共查询到20条相似文献，搜索用时 15 毫秒
1.
Jack Wisdom 《Icarus》2008,193(2):637-640
Expressions for tidal dissipation in a body in synchronous rotation at arbitrary orbital eccentricity and obliquity are derived. The rate of tidal dissipation for a synchronously rotating body is compared to that in a body in asymptotic nonsynchronous rotation. 相似文献
2.
Anthony R. Dobrovolskis 《Icarus》2009,204(1):1-1318
A previous paper [Dobrovolskis, A.R., 2007. Icarus 192, 1-23] showed that eccentricity can have profound effects on the climate, habitability, and detectability of extrasolar planets. This complementary study shows that obliquity can have comparable effects.The known exoplanets exhibit a wide range of orbital eccentricities, but those within several million kilometers of their suns are generally in near-circular orbits. This fact is widely attributed to the dissipation of tides in the planets. Tides in a planet affect its spin even more than its orbit, and such tidally evolved planets often are assumed to be in synchronous rotation, so that their rotation periods are identical to their orbital periods. The canonical example of synchronous spin is the way that our Moon always keeps nearly the same hemisphere facing the Earth.Tides also tend to reduce the planet’s obliquity (the angle between its spin and orbital angular velocities). However, orbit precession can cause the rotation to become locked in a “Cassini state”, where it retains a nearly constant non-zero obliquity. For example, our Moon maintains an obliquity of about 6.7° with respect to its orbit about the Earth. In comparison, stable Cassini states can exist for practically any obliquity up to ∼90° or more for planets of binary stars, or in multi-planet systems with high mutual inclinations, such as are produced by scattering or by the Kozai mechanism.This work considers planets in synchronous rotation with circular orbits, but arbitrary obliquity β; this affects the distribution of insolation over the planet’s surface, particularly near its poles. For β=0, one hemisphere bakes in perpetual sunshine, while the opposite hemisphere experiences eternal darkness. As β increases, the region of permanent daylight and the antipodal realm of endless night both shrink, while a more temperate area of alternating day and night spreads in longitude, and especially in latitude. The regions of permanent day or night disappear at β=90°. The insolation regime passes through several more transitions as β continues to increase toward 180°, but the surface distribution of insolation remains non-uniform in both latitude and longitude.Thus obliquity, like eccentricity, can protect certain areas of the planet from the worst extremes of temperature and solar radiation, and can improve the planet’s habitability. These results also have implications for the direct detectability of extrasolar planets, and for the interpretation of their thermal emissions. 相似文献
3.
The extrasolar planets discovered to date possess unexpected orbital elements. Most orbit their host stars with larger eccentricities and smaller semi-major axes than similarly sized planets in our own Solar System do. It is generally agreed that the interaction between giant planets and circumstellar disks (Type II migration) drives these planets inward to small radii, but the effect of these same disks on orbital eccentricity, ?, is controversial. Several recent analytic calculations suggest that disk-planet interactions can excite eccentricity, while numerical studies generally produce eccentricity damping. This paper addresses this controversy using a quasi-analytic approach, drawing on several preceding analytic studies. This work refines the current treatment of eccentricity evolution by removing several approximations from the calculation of disk torques. We encounter neither uniform damping nor uniform excitation of orbital eccentricity, but rather a function d?/dt that varies in both sign and magnitude depending on eccentricity and other Solar System properties. Most significantly, we find that for every combination of disk and planet properties investigated herein, corotation torques produce negative values of d?/dt for some range in ? within the interval [0.1, 0.5]. If corotation torques are saturated, this region of eccentricity damping disappears, and excitation occurs on a short timescale of less than 0.08 Myr. Thus, our study does not produce eccentricity excitation on a timescale of a few Myr—we obtain either eccentricity excitation on a short time scale, or eccentricity damping on a longer time scale. Finally, we discuss the implications of this result for producing the observed range in extrasolar planet eccentricity. 相似文献
4.
