Dynamical relaxation and massive extrasolar planets

Following the suggestion of Black that some massive extrasolar planets may be associated with the tail of the distribution of stellar companions, we investigate a scenario in which 5 N 100 planetary mass objects are assumed to form rapidly through a fragmentation process occuring in a disc or protostellar envelope on a scale of 100 au. These are assumed to have formed rapidly enough through gravitational instability or fragmentation that their orbits can undergo dynamical relaxation on a time-scale of ∼100 orbits.
Under a wide range of initial conditions and assumptions, the relaxation process ends with either (i) one potential 'hot Jupiter' plus up to two 'external' companions, i.e. planets orbiting near the outer edge of the initial distribution; (ii) one or two 'external' planets or even none at all; (iii) one planet on an orbit with a semi-major axis of 10 to 100 times smaller than the outer boundary radius of the inital distribution together with an 'external' companion. Most of the other objects are ejected and could contribute to a population of free-floating planets. Apart from the potential 'hot Jupiters', all the bound objects are on orbits with high eccentricity, and also with a range of inclination with respect to the stellar equatorial plane. We found that, apart from the close orbiters, the probability of ending up with a planet orbiting at a given distance from the central star increases with the distance. This is because of the tendency of the relaxation process to lead to collisions with the central star. The scenario we envision here does not impose any upper limit on the mass of the planets. We discuss the application of these results to some of the more massive extrasolar planets.     

Stable satellites around extrasolar giant planets

In this work, we study the stability of hypothetical satellites of extrasolar planets. Through numerical simulations of the restricted elliptic three-body problem we found the borders of the stable regions around the secondary body. From the empirical results, we derived analytical expressions of the critical semimajor axis beyond which the satellites would not remain stable. The expressions are given as a function of the eccentricities of the planet, e _P, and of the satellite, e _sat. In the case of prograde satellites, the critical semimajor axis, in the units of Hill's radius, is given by a _E≈ 0.4895   (1.0000 − 1.0305 e _P− 0.2738 e _sat). In the case of retrograde satellites, it is given by a _E≈ 0.9309  (1.0000 − 1.0764 e _P− 0.9812 e _sat). We also computed the satellite stability region ( a _E) for a set of extrasolar planets. The results indicate that extrasolar planets in the habitable zone could harbour the Earth-like satellites.     

Dynamical relaxation and the orbits of low-mass extrasolar planets

We consider the evolution of a system containing a population of massive planets formed rapidly through a fragmentation process occurring on a scale on the order of 100 au and a lower mass planet that assembles in a disc on a much longer time-scale. During the formation phase, the inner planet is kept on a circular orbit owing to tidal interaction with the disc, while the outer planets undergo dynamical relaxation. Interaction with the massive planets left in the system after the inner planet forms may increase the eccentricity of the inner orbit to high values, producing systems similar to those observed.     

Data analysis on the extrasolar planets using robust clustering

We use both the conventional and more recently developed methods of cluster analysis to study the data of extrasolar planets (exoplanets). Using the data set with planetary mass M , orbital period P and orbital eccentricity e , we investigate the possible clustering in the  ln  M ,  ln  P ,  ln  P –ln  M ,   e   and ln   P – e spaces. There are two main implications: (1) mass distribution is continuous, and (2) orbital population could be classified into three clusters, which correspond to the exoplanets in the regimes of tidal, ongoing tidal and disc interaction, respectively.     

Dynamics of possible Trojan planets in binary systems

This paper is devoted to the dynamical stability of possible Trojan planets in binaries and in binary systems where one of the substellar companions is not larger than a brown dwarf. Using numerical integrations, we investigated how the size of the stable region around the Lagrangian point L₄ depends on the mass parameter and the eccentricity of the secondary star. An additional goal of this work was to create a catalogue of all possible candidates, which could be useful for future observations to detect such objects.     

High-inclination planets and asteroids in multistellar systems

The Kozai mechanism often destabilizes high-inclination orbits. It couples changes in the eccentricity and inclination, and drives high inclination, circular orbits to low inclination, eccentric orbits. In a recent study of the dynamics of planetesimals in the quadruple star system HD 98800, there were significant numbers of stable particles in circumbinary polar orbits about the inner binary pair which are apparently able to evade the Kozai instability.
Here, we isolate this feature and investigate the dynamics through numerical and analytical models. The results show that the Kozai mechanism of the outer star is disrupted by a nodal libration induced by the inner binary pair on a shorter time-scale. By empirically modelling the period of the libration, a criteria for determining the high-inclination stability limits in general triple systems is derived. The nodal libration feature is interesting and, although affecting inclination and node only, shows many parallels to the Kozai mechanism. This raises the possibility that high-inclination planets and asteroids may be able to survive in multistellar systems.     

