Orbital Migration Of Giant Planets: Using Numerical Integration To Investigate Consequences For Other Bodies

A number of extrasolar planets have been detected in close orbits around nearby stars. It is probable that these planets did not form in these orbits but migrated from their formation locations beyond the ice line. Orbital migration mechanisms involving angular momentum transfer through tidal interactions between the planets and circumstellar gas-dust disks or by gravitational interaction with a residual planetesimal disk together with several means of halting inward migration have been identified. These offer plausible schemes to explain the orbits of observed extrasolar giant planets and giant planets within the Solar System. Recent advances in numerical integration methods and in the power of computer workstations have allowed these techniques to be applied to modelling directly the mechanisms and consequences of orbital migration in the Solar System. There is now potential for these techniques also to be applied to modelling the consequences of the orbital migration of planets in the observed exoplanetary systems. In particular the detailed investigation of the stability of terrestrial planets in the habitable zone of these systems and the formation of terrestrial planets after the dissipation of the gas disk is now possible. The stability of terrestrial planets in the habitable zone of selected exoplanetary systems has been established and the possibility of the accretion of terrestrial planets in these systems is being investigated by the author in collaboration with Barrie W. Jones (Open University), and with John Chambers (NASA-Ames) and Mark Bailey of Armagh Observatory, using numerical integration. The direct simulation of orbital migration by planetesimal scattering must probably await faster hardware and/or more efficient algorithms. This revised version was published online in July 2006 with corrections to the Cover Date.     

Formation of the giant planets

Observational constraints on interior models of the giant planets indicate that these planets were all much hotter when they formed and they all have rock and/or ice cores of ten to thirty earth masses. These cores are probably soluble in the envelopes above, especially in Jupiter and Saturn, and are therefore likely to be primordial. They persist despite the continual upward mixing by thermally driven convection throughout the age of the solar system, because of the inefficiency of double-diffusive convection. Thus, these planets most probably formed by the hydrodynamic collapse of a gaseous envelope onto a core rather than by direct instability of the gaseous solar nebula. Recent calculations by Mizuno (1980, Prog. Theor. Phys.64, 544) show that this formation mechanism may explain the similarity of giant planet core masses. Problems remain however, and no current model is entirely satisfactory in explaining the properties of the giant planets and simultaneously satisfying the terrestrial planet constraints. Satellite systematics and protoplanetary disk nebulae are also discussed and related to formation conditions.     

The formation and habitability of terrestrial planets in the presence of close-in giant planets

‘Hot jupiters,’ giant planets with orbits very close to their parent stars, are thought to form farther away and migrate inward via interactions with a massive gas disk. If a giant planet forms and migrates quickly, the planetesimal population has time to re-generate in the lifetime of the disk and terrestrial planets may form [P.J. Armitage, A reduced efficiency of terrestrial planet formation following giant planet migration, Astrophys. J. 582 (2003) L47-L50]. We present results of simulations of terrestrial planet formation in the presence of hot/warm jupiters, broadly defined as having orbital radii ?0.5 AU. We show that terrestrial planets similar to those in the Solar System can form around stars with hot/warm jupiters, and can have water contents equal to or higher than the Earth's. For small orbital radii of hot jupiters (e.g., 0.15, 0.25 AU) potentially habitable planets can form, but for semi-major axes of 0.5 AU or greater their formation is suppressed. We show that the presence of an outer giant planet such as Jupiter does not enhance the water content of the terrestrial planets, but rather decreases their formation and water delivery timescales. We speculate that asteroid belts may exist interior to the terrestrial planets in systems with close-in giant planets.     

