Planetary migration to large radii

There is evidence for the existence of massive planets at orbital radii of several hundred au from their parent stars where the time-scale for planet formation by core accretion is longer than the disc lifetime. These planets could have formed close to their star and then migrated outwards. We consider how the transfer of angular momentum by viscous disc interactions from a massive inner planet could cause significant outward migration of a smaller outer planet. We find that it is in principle possible for planets to migrate to large radii. We note, however, a number of effects which may render the process somewhat problematic.     

Disc–Planet Interactions: Migration and Resonances in Extrasolar Planetary Systems

We consider orbital resonances in multiplanet systems. These are expected to arise during or just after formation in a gaseous disc. Disc–planet interaction naturally produces orbital migration and circularization through the action of tidal torques which in turn may lead to an orbital resonance. The mass and angular momentum content of the disc is likely to be comparable to that in the planets so that it is essential to fully incorporate the disc in the analysis.We study the orbital evolution of two planets locked in 2:1 commensurability through migration tidally induced by the disc using both analytic methods and numerical hydrodynamic simulations. The planets are assumed to orbit in an inner cavity containing at most only a small amount of disc material. Results are found to be sensitive to initial surface density profile, planet masses and disc parameters. The evolution may range between attaining and subsequently maintaining a resonance lock with two angles librating to divergent migration with no commensurability formed. In the former case eccentricities increase monotonically with time while the system undergoes inward migration. If the migration is halted by loss of the disc leaving the planets in a final configuration, there is likely to be a low probability of seeing resonant planets at small radii as well as a sensitive dependence on past history.We have also considered a multiplanet system in secular apsidal resonance. We consider the system as being in just one secular normal mode and include the effects of a gaseous disc. It is suggested that a normal mode may be selected by adding in some weak dissipative process in the disc and that it may remain, involving only the planets, when the disc is slowly removed.     

On the migration of a system of protoplanets

The evolution of a system consisting of a protoplanetary disc with two embedded Jupiter-sized planets is studied numerically. The disc is assumed to be flat and non-self-gravitating; this is modelled by the planar (two-dimensional) Navier–Stokes equations. The mutual gravitational interaction of the planets and the star, and the gravitational torques of the disc acting on the planets and the central star are included. The planets have an initial mass of one Jupiter mass M _Jup each, and the radial distances from the star are one and two semimajor axes of Jupiter, respectively.
During the evolution a joint wide annular gap is created by the planets. Both planets increase their mass owing to accretion of gas from the disc: after about 2500 orbital periods of the inner planet it has reached a mass of 2.3  M _Jup, while the outer planet has reached a mass of 3.2  M _Jup. The net gravitational torques exerted by the disc on the planets result in an inward migration of the outer planet on time-scales comparable to the viscous evolution time of the disc. The semimajor axis of the inner planet remains constant as there is very little gas left in its vicinity to induce any migration. When the distance of close approach eventually becomes smaller than the mutual Hill radius, the eccentricities increase strongly and the system may become unstable.
If disc depletion occurs rapidly enough before the planets come too close to each other, a stable system similar to our own Solar system may remain. Otherwise the orbits may become unstable and produce systems like υ And.     

On the evolution of giant protoplanets forming in circumbinary discs

We present the results of hydrodynamic simulations of Jovian mass protoplanets that form in circumbinary discs. The simulations follow the orbital evolution of the binary plus protoplanet system acting under their mutual gravitational forces, and forces exerted by the viscous circumbinary disc. The evolution involves the clearing of the inner circumbinary disc initially, so that the binary plus protoplanet system orbits within a low density cavity. Continued interaction between disc and protoplanet causes inward migration of the planet towards the inner binary. Subsequent evolution can take three distinct paths: (i) the protoplanet enters the 4 : 1 mean motion resonance with the binary, but is gravitationally scattered through a close encounter with the secondary star; (ii) the protoplanet enters the 4 : 1 mean motion resonance, the resonance breaks, and the planet remains in a stable orbit just outside the resonance; (iii) when the binary has initial eccentricity   e _bin≥ 0.2  , the disc becomes eccentric, leading to a stalling of the planet migration, and the formation of a stable circumbinary planet.
These results have implications for a number of issues in the study of extrasolar planets. The ejection of protoplanets in close binary systems provides a source of 'free-floating planets', which have been discovered recently. The formation of a large, tidally truncated cavity may provide an observational signature of circumbinary planets during formation. The existence of protoplanets orbiting stably just outside a mean motion resonance (4 : 1) in the simulations indicate that such sites may harbour planets in binary star systems, and these could potentially be observed. Finally, the formation of stable circumbinary planets in eccentric binary systems indicates that circumbinary planets may not be uncommon.     