A model-independent test of the spatial variations of the Newtonian gravitational constant in some extrasolar planetary systems

In this paper, we directly constrain possible spatial variations of the Newtonian gravitational constant G over the range  ≈ 0.01–5 au  in various extrasolar multiplanet systems. Using the third Kepler law, we determine the quantity  Γ_XY= G _X/ G _Y  for each couple of planets X and Y located at different distances from their parent star; deviations of the measured values of Γ from unity would signal variations of G . The obtained results for  η= 1 −Γ  are found to be very compatible with zero within the experimental errors  (η/δη≈ 0.2–0.3)  . We make a comparison with an analogous test previously performed in our Solar system.     

On the global stability of single‐planet systems

In this paper, we estimate the global stability properties of single‐planet systems by using a catalogue of stability maps. The data of the catalogue were used to generate probability values on the mass parameter–eccentricity plane for the occurrence of stable orbits. We showed that the probability data can be well approximated by a second order surface. Using the resulted formula the likelihood of finding Earth‐like planets in single‐planet systems can be easily estimated. As an example, we derived estimations for four known exoplanetary systems. Our formula can be useful in selecting target stars for future space missions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Using long-term transit timing to detect terrestrial planets

We propose that the presence of additional planets in extrasolar planetary systems can be detected by long-term transit timing studies. If a transiting planet is on an eccentric orbit then the presence of another planet causes a secular advance of the transiting planet's pericentre over and above the effect of general relativity. Although this secular effect is impractical to detect over a small number of orbits, it causes long-term differences when future transits occur, much like the long-term decay observed in pulsars. Measuring this transit-timing delay would thus allow the detection of either one or more additional planets in the system or the first measurements of non-zero oblateness ( J ₂) of the central stars.     

Two-integral distribution functions for axisymmetric systems

Some formulae are presented for finding two-integral distribution functions (DFs) which depend only on the two classical integrals of the energy and the magnitude of the angular momentum with respect to the axis of symmetry for stellar systems with known axisymmetric densities. They come from a combination of the ideas of Eddington and Fricke and they are also an extension of those shown by Jiang and Ossipkov for finding anisotropic DFs for spherical galaxies. The density of the system is required to be expressed as a sum of products of functions of the potential and of the radial coordinate. The solution corresponding to this type of density is in turn a sum of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. The product of the density and its radial velocity dispersion can be also expressed as a sum of products of functions of the potential and of the radial coordinate. It can be further known that the density multiplied by its rotational velocity dispersion is equal to a sum of products of functions of the potential and of the radial coordinate minus the product of the density and the square of its mean rotational velocity. These formulae can be applied to the Binney and the Lynden-Bell models. An infinity of the odd DFs for the Binney model can be also found under the assumption of the laws of the rotational velocity.     

The Hill stability of a binary or planetary system during encounters with a third inclined body

The dynamical interaction of a binary or planetary system and a third body moving on a parabolic orbit inclined to the system is discussed in terms of Hill stability for the full three-body problem. The situation arises in binary star disruption and exchange, in extrasolar planetary system disruption, exchange and capture. It is found that increasing the inclination of the third body decreases the Hill regions of stability. This makes exchange or disruption of the component masses more likely as does increasing the eccentricity of the binary.
The stability criteria are applied to determine possible disruption and capture distances for currently known extrasolar planetary systems.     