We have investigated obliquity variations of possible terrestrial planets in habitable zones (HZs) perturbed by a giant planet(s) in extrasolar planetary systems. All the extrasolar planets so far discovered are inferred to be jovian-type gas giants. However, terrestrial planets could also exist in extrasolar planetary systems. In order for life, in particular for land-based life, to evolve and survive on a possible terrestrial planet in an HZ, small obliquity variations of the planet may be required in addition to its orbital stability, because large obliquity variations would cause significant climate change. It is known that large obliquity variations are caused by spin-orbit resonances where the precession frequency of the planet's spin nearly coincides with one of the precession frequencies of the ascending node of the planet's orbit. Using analytical expressions, we evaluated the obliquity variations of terrestrial planets with prograde spins in HZs. We found that the obliquity of terrestrial planets suffers large variations when the giant planet's orbit is separated by several Hill radii from an edge of the HZ, in which the orbits of the terrestrial planets in the HZ are marginally stable. Applying these results to the known extrasolar planetary systems, we found that about half of these systems can have terrestrial planets with small obliquity variations (smaller than 10°) over their entire HZs. However, the systems with both small obliquity variations and stable orbits in their HZs are only 1/5 of known systems. Most such systems are comprised of short-period giant planets. If additional planets are found in the known planetary systems, they generally tend to enhance the obliquity variations. On the other hand, if a large/close satellite exists, it significantly enhances the precession rate of the spin axis of a terrestrial planet and is likely to reduce the obliquity variations of the planet. Moreover, if a terrestrial planet is in a retrograde spin state, the spin-orbit resonance does not occur. Retrograde spin, or a large/close satellite might be essential for land-based life to survive on a terrestrial planet in an HZ. 相似文献
5.
Keiko Atobe 《Icarus》2007,188(1):1-17
We have investigated the obliquity evolution of terrestrial planets in habitable zones (at ∼1 AU) in extrasolar planetary systems, due to tidal interactions with their satellite and host star with wide varieties of satellite-to-planet mass ratio (m/Mp) and initial obliquity (γ0), through numerical calculations and analytical arguments. The obliquity, the angle between planetary spin axis and its orbit normal, of a terrestrial planet is one of the key factors in determining the planetary surface environments. A recent scenario of terrestrial planet accretion implies that giant impacts of Mars-sized or larger bodies determine the planetary spin and form satellites. Since the giant impacts would be isotropic, tilted spins (sinγ0∼1) are more likely to be produced than straight ones (sinγ0∼0). The ratio m/Mp is dependent on the impact parameters and impactors' mass. However, most of previous studies on tidal evolution of the planet-satellite systems have focused on a particular case of the Earth-Moon systems in which m/Mp?0.0125 and γ0∼10° or the two-body planar problem in which γ0=0° and stellar torque is neglected. We numerically integrated the evolution of planetary spin and a satellite orbit with various m/Mp (from 0.0025 to 0.05) and γ0 (from 0° to 180°), taking into account the stellar torques and precessional motions of the spin and the orbit. We start with the spin axis that almost coincides with the satellite orbit normal, assuming that the spin and the satellite are formed by one dominant impact. With initially straight spins, the evolution is similar to that of the Earth-Moon system. The satellite monotonically recedes from the planet until synchronous state between the spin period and the satellite orbital period is realized. The obliquity gradually increases initially but it starts decreasing down to zero as approaching the synchronous state. However, we have found that the evolution with initially tiled spins is completely different. The satellite's orbit migrates outward with almost constant obliquity until the orbit reaches the critical radius ∼10-20 planetary radii, but then the migration is reversed to inward one. At the reversal, the obliquity starts oscillation with large amplitude. The oscillation gradually ceases and the obliquity is reduced to ∼0° during the inward migration. The satellite eventually falls onto the planetary surface or it is captured at the synchronous state at several planetary radii. We found that the character change of precession about total angular momentum vector into that about the planetary orbit normal is responsible for the oscillation with large amplitude and the reversal of migration. With the results of numerical integration and analytical arguments, we divided the m/Mp-γ0 space into the regions of the qualitatively different evolution. The peculiar tidal evolution with initially tiled spins give deep insights into dynamics of extrasolar planet-satellite systems and discussions of surface environments of the planets. 相似文献
6.
