1. Introduction 2. Radial velocities 3. Astrometry 4. Timing 5. Microlensing 6. Transits 7. Imaging 8. Host stars 9. Brown dwarfs and free-floating planets 10. Formation and evolution 11. Interiors and atmospheres 12. The Solar System Appendixes References Index.

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The Exoplanet Handbook

PLANETARY SYSTEMS, the solar system amongst them, are believed to form as inevitable and common byproducts of star formation For orientation, an overview of the processes described in this chapter is as follows The present paradigm starts with star formation in molecular clouds Brown dwarfs are formed as the lowmass tail of this process, although some may be formed as a high-mass tail of planet formation Gas and dust in the collapsing molecular cloud which does not fall directly onto the protostar resides in a relatively long-lived accretion disk which provides the environment for the subsequent stages of planet formation Terrestrial-mass planets are formed within the disk through the progressive agglomeration of material denoted, as it grows in size, as dust, rocks, planetesimals and protoplanets A similar process typically occurring further out in the disk results in the cores of giant planets, which then gravitationally accumulate their mantles of ice and/or gas As the planet-forming bodies grow in mass, growth and dynamics become more dominated by gravitational interactions Towards the final phases, and before the remaining gas is lost through accretion or dispersal, the gas provides a viscous medium at least partially responsible for planetary migration Some migration also occurs during these later stages as a result of gravitational scattering between the (proto-)planets and the residual sea of planetesimals The final structural stabilisation of the planetary system may be affected by planet–planet interactions, until a configuration emerges which may be dynamically stable over billions of years The current observational data for exoplanet systems is broadly compatible with this overall picture Other constraints come from a substantial body of detailed observations of the solar system (Chapter 12) Context and present paradigm An understanding of howplanets formis essential in understanding and interpreting the considerable range of observed planetary system architectures and dynamics Today, the most widely considered solar nebula theory holds that planet formation in the solar system, and by inference in other exoplanet systems, follows on from the process of star formation and accretion disk formation, through the agglomeration of residual material as the protoplanetary disk collapses and evolves

The Exoplanet Handbook: Formation and evolution

The Exoplanet Handbook: Brown dwarfs and free-floating planets

Performance of the Taylor series method for ODEs/DAEs

We present a method for studying the secular gravitational dynamics of hierarchical multiple systems consisting of nested binaries, which is valid for an arbitrary number of bodies and arbitrary hierarchical structure. We derive the Hamiltonian of the system and expand it in terms of the -- assumed to be -- small ratios $x_i$ of binary separations. At the lowest nontrivial expansion order (quadrupole order, second order in $x_i$), the Hamiltonian consists of terms which, individually, depend on binary pairs. At higher orders, in addition to terms depending on binary pairs, we also find terms which, individually, depend on more than two binaries. In general, at order $n$ in $x_i$, individual terms depend on at most $n-1$ binaries. We explicitly derive the Hamiltonian including all terms up and including third order in $x_i$ (octupole order), and including the binary pairwise terms up and including fifth order in $x_i$. These terms are orbit averaged, and we present a new algorithm for efficiently solving the equations of motion. This algorithm is highly suitable for studying the secular evolution of hierarchical systems with complex hierarchies, making long-term integrations of such systems feasible. We show that accurate results are obtained for multiplanet systems with semimajor axis ratios as large as $\approx 0.4$, provided that high-order terms are included. In addition to multiplanet systems with a single star, we apply our results to multistar systems with multiple planets.

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Secular dynamics of hierarchical multiple systems composed of nested binaries, with an arbitrary number of bodies and arbitrary hierarchical structure. First applications to multiplanet and multistar systems

Stability criteria in many-body systems: I. An empirical stability criterion for co-rotational three-body systems

The analytical stability criterion applicable to coplanar hierarchical three-body systems described in the first paper of this series, Walkeret al. (1980), is modified to give an exact representation ofHill-type stability in all such cases. The dependence of the stability on all orbital parameters (in the coplanar case) is taken into account. The criterion for stability is now dependant upon the participating masses, the elements of the initial osculating Keplerian orbits of the system (viz. the orbits ofm2 aboutm1 andm3 about the mass-centre of the (m1,m2) system) and the positions within these orbits.

Stability criteria in many-body systems

In previous papers of this series the stability of hierarchical many-body dynamical systems has been considered in terms of parameters which are a measure of the perturbation imposed on the disturbed Keplerian orbit of each member of a system by the other bodies present.If these parameters are small enought then the initial hierarchical order will be undisturbed for some time i.e. the status quo will be maintained and to that extent stability will exist. In the light of this criterion the appropriate parameters for the general four-body problem are considered. Two distinct hierarchical arrangments of four-body systems are possible; these are classifid and an examination of the relevant stability parameters is made in each case.It is shown how regions may be determined within which real four-body systems can exist and may be stable. It is also shown how the various types of possible systems, (e.g.,Star-Star-Star-Star, Star-Planet-Planet-Star, etc.) within these regions may be identified.

Stability criteria in many-body systems: V: On the totality of possible hierarchical general four-body systems

In this paper two cases of hierarchical arrangement common in many types of dynamical systems are considered viz. the planetary-type arrangement and the multiple stellar system type of arrangement.The former is considered in order to investigate the applicability of the empirical stability parameters derived in the first paper of this series (Walkeret al., 1980) to planetary systems.The second type of system has not been considered thus far in the series. A coordinate system following the method of the Jacobian coordinate system is developed for these systems. Then, following a procedure similar to that of the abovementioned paper, empirical stability parameters are derived for those systems. Finally we use the method to examine the stability of the sextuple Castor system.

Stability criteria in many-body systems: IV. Empirical stability parameters for general hierarchical dynamical systems

The stability parameters developed and discussed in the first paper of this series (Walkeret al., 1980) are used to determine empirically, by means of numerical integration experiment, regions of stability for corotational, coplanar, hierarchical three-body systems. The initially circular case of these systems is studied: the components of the close binary are taken to move initially in circular orbits with respect to their common mass-centre, the third mass initially moving in a circular orbit with respect to the same mass-centre such that its orbit lies wholly outside those of the former two masses.The stability of these systems is then studied by reference to the empirical stability parameters and the initial ratio of the semi-major axes of the orbit of the close binary to that of the third mass about the binary's mass-centre, which is less than unity. For given values of the stability parameters it is determined how the stability of a system is affected by changes in the ratio of the semi-major axes. It is found that an upper limit to this ratio exists which determines the region of stability for such systems. It is also found possible, in the region of instability, to predict how unstable a system will be i.e. crudely speaking, the number of orbits it may be expected to execute before some gross instability sets in.The effect commensurabilities in mean motion have on the stability of these systems is also considered. It is generally found that these commensurabilities enhance the stability of these systems. The predictive powers of the method are then tested: using many test cases it is seen how accurately the stability or instability of a system may be predicted.

Ian W. Walker

Papers

Stability criteria in many-body systems: I. An empirical stability criterion for co-rotational three-body systems

Stability criteria in many-body systems

Stability criteria in many-body systems: V: On the totality of possible hierarchical general four-body systems

Stability criteria in many-body systems: IV. Empirical stability parameters for general hierarchical dynamical systems

Stability criteria in many-body systems: III. Empirical stability regions for corotational, coplanar, hierarchical three-body systems