The Leonard Award Address Presented 1998 July 27, Dublin,Ireland: On the difficulties of making Earth-like planets

Abstract— Here I discuss the series of events that led to the formation and evolution of our planet to examine why the Earth is unique in the solar system. A multitude of factors are involved: These begin with the initial size and angular momentum of the fragment that separated from a molecular cloud; such random factors are crucial in determining whether a planetary system or a double star develops from the resulting nebula. Another requirement is that there must be an adequate concentration of heavy elements to provide the 2% “rock” and “ice” components of the original nebula. An essential step in forming rocky planets in the inner nebula is the loss of gas and depletion of volatile elements, due to early solar activity that is linked to the mass of the central star. The lifetime of the gaseous nebula controls the formation of gas giants. In our system, fine timing was needed to form the gas giant, Jupiter, before the gas in the nebula was depleted. Although Uranus and Neptune eventually formed cores large enough to capture gas, they missed out and ended as ice giants. The early formation of Jupiter is responsible for the existence of the asteroid belt (and our supply of meteorites) and the small size of Mars, whereas the gas giant now acts as a gravitational shield for the terrestrial planets. The Earth and the other inner planets accreted long after the giant planets, from volatile-depleted planetesimals that were probably already differentiated into metallic cores and silicate mantles in a gas-free, inner nebula. The accumulation of the Earth from such planetesimals was essentially a stochastic process, accounting for the differences among the four rocky inner planets—including the startling contrast between those two apparent twins, Earth and Venus. Impact history and accretion of a few more or less planetesimals were apparently crucial. The origin of the Moon by a single massive impact with a body larger than Mars accounts for the obliquity (and its stability) and spin of the Earth, in addition to explaining the angular momentum, orbital characteristics, and unique composition of the Moon. Plate tectonics (unique among the terrestrial planets) led to the development of the continental crust on the Earth, an essential platform for the evolution of Homo sapiens. Random major impacts have punctuated the geological record, accentuating the directionless course of evolution. Thus a massive asteroidal impact terminated the Cretaceous Period, resulted in the extinction of at least 70% of species living at that time, and led to the rise of mammals. This sequence of events that resulted in the formation and evolution of our planet were thus unique within our system. The individual nature of the eight planets is repeated among the 60-odd satellites—no two appear identical. This survey of our solar system raises the question whether the random sequence of events that led to the formation of the Earth are likely to be repeated in detail elsewhere. Preliminary evidence from the “new planets” is not reassuring. The discovery of other planetary systems has removed the previous belief that they would consist of a central star surrounded by an inner zone of rocky planets and an outer zone of giant planets beyond a few astronomical units (AU). Jupiter-sized bodies in close orbits around other stars probably formed in a similar manner to our giant planets at several astronomical units from their parent star and, subsequently, migrated inwards becoming stranded in close but stable orbits as “hot Jupiters”, when the nebula gas was depleted. Such events would prevent the formation of terrestrial-type planets in such systems.     

Models of the giant planets

Models of the giant planets were constructed based on the assumption that the hydrogen to helium ratio is solar in these planets. This assumption, together with arguments about the condensation sequence in the primitive solar nebula, yields models with a central core of rock and possibly ice surrounded by an envelope of hydrogen, helium, methane, ammonia, and water. These last three volatiles may be individually enhanced due to condensation at the period of core formation. Jupiter was found to have a core of about 40 earth masses and a water enhancement in the atmosphere of about 7.5 times the solar value. Saturn was found to have a core of 20 earth masses and a water enhancement in the atmosphere of about 25 times the solar value. Rock plus ice constitute 75–85% of the mass of Uranus and Neptune. Temperatures in the interiors of these planets are probably above the melting points, if there is an adiabatic relation throughout the interiors. Some aspects of the sensitivities of these results to uncertainties in rotational flattening are discussed.     

内日冕的绿线强度分布研究

日冕是太阳大气活动的关键区域,是日地空间天气的源头.受观测限制,对日冕低层大气等离子体结构和磁场状态的研究非常欠缺,国际上对于可见光波段日冕低层大气的亮度分层研究很少.利用丽江日冕仪YOGIS(Yunnan Green-line Imaging System)的日冕绿线(Fe_ⅩⅣ5303?)观测资料,对内日冕区域(1.03R_☉-1.25R_☉,R_☉表示太阳半径)亮结构及其中冕环进行了有效的强度衰减分析.对亮结构的强度在太阳径向高度上进行了指数衰减拟合,比较这些拟合结果发现所得到的静态内冕环的衰减指数在一固定值附近.然后将比较明显的冕环提取出来,通过对不同高度的绿线强度进行指数拟合,得出的衰减指数与亮结构中也比较相近,这对进一步研究日冕中的各项物理参数演化提供了参考.     