Runaway growth ends when the largest protoplanets dominate the dynamics of the planetesimal disk; the subsequent self-limiting accretion mode is referred to as “oligarchic growth.” Here, we begin by expanding on the existing analytic model of the oligarchic growth regime. From this, we derive global estimates of the planet formation rate throughout a protoplanetary disk. We find that a relatively high-mass protoplanetary disk (∼10 × minimum-mass) is required to produce giant planet core-sized bodies (∼10 M⊕) within the lifetime of the nebular gas (?10 million years). However, an implausibly massive disk is needed to produce even an Earth mass at the orbit of Uranus by 10 Myrs. Subsequent accretion without the dissipational effect of gas is even slower and less efficient. In the limit of noninteracting planetesimals, a reasonable-mass disk is unable to produce bodies the size of the Solar System’s two outer giant planets at their current locations on any timescale; if collisional damping of planetesimal random velocities is sufficiently effective, though, it may be possible for a Uranus/Neptune to form in situ in less than the age of the Solar System. We perform numerical simulations of oligarchic growth with gas and find that protoplanet growth rates agree reasonably well with the analytic model as long as protoplanet masses are well below their estimated final masses. However, accretion stalls earlier than predicted, so that the largest final protoplanet masses are smaller than those given by the model. Thus the oligarchic growth model, in the form developed here, appears to provide an upper limit for the efficiency of giant planet formation. 相似文献
7.
We study the influence of mutual planetary perturbations on the process of eccentricity excitation by jet acceleration suggested by Namouni (Astron. J. 130, 280–294). Modeling the jet’s action by a constant-direction acceleration, we solve the linear secular equations of the combined planetary perturbations and the jet acceleration of the host star for a two-planet system. The effects of the acceleration’s strength, relative mass ratio and the relative distance of the two planets are investigated. The model is applied to the extrasolar planetary systems of HD 108874, 47 Uma, and HD 12661. 相似文献
8.
The final stage in the formation of terrestrial planets consists of the accumulation of ∼1000-km “planetary embryos” and a swarm of billions of 1-10 km “planetesimals.” During this process, water-rich material is accreted by the terrestrial planets via impacts of water-rich bodies from beyond roughly 2.5 AU. We present results from five high-resolution dynamical simulations. These start from 1000-2000 embryos and planetesimals, roughly 5-10 times more particles than in previous simulations. Each simulation formed 2-4 terrestrial planets with masses between 0.4 and 2.6 Earth masses. The eccentricities of most planets were ∼0.05, lower than in previous simulations, but still higher than for Venus, Earth and Mars. Each planet accreted at least the Earth's current water budget. We demonstrate several new aspects of the accretion process: (1) The feeding zones of terrestrial planets change in time, widening and moving outward. Even in the presence of Jupiter, water-rich material from beyond 2.5 AU is not accreted for several millions of years. (2) Even in the absence of secular resonances, the asteroid belt is cleared of >99% of its original mass by self-scattering of bodies into resonances with Jupiter. (3) If planetary embryos form relatively slowly, then the formation of embryos in the asteroid belt may have been stunted by the presence of Jupiter. (4) Self-interacting planetesimals feel dynamical friction from other small bodies, which has important effects on the eccentricity evolution and outcome of a simulation. 相似文献
9.
Althea V. Moorhead 《Icarus》2005,178(2):517-539
This paper presents a parametric study of giant planet migration through the combined action of disk torques and planet-planet scattering. The torques exerted on planets during Type II migration in circumstellar disks readily decrease the semi-major axes a, whereas scattering between planets increases the orbital eccentricities ?. This paper presents a parametric exploration of the possible parameter space for this migration scenario using two (initial) planetary mass distributions and a range of values for the time scale of eccentricity damping (due to the disk). For each class of systems, many realizations of the simulations are performed in order to determine the distributions of the resulting orbital elements of the surviving planets; this paper presents the results of ∼8500 numerical experiments. Our goal is to study the physics of this particular migration mechanism and to test it against observations of extrasolar planets. The action of disk torques and planet-planet scattering results in a distribution of final orbital elements that fills the a-? plane, in rough agreement with the orbital elements of observed extrasolar planets. In addition to specifying the orbital elements, we characterize this migration mechanism by finding the percentages of ejected and accreted planets, the number of collisions, the dependence of outcomes on planetary masses, the time spent in 2:1 and 3:1 resonances, and the effects of the planetary IMF. We also determine the distribution of inclination angles of surviving planets and the distribution of ejection speeds for exiled planets. 相似文献
10.