The total number of giant planets in debris discs with central clearings

Infrared spectra from the Spitzer Space Telescope ( SSC ) of many debris discs are well fit with a single blackbody temperature which suggest clearings within the disc. We assume that clearings are caused by orbital instability in multiple planet systems with similar configurations to our own. These planets remove dust-generating planetesimal belts as well as dust generated by the outer disc that is scattered or drifts into the clearing. From numerical integrations, we estimate a minimum planet spacing required for orbital instability (and so planetesimal and dust removal) as a function of system age and planet mass. We estimate that a 10⁸ yr old debris disc with a dust disc edge at a radius of 50 au hosted by an A star must contain approximately five Neptune mass planets between the clearing radius and the iceline in order to remove all primordial objects within it. We infer that known debris disc systems contain at least a fifth of a Jupiter mass in massive planets. The number of planets and spacing required is insensitive to the assumed planet mass. However, an order of magnitude higher total mass in planets could reside in these systems if the planets are more massive.     

Eccentric Extrasolar Planets: The Jumping Jupiter Model

Most extrasolar planets discovered to date are more massive than Jupiter, in surprisingly small orbits (semimajor axes less than 3 AU). Many of these have significant orbital eccentricities. Such orbits may be the product of dynamical interactions in multiplanet systems. We examine outcomes of such evolution in systems of three Jupiter-mass planets around a solar-mass star by integration of their orbits in three dimensions. Such systems are unstable for a broad range of initial conditions, with mutual perturbations leading to crossing orbits and close encounters. The time scale for instability to develop depends on the initial orbital spacing; some configurations become chaotic after delays exceeding 10⁸ y. The most common outcome of gravitational scattering by close encounters is hyperbolic ejection of one planet. Of the two survivors, one is moved closer to the star and the other is left in a distant orbit; for systems with equal-mass planets, there is no correlation between initial and final orbital positions. Both survivors may have significant eccentricities, and the mutual inclination of their orbits can be large. The inner survivor's semimajor axis is usually about half that of the innermost starting orbit. Gravitational scattering alone cannot produce the observed excess of “hot Jupiters” in close circular orbits. However, those scattered planets with large eccentricities and small periastron distances may become circularized if tidal dissipation is effective. Most stars with a massive planet in an eccentric orbit should have at least one additional planet of comparable mass in a more distant orbit.     

HD 12661行星系统构形的形成与稳定性

最近的多普勒观测表明恒星HD 12661周围存在两颗中等偏心率轨道上运行的行星,内行星的最小质量为2．3木星质量,轨道周期为263．6天;外行星的最小质量为1．57木星质量,轨道周期为1444．5天．该系统的稳定性要求两颗行星处在平运动轨道共振．用数值方法研究了该系统形成初期在恒星气体盘作用下的轨道迁移与稳定性,计算了行星在迁移中被平运动共振俘获的概率．发现这两颗行星目前很可能正处在11:2平运动共振边缘,且运动是混沌的,从而澄清了关于系统目前构形的不同说法,并且很可能在系统形成后行星迁移到目前构形时,气体盘几乎消失了．     

The formation of an eccentric gap in a gas disc by a planet in an eccentric orbit

We investigate the effect of a planet on an eccentric orbit on a two-dimensional low-mass gaseous disc. At a planet eccentricity above the planet's Hill radius divided by its semimajor axis, we find that the disc morphology differs from that exhibited by a disc containing a planet in a circular orbit. An eccentric gap is created with eccentricity that can exceed the planet's eccentricity and precesses with respect to the planet's orbit. We find that a more massive planet is required to open a gap when the planet is on an eccentric orbit. We attribute this behaviour to spiral density waves excited at corotation resonances by the eccentric planet. These act to increase the disc's eccentricity and exert a torque opposite in sign to that exerted by the Lindblad resonances. The reduced torque makes it more difficult for waves driven by the planet to overcome viscous inflow in the disc.     