Young Jupiters are faint: new models of the early evolution of giant planets

Here we show preliminary calculations of the cooling and contraction of a 2 M_J planet. These calculations, which are being extended to 1–10 M_J, differ from other published “cooling tracks” in that they include a core accretion‐gas capture formation scenario, the leading theory for the formation of gas giant planets.We find that the initial post‐accretionary intrinsic luminosity of the planet is ∼3 times less than previously published models which use arbitrary initial conditions. These differences last a few tens of millions of years. Young giant planets are intrinsically fainter than has been previously appreciated. We also discuss how uncertainties in atmospheric chemistry and the duration of the formation time of giant planets lead to challenges in deriving planetary physical properties from comparison with tabulated model values. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The interiors of the giant planets — 1983

The last few years brought progress in our understanding of the interiors of the giant planets especially of the two larger ones which have been visited by Pioneer and Voyager spacecraft. An analysis of the formation of the giant planets also heped to clarify certain important common features. The presently available model of Jupiter is still based on certain somewhat bothersome approximations but it appears to satisfy the main observational constraints. Saturn's interior is much better understood than it was previously although the quantitative aspects of the role of the miscibility gap in the hydrogen-helium system have not yet been entirely resolved. Much attention has been directed at the interiors of Uranus and Neptune and the outstanding question appears to be the location and the amount of ices and methane present in their outer layers. Both the two-and the three-layer models are moderately successful. Serious difficulties arise from the considerable uncertainties concerning the rotational periods of both planets. Also the estimates of the internal heat fluxes and of the magnetic fields of both planets are not sufficiently certain. It is hoped that the forthcoming flyby of these two planets by a Voyager spacecraft will provide important new data for a future study of their interiors.     

Late stages of planetary accretion

We derive first-order differential equations for the late stages of planetary accretion (planetesimal mass >10¹³ g). The effect of gravitational encounters, energy exchange, collisions, and gas drag has been included. Two simple models are discussed, namely, (i) when all planetesimals have the same mass and (ii) when there is one large planetesimal and numerous small planetesmals. Gravitational two-body encounters are modeled according to Chandrasekhar's classical theory from stellar dynamics. It is shown that the velocity increase due to mutual encounters can be modeled according to the simple theory of random flights. We find analytical equations for the average velocity decrease due to collisions. Gas drag, if present, is modeled in averaged form up to the first order in the eccentricities and inclinations of the planetesimals. Characteristic time scales for the formation of the terrestrial planets are found for the most favorable models to be of order 10⁸ year. The calculated mass of rock and ice of the giant planets is too low as compared to the observed one. This difficulty of our model could be overcome by assuming a several times larger surface density, an enlarged accretion cross section, and gas accretion during the final stages of accretion of the solid cores of the giant planets. Analytical and numerical results are presebted, the evolutionary tracks showing satisfactory agreement with observations for some models.     

Practical SPH models for major planets

Modelling planets is done for two main reasons – the first to further understanding of their internal structure and the second to provide models to explore astrophysical situations in which planets play a role. For the latter reason, the requirements on accuracy are less severe, although the planet must be realistic in its major features. A numerical model of a layered giant planet is developed with an iron core, a silicate mantle, an ice region and a hydrogen–helium atmosphere. The Tillotson equation of state is used and examples of two model planets are given, one reproducing the mass and radius of Jupiter quite closely and the other with two Jupiter masses. Transferring these results into a smoothed particle hydrodynamics (SPH) model presents two main difficulties. A uniform distribution of SPH points leads to too few points representing the non-atmospheric component. It is shown that using a distorted lattice enables the core + silicate + ice to be represented by several hundred points so that the evolution of these regions can be followed in detail. Another difficulty concerns the density discontinuities attendant on a layered structure. Density estimates of SPH points are either too large or too small near material interfaces leading to unrealistic pressure gradients and, consequently, to large and unphysical local forces. Algorithms are described for avoiding this difficulty both at material interfaces and near the surface of the planet. In some astrophysical situations involving SPH-modelled planets, the main bulk of the planet is so opaque that internal heat transfer can be neglected. However, surface regions should radiate and a convenient way for including radiation from a planetary surface is described.     