F. Selsis P. Bordé F. Brachet F. Bouchy J.-M. Grießmeier C. Sotin C. Moutou M. Deleuil D. Despois A. Léger 《Icarus》2007,191(2):453-468
Planets less massive than about 10 MEarth are expected to have no massive H-He atmosphere and a cometary composition (∼50% rocks, 50% water, by mass) provided they formed beyond the snowline of protoplanetary disks. Due to inward migration, such planets could be found at any distance between their formation site and the star. If migration stops within the habitable zone, this may produce a new kind of planets, called ocean-planets. Ocean-planets typically consist in a silicate core, surrounded by a thick ice mantle, itself covered by a 100 km-deep ocean. The possible existence of ocean-planets raises important astrobiological questions: Can life originate on such body, in the absence of continent and ocean-silicate interfaces? What would be the nature of the atmosphere and the geochemical cycles? In this work, we address the fate of hot ocean-planets produced when migration ends at a closer distance. In this case the liquid/gas interface can disappear, and the hot H2O envelope is made of a supercritical fluid. Although we do not expect these bodies to harbor life, their detection and identification as water-rich planets would give us insight as to the abundance of hot and, by extrapolation, cool ocean-planets. The water reservoir of these planets seems to be weakly affected by gravitational escape, provided that they are located beyond some minimum distance, e.g. 0.04 AU for a 5-Earth-mass planet around a Sun-like star. The swelling of their water atmospheres by the high stellar flux is expected not to significantly increase the planets' radii. We have studied the possibility of detecting and characterizing these hot ocean-planets by measuring their mean densities using transit missions in space—CoRoT (CNES) and Kepler (NASA)—in combination with Doppler velocimetry from the ground—HARPS (ESO) and possible future instruments. We have determined the domain in the [stellar magnitude, orbital distance] plane where discrimination between ocean-planets and rocky planets is possible with these instruments. The brightest stars of the mission target lists and the planets closest to their stars are the most favorable cases. Full advantage of high precision photometry by CoRoT, and particularly Kepler, can be obtained only if a new generation of Doppler instruments is built. 相似文献
11.
In Simpson’s (Simpson, G.C. [1927]. Mem. R. Meteorol. Soc. II (16), 69–95) classical derivation of the temperature of the Earth in the semi-gray model, the surface temperature diverges as the fourth root of the thermal radiation’s optical depth. No resolution to this apparent paradox was yet obtained under the strict semi-gray approximation. Using this approximation and a simplified approach, we study the saturation of the runaway greenhouse effect.First we generalize the problem of the semi-gray model to cases in which a non-negligible fraction of the stellar radiation falls on the long-wavelength range, and/or that the planetary long-wavelength emission penetrates into the transparent short wavelength domain of the absorption.Second, applying the most general assumptions and independently of any particular properties of an absorber, we show that the greenhouse effect saturates and that any Earth-like planet has a maximal temperature which depends on the type of and distance to its main-sequence star, its albedo and the primary atmospheric components which determine the cutoff frequency below which the atmosphere is optically thick. For example, a hypothetical convection-less planet similar to Venus, that is optically thin in the visible, could have at most a surface temperature of 1200–1300 K irrespective of the nature of the greenhouse gas.We show that two primary mechanisms are responsible for the saturation of the runaway greenhouse effect, depending on the value of λcut, the wavelength above which the atmosphere becomes optically thick. Unless λcut is small and resides in the optical region, saturation is achieved by radiating the thermal flux of the planet through the short wavelength tail of the thermal distribution. This has an interesting observational implication, the radiation from such a planet should be skewed towards the NIR. Otherwise, saturation takes place by radiating through windows in the FIR. 相似文献
12.