The formation and stability of the configuration of the planetary system HD 126611

Recent Doppler velocity measurements have revealed the existence of two planets orbiting the star HD 12661 on medium-eccentricity orbits. The inner planet has a period of 263.6 d and a mass of 2.3 Jupiter masses, and the outer planet has a period of 1444.5d and a mass of 1.57 Jupiter masses. The stability of this system requires the two planets to be in a state of mean-motion orbit resonances. By numerical method we have studied the orbit migration and stability of the system in its early ages under the action of the proto-stellar disk, and calculated the probabilities of the planets being captured into the mean -motion resonances during their migrations. It is found that at present the two planets are probably situated at the edge of the 11:2 mean-motion resonance and are in chaotic motions. This result may be helpful to clarify the arguments on the present configuration. Besides, it is indicated that very probably, after the formation of the system, the gaseous disk has almost disappeared before the planets migrated to the present configuration.     

Three-dimensional calculations of high- and low-mass planets embedded in protoplanetary discs

We analyse the non-linear, three-dimensional response of a gaseous, viscous protoplanetary disc to the presence of a planet of mass ranging from 1 Earth mass (1 M_⊕) to 1 Jupiter mass (1 M_J) by using the zeus hydrodynamics code. We determine the gas flow pattern, and the accretion and migration rates of the planet. The planet is assumed to be in a fixed circular orbit about the central star. It is also assumed to be able to accrete gas without expansion on the scale of its Roche radius. Only planets with masses   M _p≳ 0.1 M_J  produce significant perturbations in the surface density of the disc. The flow within the Roche lobe of the planet is fully three-dimensional. Gas streams generally enter the Roche lobe close to the disc mid-plane, but produce much weaker shocks than the streams in two-dimensional models. The streams supply material to a circumplanetary disc that rotates in the same sense as the orbit of the planet. Much of the mass supply to the circumplanetary disc comes from non-coplanar flow. The accretion rate peaks with a planet mass of approximately 0.1 M_J and is highly efficient, occurring at the local viscous rate. The migration time-scales for planets of mass less than 0.1 M_J, based on torques from disc material outside the Roche lobes of the planets, are in excellent agreement with the linear theory of type I (non-gap) migration for three-dimensional discs. The transition from type I to type II (gap) migration is smooth, with changes in migration times of about a factor of 2. Starting with a core which can undergo runaway growth, a planet can gain up to a few M_J with little migration. Planets with final masses of the order of 10 M_J would undergo large migration, which makes formation and survival difficult.     

The minimum gap-opening planet mass in an irradiated circumstellar accretion disc

We consider the minimum mass planet, as a function of radius, that is capable of opening a gap in an α-accretion disc. We estimate that a half-Jupiter mass planet can open a gap in a disc with accretion rate     for viscosity parameter  α= 0.01  , and solar mass and luminosity. The minimum mass is approximately proportional to     . This estimate can be used to rule out the presence of massive planets in gapless accretion discs. We identify two radii at which an inwardly migrating planet may become able to open a gap and so slow its migration; the radius at which the heating from viscous dissipation is similar to that from stellar radiation in a flared disc, and the radius at which the disc becomes optically thin in a self-shadowed disc. In the inner portions of the disc, we find that the minimum planet mass required to open a gap is only weakly dependent on radius. If a migrating planet is unable to open a gap by the time it reaches either of the transition radii, then it is likely to be lost on to the star. If a gap-opening planet cuts off disc accretion allowing the formation of a central hole or clearing in the disc then we would estimate that the clearing radius would approximately be proportional to the stellar mass.     

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.     