Observable consequences of planet formation models in systems with close-in terrestrial planets

To date, two planetary systems have been discovered with close-in, terrestrial-mass planets     . Many more such discoveries are anticipated in the coming years with radial velocity and transit searches. Here we investigate the different mechanisms that could form 'hot Earths' and their observable predictions. Models include: (1) in situ accretion; (2) formation at larger orbital distance followed by inward 'type 1' migration; (3) formation from material being 'shepherded' inward by a migrating gas giant planet; (4) formation from material being shepherded by moving secular resonances during dispersal of the protoplanetary disc; (5) tidal circularization of eccentric terrestrial planets with close-in perihelion distances and (6) photoevaporative mass-loss of a close-in giant planet. Models 1–4 have been validated in previous work. We show that tidal circularization can form hot Earths, but only for relatively massive planets     with very close-in perihelion distances (≲0.025 au), and even then the net inward movement in orbital distance is at most only 0.1–0.15 au. For planets of less than     , photoevaporation can remove the planet's envelope and leave behind the solid core on a Gyr time-scale, but only for planets inside 0.025–0.05 au. Using two quantities that are observable by current and upcoming missions, we show that these models each produce unique signatures, and can be observationally distinguished. These observables are the planetary system architecture (detectable with radial velocities, transits and transit timing) and the bulk composition of transiting close-in terrestrial planets (measured by transits via the planet's radius).     

Metallicity, debris discs and planets

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.     

The migrating embryo model for disk evolution

Abstract– A new view of disk evolution is emerging from self‐consistent numerical simulation modeling of the formation of circumstellar disks from the direct collapse of prestellar cloud cores. This has implications for many aspects of star and planet formation, including the growth of dust and high‐temperature processing of materials. A defining result is that the early evolution of a disk is crucially affected by the continuing mass loading from the core envelope, and is driven into recurrent phases of gravitational instability. Nonlinear spiral arms formed during these episodes fragment to form gaseous clumps in the disk. These clumps generally migrate inward due to gravitational torques arising from their interaction with a trailing spiral arm. Occasionally, a clump can open up a gap in the disk and settle into a stable orbit, revealing a direct pathway to the formation of companion stars, brown dwarfs, or giant planets. At other times, when multiple clumps are present, a low mass clump may even be ejected from the system, providing a pathway to the formation of free‐floating brown dwarfs and giant planets in addition to low mass stars. Finally, it has been suggested that the inward migration of gaseous clumps can provide the proper conditions for the transport of high‐temperature processed solids from the outer disk to the inner disk, and even possibly accelerate the formation of terrestrial planets in the inner disk. All of these features arising from clump formation and migration can be tied together conceptually in a migrating embryo model for disk evolution that can complement the well‐known core accretion model for planet formation.     

The origin and nature of Neptune-like planets orbiting close to solar type stars

The sample of known exoplanets is strongly biased to masses larger than the ones of the giant gaseous planets of the Solar System. Recently, the discovery of two extrasolar planets of considerably lower masses around the nearby Stars GJ 436 and ρ Cancri was reported. They are like our outermost icy giants, Uranus and Neptune, but in contrast, these new planets are orbiting at only some hundredth of the Earth-Sun distance from their host stars, raising several new questions about their origin and constitution. Here we report numerical simulations of planetary accretion that show, for the first time through N-body integrations that the formation of compact systems of Neptune-like planets close to the hosts stars could be a common by-product of planetary formation. We found a regime of planetary accretion, in which orbital migration accumulates protoplanets in a narrow region around the inner edge of the nebula, where they collide each other giving rise to Neptune-like planets. Our results suggest that, if a protoplanetary solar environment is common in the Galaxy, the discovery of a vast population of this sort of ‘hot cores’ should be expected in the near future.     

