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1.
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.  相似文献   

2.
The migration and growth of protoplanets in protostellar discs   总被引：1，自引：0，他引：1  
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.  相似文献   

3.
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.  相似文献   

4.
Planets orbiting a planetesimal circumstellar disc can migrate inward from their initial positions because of dynamical friction between planets and planetesimals. The migration rate depends on the disc mass and on its time evolution. Planets that are embedded in long-lived planetesimal discs, having total mass of 10−4– 0.01 M⊙ , can migrate inward a large distance and can survive only if the inner disc is truncated or as a result of tidal interaction with the star. In this case the semimajor axis, a , of the planetary orbit is less than 0.1 au. Orbits with larger a are obtained for smaller values of the disc mass or for a rapid evolution (depletion) of the disc. This model may explain not only several of the orbital features of the giant planets that have been discovered in recent years orbiting nearby stars, but also the metallicity enhancement found in several stars associated with short-period planets.  相似文献   

5.
We have performed N -body numerical simulations of the exchange of angular momentum between a massive planet and a 3D Keplerian disc of planetesimals. Our interest is directed at the study of the classical analytical expressions of the lineal theory of density waves, as representative of the dynamical friction in discs 'dominated by the planet' and the orbital migration of the planets with regard to this effect. By means of a numerical integration of the equations of motion, we have carried out a set of numerical experiments with a large number of particles  ( N ≥10 000)  , and planets with the mass of Jupiter, Saturn and one core mass of the giant planets in the Solar system  ( M c=10 M⊕)  . The torque, measured in a phase in which a 'steady forcing' is clearly measurable, yields inward migration in a minimum-mass solar disc  (Σ∼10 g cm-2  ), with a characteristic drift time of ∼ a few 106 yr. The planets predate the disc, but the orbital decay rate is not sufficient to allow accretion in a time-scale relevant to the formation of giant planets. We found reductions of the measured torque on the planet, with respect to the linear theory, by a factor of 0.38 for M c, 0.04 for Saturn and 0.01 for Jupiter, due to the increase in the perturbation on the disc. The behaviour of planets whose mass is larger than M c is similar to the one of type II migrators in gaseous discs. Our results suggest that, in a minimum mass, solar planetesimals disc, type I migrations occur for masses smaller than M c, whereas for this mass value it could be a transition zone between the two types of migration.  相似文献   

6.
We investigate the populations of main-sequence stars within 25 pc that have debris discs and/or giant planets detected by Doppler shift. The metallicity distribution of the debris sample is a very close match to that of stars in general, but differs with >99 per cent confidence from the giant planet sample, which favours stars of above average metallicity. This result is not due to differences in age of the two samples. The formation of debris-generating planetesimals at tens of au thus appears independent of the metal fraction of the primordial disc, in contrast to the growth and migration history of giant planets within a few au. The data generally fit a core accumulation model, with outer planetesimals forming eventually even from a disc low in solids, while inner planets require fast core growth for gas to still be present to make an atmosphere.  相似文献   

7.
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 MJ) 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 MJ  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 MJ and is highly efficient, occurring at the local viscous rate. The migration time-scales for planets of mass less than 0.1 MJ, 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 MJ with little migration. Planets with final masses of the order of 10 MJ would undergo large migration, which makes formation and survival difficult.  相似文献   

8.
In this paper we develop further the model for the migration of planets introduced in Del Popolo et al. We first model the protoplanetary nebula as a time-dependent accretion disc, and find self-similar solutions to the equations of the accretion disc that give us explicit formulae for the spatial structure and the temporal evolution of the nebula. These equations are then used to obtain the migration rate of the planet in the planetesimal disc, and to study how the migration rate depends on the disc mass, on its time evolution and on some values of the dimensionless viscosity parameter α . We find that planets that are embedded in planetesimal discs, having total mass of  10-4-0.1 M⊙  , can migrate inward a large distance for low values of α (e.g.,   α ≃10-3-10-2)  and/or large disc mass, and can survive only if the inner disc is truncated or because of tidal interaction with the star. Orbits with larger a are obtained for smaller values of the disc mass and/or for larger values of α . This model may explain several orbital features of the recently discovered giant planets orbiting nearby stars.  相似文献   

9.
Recent observations point to the presence of structured dust grains in the discs surrounding young brown dwarfs, thus implying that the first stages of planet formation take place also in the substellar regime. Here, we investigate the potential for planet formation around brown dwarfs and very low-mass stars according to the sequential core accretion model of planet formation. We find that, for a brown dwarf mass 0.05 M⊙, our models predict a maximum planetary mass of  ∼5   M⊕  , orbiting with semimajor axis ∼ 1 au. However, we note that the predictions for the mass–semimajor axis distribution are strongly dependent upon the models chosen for the disc surface density profiles and the assumed distribution of disc masses. In particular, if brown dwarf disc masses are of the order of a few Jupiter masses, Earth-mass planets might be relatively frequent, while if typical disc masses are only a fraction of Jupiter mass, we predict that planet formation would be extremely rare in the substellar regime. As the observational constraints on disc profiles, mass dependencies and their distributions are poor in the brown dwarf regime, we advise caution in validating theoretical models only on stars similar to the Sun and emphasize the need for observational data on planetary systems around a wide range of stellar masses. We also find that, unlike the situation around solar-like stars, Type II migration is totally absent from the planet formation process around brown dwarfs, suggesting that any future observations of planets around brown dwarfs would provide a direct measure of the role of other types of migration.  相似文献   

10.
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.  相似文献   

11.
Dynamical relaxation and the orbits of low-mass extrasolar planets   总被引：1，自引：0，他引：1  
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.  相似文献   