Obliquity evolution of extrasolar terrestrial planets

We have investigated the obliquity evolution of terrestrial planets in habitable zones (at ∼1 AU) in extrasolar planetary systems, due to tidal interactions with their satellite and host star with wide varieties of satellite-to-planet mass ratio (m/Mp) and initial obliquity (γ₀), through numerical calculations and analytical arguments. The obliquity, the angle between planetary spin axis and its orbit normal, of a terrestrial planet is one of the key factors in determining the planetary surface environments. A recent scenario of terrestrial planet accretion implies that giant impacts of Mars-sized or larger bodies determine the planetary spin and form satellites. Since the giant impacts would be isotropic, tilted spins (sinγ₀∼1) are more likely to be produced than straight ones (sinγ₀∼0). The ratio m/Mp is dependent on the impact parameters and impactors' mass. However, most of previous studies on tidal evolution of the planet-satellite systems have focused on a particular case of the Earth-Moon systems in which m/Mp?0.0125 and γ₀∼10° or the two-body planar problem in which γ₀=0° and stellar torque is neglected. We numerically integrated the evolution of planetary spin and a satellite orbit with various m/Mp (from 0.0025 to 0.05) and γ₀ (from 0° to 180°), taking into account the stellar torques and precessional motions of the spin and the orbit. We start with the spin axis that almost coincides with the satellite orbit normal, assuming that the spin and the satellite are formed by one dominant impact. With initially straight spins, the evolution is similar to that of the Earth-Moon system. The satellite monotonically recedes from the planet until synchronous state between the spin period and the satellite orbital period is realized. The obliquity gradually increases initially but it starts decreasing down to zero as approaching the synchronous state. However, we have found that the evolution with initially tiled spins is completely different. The satellite's orbit migrates outward with almost constant obliquity until the orbit reaches the critical radius ∼10-20 planetary radii, but then the migration is reversed to inward one. At the reversal, the obliquity starts oscillation with large amplitude. The oscillation gradually ceases and the obliquity is reduced to ∼0° during the inward migration. The satellite eventually falls onto the planetary surface or it is captured at the synchronous state at several planetary radii. We found that the character change of precession about total angular momentum vector into that about the planetary orbit normal is responsible for the oscillation with large amplitude and the reversal of migration. With the results of numerical integration and analytical arguments, we divided the m/Mp-γ₀ space into the regions of the qualitatively different evolution. The peculiar tidal evolution with initially tiled spins give deep insights into dynamics of extrasolar planet-satellite systems and discussions of surface environments of the planets.     

Two-integral distribution functions for axisymmetric stellar systems with separable densities

We show different expressions of distribution functions (DFs) which depend only on the two classical integrals of the energy and the magnitude of the angular momentum with respect to the axis of symmetry for stellar systems with known axisymmetric densities. The density of the system is required to be a product of functions separable in the potential and the radial coordinates, where the functions of the radial coordinate are powers of a sum of a square of the radial coordinate and its unit scale. The even part of the two-integral DF corresponding to this type of density is in turn a sum or an infinite series of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. A similar expression of its odd part can be also obtained under the assumption of the rotation laws. It can be further shown that these expressions are in fact equivalent to those obtained by using Hunter & Qian's contour integral formulae for the system. This method is generally computationally preferable to the contour integral method. Two examples are given to obtain the even and odd parts of their two-integral DFs. One is for the prolate Jaffe model and the other for the prolate Plummer model.
It can be also found that the Hunter–Qian contour integral formulae of the two-integral even DF for axisymmetric systems can be recovered by use of the Laplace–Mellin integral transformation originally developed by Dejonghe.     

Gravothermal catastrophe in anisotropic spherical systems

In this paper we investigate the gravothermal instability of spherical stellar systems endowed with a radially anisotropic velocity distribution. We focus our attention on the effects of anisotropy on the conditions for the onset of instability and in particular we study the dependence of the spatial structure of critical models on the amount of anisotropy present in a system. The investigation has been carried out by the method of linear series which has already been used in the past to study the gravothermal instability of isotropic systems._   We consider models described by King, Wilson and Woolley–Dickens distribution functions. In the case of King and Woolley–Dickens models, our results show that, for quite a wide range of the amount of anisotropy in the system, the critical value of the concentration of the system (defined as the ratio of the tidal to the King core radius of the system) is approximately constant and equal to the corresponding value for isotropic systems. Only for very anisotropic systems does the critical value of the concentration start to change and it decreases significantly as the anisotropy increases and penetrates the inner parts of the system. For Wilson models the decrease of the concentration of critical models is preceded by an intermediate regime in which critical concentration increases, reaches a maximum and then starts to decrease. The critical value of the central potential always decreases as the anisotropy increases.     

Maximum-likelihood method for estimating the mass and period distributions of extrasolar planets

We investigate the distribution of mass M and orbital period P of extrasolar planets, taking account of selection effects caused by the limited velocity precision and duration of existing surveys. We fit the data on 72 planets to a power-law distribution of the form  d n = CM ^−α P ^−β(d M / M )(d P / P )  , and find  α= 0.11 ± 0.10  ,  β=−0.27 ± 0.06  for   M ≲ 10 M _J  , where   M _J  is the mass of Jupiter. The correlation coefficient between these two exponents is −0.31, indicating that uncertainties in the two distributions are coupled. We estimate that 4 per cent of solar-type stars have companions in the range  1 M _J < M < 10 M _J  ,  2_d < P < 10 yr  .