$, has been confirmed by observations.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Results of the photographic observations of 61 Cyg are given. The orbital elements and mass ratio of the components of the pair are calculated. The preliminary orbit of a hypothetical satellite with a period of 6.5 yr is given.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

The early gas and dust protosolar nebula of the solar composition is considered analytically. A simultaneous formation of the sun and all the planets around it ($\approx 5 \times 10^9$ yr ago) through a local gravitational Jeans-type instability of small-amplitude gravity perturbations in the nebula disk is suggested. It is shown that a collective process, forming the basis of the disk instability hypothesis, solves with surprising simplicity the two main problems of the dynamical characteristics of the system, which are associated with its observed spacing and orbital momentum distribution, namely, Bode's law on planet spacing and the concentration of angular momentum in the planets and mass in the sun. Besides, the analysis is found to imply the existence of new planets or other Kuiper-type belts of comets at mean distances from the sun of 87 AU, 151 AU, 261 AU, 452 AU, 781 AU (Mercury, Venus, $\ldots$, Asteroid belt, $\ldots$, Neptune, Kuiper belt, new planets or other Kuiper-type belts).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

The orbital evolution of planetary systems similar to our Solar one represents one of the most important problems of Celestial Mechanics. In the present work we use Jacobian coordinates, introduce two systems of osculating elements, construct the Hamiltonian expansions in Poisson series for all the elements for the planetary three-body problem (including the problem Sun–Jupiter–Saturn). Further we construct the averaged Hamiltonian by the Hori–Deprit method with accuracy up to second order with respect to the small parameter, the generating function, the change of variables formulae, and the right-hand sides of the averaged equations. The averaged equations for the Sun–Jupiter–Saturn system are integrated numerically over a time span of 10 Gyr. The Liapunov Time turns out to be 14 Myr (Jupiter) and 10 Myr (Saturn).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

We are witnessing tremendous progress in the nascent multidisciplinary field of astrobiology, encompassing the origin and evolution of life in the cosmic context. One of the key concepts recently introduced in this field is the Galactic Habitable Zone (GHZ): an interval of galactocentric distances convenient for formation of stars possessing habitable planets. The boundaries of the GHZ are still poorly understood, however. Here we present a comparative analysis of various proposals for the mechanisms determining the GHZ boundaries, as well as different numerical values obtained. When joined with the models of Galactic stellar distribution, this gives us a better handle on the number of potential life-bearing sites.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

In this work, we review the analytical and semi-analytical tools introduced to deal with resonant proper elements and their applications to the Trojan asteroids, the numerical computation of synthetic proper elements for resonant and non resonant asteroids, and the introduction of proper elements for planet crossing asteroids. We discuss the applications and accuracy of these methods and present some comparisons between them.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

The paper presents a review of the recent contributions and open questions concerning the families of asteroids. Due to the availability of very large catalogues (synthetic and analytical proper elements of the asteroids and large observational surveys of their spectra) and to the introduction of non gravitational forces in their determination, the concept of static family has disappeared, to be replaced by this of dynamical families. The proper elements are not constant anymore but are ageing on very long timescales. The size distributions of the populations of asteroids, in and out the families, their ages, the ejection velocities of the fragments after an impact, have been reconsidered by several teams of research, with this new approach. Parallel numerical simulations of collisions and fragmentations of bodies have showed that most of the asteroids are likely rubble piles or agglomerates than monolithic blocks. The methods of classification have been refined and combine, in their newest versions, the dynamics and the observations, working now on 5 dimensional space instead of 3. A series of sub families of the large well-known families have been recently identified, using catalogues with more than 100 000 asteroids (the cluster Karin for example).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Gravity is the most important force to affect the motion of bodies in the Solar system. At small sizes, however, additional forces must be taken into account to explain fine details of their translational and rotational motion, as well as parameters of their populations. This is because the strength of the non-gravitational perturbations typically increases as $\simeq 1/D$ toward small sizes $D$. The principal perturbation acting on macroscopic main-belt bodies (sizes up to several kilometers for timescales of about billion years) is due to the anisotropic thermal re-radiation of the absorbed sunlight. In orbital dynamics this is known as the Yarkovsky effect, while in rotational dynamics the same physical phenomenon is called the YORP (Yarkovsky-O'Keefe-Radzievskii-Paddack) effect. We review the main observational implications of the Yarkovsky/YORP effects as understood and evidenced today.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

In the beginning we review briefly the evolution of the ideas on the motion of the bodies in our solar system, from Newton's clockwork Universe to the presently accepted ubiquity of chaotic transport in the asteroid belt. Then we discuss the result of chaotic motion, which is transport in phase space, and we introduce the concept of diffusion of an asteroid in action space. We proceed by reviewing recent work on numerical as well as analytical study of asteroids following chaotic trajectories and we summarize the main results. We present several applications of the theoretical modelling of asteroid motion as diffusion in action space, to problems of specific interest.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Yarkovsky effect (YE), a tiny nongravitational force due to radiative recoil of the anisotropic thermal emission, is known to secularly affect the orbital semimajor axis. Therefore, angular phases such as longitude in orbit or proper longitude of node undergo a quadratic perturbation. This is fast enough to allow direct detection of the YE. The first positive case was obtained for (6489) Golevka in 2003 and prospects are very good for many more detections in the near future. To make productive scientific use of the YE detections, we need to accurately compute its strength for a given body. Simple models, available so far, will likely not be adequate in many of the forthcoming YE detection possibilities. We thus developed a complex numerical approach capable of treating most of them. Here we illustrate its power by discussing the cases of: (i) Toutatis, with a tumbling (non-principal-axis) rotation state, and (ii) 2000 DP107, a binary system.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html