Gap formation in a self-gravitating disk and the associated migration of the embedded giant planet

We present the results of our recent study on the interactions between a giant planet and a self-gravitating gas disk. We investigate how the disk's self-gravity affects the gap formation process and the migration of the giant planet. Two series of 1-D and 2-D hydrodynamic simulations are performed. We select several surface densities and focus on the gravitationally stable region. To obtain more reliable gravity torques exerted on the planet, a refined treatment of the disk's gravity is adopted in the vicinity of the planet. Our results indicate that the net effect of the disk's selfgravity on the gap formation process depends on the surface density of the disk. We notice that there are two critical values, ΣIand ΣII. When the surface density of the disk is lower than the first one, Σ0 ΣI, the effect of self-gravity suppresses the formation of a gap. When Σ0 ΣI, the self-gravity of the gas tends to benefit the gap formation process and enlarges the width/depth of the gap. According to our 1-D and2-D simulations, we estimate the first critical surface density to be ΣI≈ 0.8 MMSN.This effect increases until the surface density reaches the second critical value ΣII.When Σ0 ΣII, the gravitational turbulence in the disk becomes dominant and the gap formation process is suppressed again. Our 2-D simulations show that this critical surface density is around 3.5 MMSN. We also study the associated orbital evolution of a giant planet. Under the effect of the disk's self-gravity, the migration rate of the giant planet increases when the disk is dominated by gravitational turbulence. We show that the migration timescale correlates with the effective viscosity and can be up to 104yr.     

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.     

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  .     

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.     

Stability of fictitious Trojan planets in extrasolar systems

Our work deals with the dynamical possibility that in extrasolar planetary systems a terrestrial planet may have stable orbits in a 1:1 mean motion resonance with a Jovian like planet. We studied the motion of fictitious Trojans around the Lagrangian points L₄/L₅ and checked the stability and/or chaoticity of their motion with the aid of the Lyapunov Indicators and the maximum eccentricity. The computations were carried out using the dynamical model of the elliptic restricted three‐body problem that consists of a central star, a gas giant moving in the habitable zone, and a massless terrestrial planet. We found 3 new systems where the gas giant lies in the habitable zone, namely HD99109, HD101930, and HD33564. Additionally we investigated all known extrasolar planetary systems where the giant planet lies partly or fully in the habitable zone. The results show that the orbits around the Lagrangian points L₄/L₅ of all investigated systems are stable for long times (10⁷ revolutions). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Predictions for the frequency and orbital radii of massive extrasolar planets

We investigate the migration of massive extrasolar planets caused by gravitational interaction with a viscous protoplanetary disc. We show that a model in which planets form at 5 au at a constant rate, before migrating, leads to a predicted distribution of planets that is a steeply rising function of log( a ), where a is the orbital radius. Between 1 and 3 au, the expected number of planets per logarithmic interval in a roughly doubles. We demonstrate that, once selection effects are accounted for, this is consistent with current data, and then extrapolate the observed planet fraction to masses and radii that are inaccessible to current observations. In total, approximately 15 per cent of stars targeted by existing radial velocity searches are predicted to possess planets with masses  0.3< M _p sin( i )<10 M _J  and radii  0.1< a <5 au  . A third of these planets (around 5 per cent of the target stars) lie at the radii most amenable to detection via microlensing. A further  5–10  per cent of stars could have planets at radii of  5< a <8 au  that have migrated outwards. We discuss the probability of forming a system (akin to the Solar system) in which significant radial migration of the most massive planet does not occur. Approximately  10–15  per cent of systems with a surviving massive planet are estimated to fall into this class. Finally, we note that a smaller fraction of low-mass planets than high-mass planets is expected to survive without being consumed by the star. The initial mass function for planets is thus predicted to rise more steeply towards small masses than the observed mass function.     

Symmetric and asymmetric librations in extrasolar planetary systems: a global view

We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, ...), (2) the resonances (2n + 1):(2n-1), (3:1, 5:3, ...) and (3) all other resonances. The topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits.     

Giant Metrewave Radio Telescope low-frequency observations of extrasolar planetary systems

Extrasolar planets are expected to emit detectable low-frequency radio emission. In this paper, we present results from new low-frequency observations of two extrasolar planetary systems (Epsilon Eridani and HD 128311) taken at 150 MHz with the Giant Metrewave Radio Telescope (GMRT). These two systems have been chosen because the stars are young (with ages <1 Gyr) and are likely to have strong stellar winds, which will increase the expected radio flux. The planets are massive (presumably) gas giant planets in longer period orbits, and hence will not be tidally locked to their host star (as is likely to be the case for short-period planets) and we would expect them to have a strong planetary dynamo and magnetic field. We do not detect either system, but are able to place tight upper limits on their low-frequency radio emission, at levels comparable to the theoretical predictions for these systems. From these observations, we have a 2.5σ limit of 7.8 mJy for ε Eri and 15.5 mJy for HD 128311. In addition, these upper limits also provide limits on the low-frequency radio emission from the stars themselves. These results are discussed and also the prospects for the future detection of radio emission from extrasolar planets.