We present calculations for the evolution and surviving mass of highly-irradiated extrasolar giant planets (EGPs) at orbital semimajor axes ranging from 0.023 to 0.057 AU using a generalized scaled theory for mass loss, together with new surface-condition grids for hot EGPs and a consistent treatment of tidal truncation. Theoretical estimates for the rate of energy-limited hydrogen escape from giant-planet atmospheres differ by two orders of magnitude, when one holds planetary mass, composition, and irradiation constant. Baraffe et al. [Baraffe, I., Selsis, F., Chabrier, G., Barman, T.S., Allard, F., Hauschildt, P.H., Lammer, H., 2004. Astron. Astrophys. 419, L13-L16] predict the highest rate, based on the theory of Lammer et al. [Lammer, H., Selsis, F., Ribas, I., Guinan, E.F., Bauer, S.J., Weiss, W.W., 2003. Astrophys. J. 598, L121-L124]. Scaling the theory of Watson et al. [Watson, A.J., Donahue, T.M., Walker, J.C.G., 1981. Icarus 48, 150-166] to parameters for a highly-irradiated exoplanet, we find an escape rate ∼102 lower than Baraffe's. With the scaled Watson theory we find modest mass loss, occurring early in the history of a hot EGP. In this theory, mass loss including the effect of Roche-lobe overflow becomes significant primarily for masses below a Saturn mass, for semimajor axes ?0.023 AU. This contrasts with the Baraffe model, where hot EGPs are claimed to be remnants of much more massive bodies, originally several times Jupiter and still losing substantial mass fractions at present. 相似文献
13.
Christopher H. Broeg 《Icarus》2009,204(1):15-31
Context
Current planet search programs are detecting extrasolar planets at a rate of 60 planets per year. These planets show more diverse properties than was expected.Aims
We try to get an overview of possible gas giant (proto-) planets for a full range of orbital periods and stellar masses. This allows the prediction of the full range of possible planetary properties which might be discovered in the near future.Methods
We calculate the purely hydrostatic structure of the envelopes of proto-planets that are embedded in protoplanetary disks for all conceivable locations: combinations of different planetesimal accretion rates, host star masses, and orbital separations. At each location all hydrostatic equilibrium solutions to the planetary structure equations are determined by variation of core mass and pressure over many orders of magnitude. For each location we analyze the distribution of planetary masses.Results
We get a wide spectrum of core-envelope structures. However, practically all calculated proto-planets are in the planetary mass range. Furthermore, the planet masses show a characteristic bimodal, sometimes trimodal, distribution. For the first time, we identify three physical processes that are responsible for the three characteristic planet masses: self-gravity in the Hill sphere, compact objects, and a region of very low adiabatic pressure gradient in the hydrogen equation of state. Using these processes, we can explain the dependence of the characteristic masses on the planet’s location: orbital period, host star mass, and planetesimal accretion rate (luminosity). The characteristic mass caused by the self-gravity effect at close proximity to the host star is typically one Neptune mass, thus producing the so-called hot Neptunes.Conclusions
Our results suggest that hot Jupiters with orbital period less than 64 days (the exact location of the boundary depends on stellar type and accretion rate) have quite distinct properties which we expect to be reflected in a different mass distribution of these planets when compared to the “normal” planetary population. We use our theoretical survey to produce an upper mass limit for embedded planets: the maximum embedded equilibrium mass (MEEM). This naturally explains the lack of high mass planets between 3 and 64 days orbital period. 相似文献14.
15.
E. Mysen 《Monthly notices of the Royal Astronomical Society》2007,381(1):301-307
It is demonstrated how globally distributed outgassing activity on a triaxial comet nucleus bridges the gap between the intuitive Sekanina model, used for comet orbit solutions, and the physics of the problem. In this activity and shape limit, it is shown how a recoil force component, which originates from a day-side restricted sublimation process, is necessary to describe the comet's rotational evolution. Modifications of the non-gravitational force cosines are suggested, with a fundamentally different interpretation than before. Applications to asteroid rotation yield that the ability of specular reflection, of solar photons on an asteroid's surface, to change the asteroid's rotation period and equatorial obliquity, is not dependent on the overall shape of the asteroid. 相似文献
16.