The migration and growth of protoplanets in protostellar discs

We investigate the gravitational interaction of a Jovian-mass protoplanet with a gaseous disc with aspect ratio and kinematic viscosity expected for the protoplanetary disc from which it formed. Different disc surface density distributions are investigated. We focus on the tidal interaction with the disc with the consequent gap formation and orbital migration of the protoplanet. Non-linear two-dimensional hydrodynamic simulations are employed using three independent numerical codes.
A principal result is that the direction of the orbital migration is always inwards and such that the protoplanet reaches the central star in a near-circular orbit after a characteristic viscous time‐scale of ∼10⁴ initial orbital periods. This is found to be independent of whether the protoplanet is allowed to accrete mass or not. Inward migration is helped by the disappearance of the inner disc, and therefore the positive torque it would exert, because of accretion on to the central star. Maximally accreting protoplanets reach about 4 Jovian masses on reaching the neighbourhood of the central star. Our results indicate that a realistic upper limit for the masses of closely orbiting giant planets is ∼5 Jupiter masses, if they originate in protoplanetary discs similar to the minimum-mass solar nebula. This is because of the reduced accretion rates obtained for planets of increasing mass.
Assuming that some process such as termination of the inner disc through a magnetospheric cavity stops the migration, the range of masses estimated for a number of close orbiting giant planets as well as their inward orbital migration can be accounted for by consideration of disc–protoplanet interactions during the late stages of giant planet formation.     

On the stability of a planet between Mars and the asteroid belt: Implications for the Planet V hypothesis

The stability of an additional planet between the orbit of Mars and the asteroid belt is examined in the context of the Planet V hypothesis. In this model, the Solar System initially contained a fifth terrestrial planet, “Planet V,” which was removed after ∼700 Myr, a possible trigger for the late heavy bombardment on the inner planets. The model is investigated using 96 N-body integrations of the 8 major planets with an additional body between Mars and the asteroid belt. In more than 1/4 of simulations, Planet V survives for 1000 Myr. In most other cases, Planet V collides with the Sun or hits another planet after several hundred Myr, leaving 4 surviving terrestrial planets. In 24/96 simulations, Planet V is lost by ejection or collision with the Sun while the other four terrestrial planets survive without undergoing a collision. In 18 cases, Planet V is removed at least 200 Myr after the beginning of the simulation. The endstate depends sensitively on the mass of Planet V. Collision with the Sun is likely when Planet V's mass is 0.25 Mars masses or less. When Planet V is more massive than this, collisions involving it and/or other terrestrial planets become commonplace. In unstable systems, the times of first encounter and first collision/ejection depend on the initial aphelion distance of Mars. Reducing Mars's aphelion distance increases these times and also increases the fraction of systems surviving for 1000 Myr. When Mars's current orbit is used, the stability of Planet V increases when these two planets are widely separated initially. Planet V's aphelion distance Q typically begins to cross the asteroid belt within a few tens to a few hundred Myr, and its orbit last leaves the belt several hundred Myr later in most cases. The total time spent with Q>2.1 AU is typically less than 200 Myr.     

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.     

A study on dynamic processes at late stages in the formation of planetary systems in gas and dust disks

The discovered exoplanetary systems have highly diverse dynamic properties, which differ from those of the Solar System. A single model including planet migration effects and their gravitational interaction is used to investigate the features of dynamic processes that lead to the formation of giant-planet systems with different orbital characteristics. It is shown for a system of four giant planets similar to the Solar System how Type I migration could lead to all the planets being captured into resonant configurations. The resonant motion can continue for a long period of time after the transition to Type II migration and after the dissipation of the gas-and-dust disk. The three-planet system of GJ 876 is used to investigate the migration of the planets inward the orbit of the most massive planet and their capture into low-order resonant configurations under the conditions of Type II migration. A system similar to the exoplanetary system of HD 102272 is used to study the capture into high-order resonances followed by an increase in the orbit’s eccentricity.     