On the internal energy source of the large planets

Based on spacecraft observations of the giant planets we call attention to an apparent regular mass-luminosity relation for the large planets (giant planets plus Earth). It can provide important clues and constraints on the question of their energy source, including various relevant theories and their difficulties. We describe in some detail a new thermonuclear reaction mechanism, valid for plasma fusions at both super—high and low temperatures, and including the original thermonuclear mechanism as a particular case. It can explain not only the internal energy souce of the large planets, but also many anomalous fusion phenomena observed in the laboratory and in astronomical bodies. Although these appear in various guises, their basic principle is the same, that is, they can all be understood in terms of the new mechanism.     

Transits of extrasolar planets

In this paper, we consider the physical properties and characteristic features of extrasolar planets and planetary systems, those, for which the passage of low-orbit giant planets across the stellar disk (transits) are observed. The paper is mostly a review. The peculiarities of the search for transits are briefly considered. The main attention in this paper is given to the difference in the physical properties of low-orbit giant planets. A comparison of the data obtained during the transits of “hot Jupiters” points to the probable existence of several distinct subtypes of low-orbit extrasolar planets. “Hot Jupiters” of low density (HD 209458b), “hot Jupiters” with massive cores composed of heavy elements (HD 149026b), and “very hot Jupiters” (HD 189733b) are bodies that probably fall into different categories of exoplanets. Dissipation of the atmospheres of low-orbit giant planets estimated from the experimental data is compared with the calculated Jeans atmospheric losses. For “hot Jupiters”, the expected Jeans mass losses due to atmospheric escape on a cosmogonic time scale hardly exceed a few percent. Low-orbit giant planets should have a strong magnetic field. Since the orbital velocity of “hot Jupiters” is close to the magnetosonic velocity (or can even exceed it), the moving planet should actively interact with the “stellar wind” plasma. The possession of a magnetic field by extrasolar planets and the effects of their interaction with plasma can be used to search for extrasolar planets.     

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.     

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.     

The formation of a gas giant planet in a viscously evolved protoplanetary disk within the core accretion model

The gas giant planets’ formation processes in a viscously evolved protoplanetary disk are studied in the context of the core accretion model. In this paper, we follow the entire formation process of the core accretion model (the three stages). We find that the gas giant planets’ final masses and formation regions have strong dependence on the molecular cloud core’s properties (angular velocity \(\omega \) and mass \(M _{c d}\)) and the \(\alpha _{ \mathit{min} }\) parameter. We find and build the relationship between gas giant planets’ properties and molecular cloud core’s properties. In contrast to the previous works, we find that the formation process can be finished within the protoplanetary disk’s lifetime (4×10⁶ yr) in our disk model. This is because the mass influx produced by the molecular cloud core can provide enough material to the protoplanetary disk. We also find that the gas giant planets’ final masses increase generally with the viscosity coefficient \(\alpha \). This is because most of the gas giant planet’s mass is captured during the rapid gas accretion phase (the third stage of the core accretion model), and furthermore the accretion of gas in this phase is dominated by the “gap limiting case”. And our numerical results can also be compared with the observed data of exoplanet systems.     

Colour-differential interferometry for the observation of extrasolar planets

We present the high angular resolution technique of colour-differential interferometry for direct detection of extrasolar giant planets (EGPs). The measurement of differential phase with long-baseline ground-based interferometers in the near-infrared could allow the observation of several hot giant extrasolar planets in tight orbit around the nearby stars, and thus yield their low- or mid-resolution spectroscopy, complete orbital data set and mass. Estimates of potentially achievable signal-to-noise ratios are presented for a number of planets already discovered by indirect methods. The limits from the instrumental and atmospheric instability are discussed, and a subsequent observational strategy is proposed.