12.
Giant planets embedded in circumstellar discs are expected to open gaps in these discs. We examine the vertical structure of the gap edges. We find that the planet excites spiral arms with significant (Mach number of a half) vertical motion of the gas, and discuss the implications of these motions. In particular, the spiral arms will make the edge appear 'puffed up' relative to the bulk of the disc. Infrared observations (sensitive to dust) would be dominated by the light from the thick inner edge of the disc. Submillimetre observations (sensitive to gas velocities) would appear to be hot in 'turbulent' motions (actually the ordered motion caused by the passage of the spiral arms), but cold in chemistry. Resolved submillimetre maps of circumstellar discs might even be able to detect the spiral arms directly.  相似文献   

13.
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.  相似文献   

14.
On the migration of a system of protoplanets   总被引：1，自引：0，他引：1  
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.  相似文献   

15.
Detectable debris discs are thought to require dynamical excitation ('stirring'), so that planetesimal collisions release large quantities of dust. We investigate the effects of the secular perturbations of a planet, which may lie at a significant distance from the planetesimal disc, to see if these perturbations can stir the disc, and if so over what time-scale. The secular perturbations cause orbits at different semimajor axes to precess at different rates, and after some time   t cross  initially non-intersecting orbits begin to cross. We show that   t cross∝ a 9/2disc/( m pl e pl a 3pl)  , where   m pl, e pl  and   a pl  are the mass, eccentricity and semimajor axis of the planet, and   a disc  is the semimajor axis of the disc. This time-scale can be faster than that for the growth of planetesimals to Pluto's size within the outer disc. We also calculate the magnitude of the relative velocities induced among planetesimals and infer that a planet's perturbations can typically cause destructive collisions out to 100 s of au. Recently formed planets can thus have a significant impact on planet formation in the outer disc which may be curtailed by the formation of giant planets much closer to the star. The presence of an observed debris disc does not require the presence of Pluto-sized objects within it, since it can also have been stirred by a planet not in the disc. For the star ε Eridani, we find that the known radial velocity planet can excite the planetesimal belt at 60 au sufficiently to cause destructive collisions of bodies up to 100 km in size, on a time-scale of 40 Myr.  相似文献   

16.
We study the coorbital flow for embedded, low-mass planets. We provide a simple semi-analytic model for the corotation region, which is subsequently compared to high-resolution numerical simulations. The model is used to derive an expression for the half-width of the horseshoe region, x s, which in the limit of zero softening is given by   x s/ r p= 1.68( q / h )1/2  , where q is the planet to central star mass ratio, h is the disc aspect ratio and   r p  is the orbital radius. This is in very good agreement with the same quantity measured from simulations. This result is used to show that horseshoe drag is about an order of magnitude larger than the linear corotation torque in the zero-softening limit. Thus, the horseshoe drag, the sign of which depends on the gradient of specific vorticity, is important for estimates of the total torque acting on the planet. We further show that phenomena, such as the Lindblad wakes, with a radial separation from corotation of approximately a pressure scaleheight H can affect x s, even though for low-mass planets   x s≪ H   . The effect is to distort streamlines and reduce x s through the action of a back pressure. This effect is reduced for smaller gravitational softening parameters and planets of higher mass, for which x s becomes comparable to H .  相似文献   

17.
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.  相似文献   

18.
Dynamical relaxation and massive extrasolar planets   总被引：1，自引：0，他引：1  
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.  相似文献   

19.
The irradiation of protoplanetary discs by central stars is the main heating mechanism for discs, resulting in their flared geometric structure. In a series of papers, we investigate the deep links between two-dimensional self-consistent disc structure and planetary migration in irradiated discs, focusing particularly on those around M stars. In this first paper, we analyse the thermal structure of discs that are irradiated by an M star by solving the radiative transfer equation by means of a Monte Carlo code. Our simulations of irradiated hydrostatic discs are realistic and self-consistent in that they include dust settling with multiple grain sizes  ( N = 15)  , the gravitational force of an embedded planet on the disc and the presence of a dead zone (a region with very low levels of turbulence) within it. We show that dust settling drives the temperature of the mid-plane from an   r −3/5  distribution (well mixed dust models) towards an   r −3/4  . The dead zone, meanwhile, leaves a dusty wall at its outer edge because dust settling in this region is enhanced compared to the active turbulent disc at larger disc radii. The disc heating produced by this irradiated wall provides a positive gradient region of the temperature in the dead zone in front of the wall. This is crucially important for slowing planetary migration because Lindblad torques are inversely proportional to the disc temperature. Furthermore, we show that low turbulence of the dead zone is self-consistently induced by dust settling, resulting in the Kelvin–Helmholtz instability (KHI). We show that the strength of turbulence arising from the KHI in the dead zone is  α= 10−5  .  相似文献   

20.
We present results from a search for additional transiting planets in 24 systems already known to contain a transiting planet. We model the transits due to the known planet in each system and subtract these models from light curves obtained with the SuperWASP (Wide Angle Search for Planets) survey instruments. These residual light curves are then searched for evidence of additional periodic transit events. Although we do not find any evidence for additional planets in any of the planetary systems studied, we are able to characterize our ability to find such planets by means of Monte Carlo simulations. Artificially generated transit signals corresponding to planets with a range of sizes and orbital periods were injected into the SuperWASP photometry and the resulting light curves searched for planets. As a result, the detection efficiency as a function of both the radius and orbital period of any second planet is calculated. We determine that there is a good (>50 per cent) chance of detecting additional, Saturn-sized planets in   P ∼  10 d orbits around planet-hosting stars that have several seasons of SuperWASP photometry. Additionally, we confirm previous evidence of the rotational stellar variability of WASP-10, and refine the period of rotation. We find that the period of the rotation is  11.91 ± 0.05  d, and the false alarm probability for this period is extremely low  (∼10−13)  .  相似文献   

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