Jupiter and Super-Earth embedded in a gaseous disc

In this paper we investigate the evolution of a pair of interacting planets – a Jupiter-mass planet and a Super-Earth with a mass of  5.5 M _⊕  – orbiting a Solar-type star and embedded in a gaseous protoplanetary disc. We focus on the effects of type I and II orbital migrations, caused by the planet–disc interaction, leading to the capture of the Super-Earth in first-order mean-motion resonances by the Jupiter. The stability of the resulting resonant system in which the Super-Earth is on the internal orbit relative to the Jupiter is studied numerically by means of full 2D hydrodynamical simulations. Our main aim is to determine the Super-Earth behaviour in the presence of the gas giant in the system. It is found that the Jupiter captures the Super-Earth into the interior 3:2 or 4:3 mean-motion resonance, and that the stability of such configurations depends on the initial positions of the planets and on the evolution of the eccentricity. If the initial separation of the orbits of the planets is larger than or close to that required for the exact resonance, the final outcome is the migration of the pair of planets at a rate similar to that of the gas giant, at least for the time of our simulations. Otherwise, we observe a scattering of the Super-Earth from the disc. The evolution of planets immersed in a gaseous disc is compared with their behaviour in the case of the classical three-body problem when the disc is absent.     

Migration and evolution of extrasolar planets

Giant planets in circumstellar disks can migrate inward from their initial (formation) positions at several AUs. Inward radial migration of the planet is caused by torques between the planet and the disk; outward radial migration of the planet is caused by torques between the planet and the spinning star, and by torques due to Roche lobe overflow and consequent mass loss from the planet. We present self-consistent numerical considerations of the problem of migrating giant planets by summing torques on planets for various physical parameters of the disk and of planets. We find that Jupiter-mass planets can stably arrive and survive at small heliocentric distances, thus reproducing observed properties of some of the recently discovered extra-solar planets. The range of fates of massive planets is broad, and some perish by losing all their mass onto the central star during Roche lobe overflow, while others survive for the lifetime of the central star. Surviving planets cluster into two groups when examined in terms of final mass and final heliocentric distance: those which have lost mass and those which have not. Some of the observed extrasolar planets fall into each of these two exclusive classes. We also find that there is an inner boundary for planets' final heliocentric distances, caused by tidal torques with the central star. Planets in small orbits are shown to be stable against atmospheric loss.     

系外行星系统掩星周期变化研究进展

在掩星法发现的系外行星系统中,如果存在其他未知的伴星绕同一颗恒星运动,掩星行星由于受到伴星引力的影响,运动轨道将发生变化,轨道周期不再是常数,而是变化的。利用这种变化探测掩星系统中的其他行星,已成为一种新的方法。主要介绍了未知行星与掩星行星之间的引力作用引起的掩星周期变化效应,以及掩星周期变化法探测系外行星的理论和研究进展状况,最后简要讨论了几种影响掩星周期变化的其他因素:共轨行星、卫星、潮汐效应、相对论效应及恒星的引力四极矩等。     

Weak lensing signal in unified dark matter models

This paper is an extension of the work done by Pierens & Nelson in which they investigated the behaviour of a two-planet system embedded in a protoplanetary disc. They put a Jupiter mass gas giant on the internal orbit and a lower mass planet on the external one. We consider here a similar problem taking into account a gas giant with mass in the range 0.5 to  1 M _J  and a Super-Earth (i.e. a planet with mass  ≤10 M _⊕  ) as the outermost planet. By changing disc parameters and planet masses, we have succeeded in getting the convergent migration of the planets which allows for the possibility of their resonant locking. However, in the case in which the gas giant has the mass of Jupiter, before any mean-motion first-order commensurability could be achieved, the Super-Earth is caught in a trap when it is very close to the edge of the gap opened by the giant planet. This confirms the result obtained by Pierens & Nelson in their simulations. Additionally, we have found that, in a very thin disc, an apsidal resonance is observed in the system if the Super-Earth is captured in the trap. Moreover, the eccentricity of the small planet remains low, while the eccentricity of the gas giant increases slightly due to the imbalance between Lindblad and corotational resonances. We have also extended the work of Pierens & Nelson by studying analogous systems in which the gas giant is allowed to take sub-Jupiter masses. In this case, after conducting an extensive survey over all possible parameters, we have succeeded in getting the 1:2 mean-motion resonant configuration only in a disc with low aspect ratio and low surface density. However, the resonance is maintained just for a few thousand orbits. Thus, we conclude that for typical protoplanetary discs the mean-motion commensurabilities are rare if the Super-Earth is located on the external orbit relative to the gas giant.