Matija ?uk 《Icarus》2011,211(1):97-100
The Moon has long been known to have an overall shape not consistent with expected past tidal forces. It has recently been suggested (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) that the present lunar moments of inertia indicate a past high-eccentricity orbit and, possibly, a past non-synchronous spin-orbit resonance. Here I show that the match between the lunar shape and the proposed orbital and spin states is much less conclusive than initially proposed. Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) spin and shape evolution scenarios also completely ignore the physics of the capture into such resonances, which require prior permanent deformation, as well as tidal despinning to the relevant resonance. If the early lunar orbit was eccentric, the Moon would have been rotating at an equilibrium non-synchronous rate determined by it eccentricity. This equilibrium supersynchronous rotation would be much too fast to allow a synchronous spin-orbit lock at e = 0.49, while the capture into the 3:2 resonance is possible only for a very constrained lunar eccentricity history and assuming some early permanent lunar tri-axiality. Here I show that large impacts in the early history of the Moon would have frequently disrupted this putative resonant rotation, making the rotation and eccentricity solutions of Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) unstable. I conclude that the present lunar shape cannot be used to support the hypothesis of an early eccentric lunar orbit. 相似文献
17.
Adrián Brunini 《Icarus》2005,177(1):264-268
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. 相似文献
18.
Zonal winds simulated in two-dimensional computer models of turbulent convection in the equatorial plane of giant planets have greater surface amplitudes for cases with smaller solid cores, and therefore larger buoyancy driving, all other properties being equal. This differential rotation in radius is maintained by the convergence of angular momentum flux, which occurs because of the convective flow that develops due to the effects of planetary rotation and density stratification. The superposition of the convective flow and the stronger zonal flow produces wave-like, instead of cellular convection. 相似文献
19.
As the obliquity of Mars is strongly chaotic, it is not possible to give a solution for its evolution over more than a few million years. Using the most recent data for the rotational state of Mars, and a new numerical integration of the Solar System, we provide here a precise solution for the evolution of Mars' spin over 10 to 20 Myr. Over 250 Myr, we present a statistical study of its possible evolution, when considering the uncertainties in the present rotational state. Over much longer time span, reaching 5 Gyr, chaotic diffusion prevails, and we have performed an extensive statistical analysis of the orbital and rotational evolution of Mars, relying on Laskar's secular solution of the Solar System, based on more than 600 orbital and 200,000 obliquity solutions over 5 Gyr. The density functions of the eccentricity and obliquity are specified with simple analytical formulas. We found an averaged eccentricity of Mars over 5 Gyr of 0.0690 with standard deviation 0.0299, while the averaged value of the obliquity is 37.62° with a standard deviation of 13.82°, and a maximal value of 82.035°. We find that the probability for Mars' obliquity to have reached more than 60° in the past 1 Gyr is 63.0%, and 89.3% in 3 Gyr. Over 4 Gyr, the position of Mars' axis is given by a uniform distribution on a spherical cap limited by the obliquity 58.62°, with the addition of a random noise allowing a slow diffusion beyond this limit. We can also define a standard model of Mars' insolation parameters over 4 Gyr with the most probable values 0.068 for the eccentricity and 41.80° for the obliquity. 相似文献
20.
We calculate the evolution of planets undergoing a strong tidal encounter using smoothed particle hydrodynamics (SPH), for a range of periastron separations. We find that outside the Roche limit, the evolution of the planet is well-described by the standard model of linear, non-radial, adiabatic oscillations. If the planet passes within the Roche limit at periastron, however, mass can be stripped from it, but in no case do we find enough energy transferred to the planet to lead to complete disruption. In light of the three new extrasolar planets discovered with periods shorter than two days, we argue that the shortest-period cases observed in the period-mass relation may be explained by a model whereby planets undergo strong tidal encounters with stars, after either being scattered by dynamical interactions into highly eccentric orbits, or tidally captured from nearly parabolic orbits. Although this scenario does provide a natural explanation for the edge found for planets at twice the Roche limit, it does not explain how such planets will survive the inevitable expansion that results from energy injection during tidal circularization. 相似文献