Statistical approach to triple systems in three-dimensional motion

This paper considers disruption of triple close approaches with low initial velocities and equal masses in the framework of statistical escape theory in a three-dimensional space. The statistical escape theory is based on the assumption that the phase trajectory of a triple system is quasi-ergodic. This system is described by allowing for both energy and angular momentum conservation in the phase space. In this paper, “possibility of escape” is derived with the formation of a binary on the basis of relative distances of the participating bodies. The complete statistical solutions (i.e. the semi-major axis \(a\), the distributions of eccentricity \(e\) of the binary, binary energy \({E}_{{b}}\), escape energy \({E}_{{s}}\) of escaper, and its escape velocity \({v}_{{s}}\)) of the system are derived from the allowable phase space volumes and are in good agreement with the numerical results in the range of perturbing velocities \({v}_{{i}}\)(\(10^{ - 1} \le {v}_{{i}} \le 10^{ - 10}\)) and directions of \({v}_{{i}}(0 \le \alpha _{{i}},\beta _{{i}},\gamma _{{i}} \le \pi )\), \({i} = 1,2,3\). In this paper, the double limit process has been applied to approximate the escape probability. Through this process, it is observed that the perturbing velocity \({v}_{{i}} \to 0^{ +} \), as the product of the semi-major axis \(a\) of the final binary and the square of the escape velocity \({v}_{{s}}\) approach 2/3, i.e. \({a} {v}_{{s}}^{2} \to 2 / 3\), whatever direction of \(\mathbf{v}_{{i}}\) may be.     

Model for the explosion of a critical-mass neutron star in a binary system

A series of numerical models has been constructed for the three-dimensional explosion dynamics of a low-mass neutron star in a binary system that results from the collapse of the rotating iron core of a massive supernova. The numerical solution has been obtained by the particle method with an adaptive time step that allows the computational accuracy to be controlled automatically. The constructed numerical models include the proper motion of the massive component in the binary system of neutron stars, their finite sizes, the graduality of energy release during the explosive disruption of a critical-mass neutron star, and the nonuniform expansion velocity distribution of iron ejecta. The extent to which each of the listed parameters affects the explosion characteristics has been determined. The total explosion energy and the pulsar escape velocity have been estimated. A sizable fraction of the material of the exploded neutron star has been found to remain gravitationally bound to the massive component of the binary system. A further study of its dynamics is of interest in its own right, because the captured material can be considered as an additional source of muon neutrinos.     

Propagation of sprays with deceleration

The dependence of the maximum height achieved by sprays showing deceleration and stopping at the end of their propagation, on the velocity has been examined. It was assumed that the velocity is greatest at the beginning, so this maximum velocity defines the kinetic energy of each investigated knot at its start. The relation between the maximum height and maximum velocity was examined in 32 knots from 23 sprays. The problem of whether the maximum height achieved by a knot is determined by the gravitational field alone has been studied. The problem of stopping is important especially for knots which have initial velocities higher than the velocity of escape from the Sun.     

Earth–Moon Weak Stability Boundaries in the restricted three and four body problem

This work deals with the structure of the lunar Weak Stability Boundaries (WSB) in the framework of the restricted three and four body problem. Geometry and properties of the escape trajectories have been studied by changing the spacecraft orbital parameters around the Moon. Results obtained using the algorithm definition of the WSB have been compared with an analytical approximation based on the value of the Jacobi constant. Planar and three-dimensional cases have been studied in both three and four body models and the effects on the WSB structure, due to the presence of the gravitational force of the Sun and the Moon orbital eccentricity, have been investigated. The study of the dynamical evolution of the spacecraft after lunar capture allowed us to find regions of the WSB corresponding to stable and safe orbits, that is orbits that will not impact onto lunar surface after capture. By using a bicircular four body model, then, it has been possible to study low-energy transfer trajectories and results are given in terms of eccentricity, pericenter altitude and inclination of the capture orbit. Equatorial and polar capture orbits have been compared and differences in terms of energy between these two kinds of orbits are shown. Finally, the knowledge of the WSB geometry permitted us to modify the design of the low-energy capture trajectories in order to reach stable capture, which allows orbit circularization using low-thrust propulsion systems.     

Mass effects in the problem of three bodies

This is a study of the dynamical behavior of three point masses moving under their mutual gravitational attraction in a plane. The initial positions and velocities are identical for all cases studied and only the masses of the participating bodies change in the series of numerical experiments. In this way the effect of the coupling terms in the differential equations of motion are investigated. The motion in all 125 cases begins with an interplay between the three bodies, followed by temporary ejections or by an eventual escape. The total mass of the system is kept constant while the massratios change from 1 to 5. The initial velocities being zero, the total energy is negative in all cases.Approximately 74% of the cases disintegrated (i.e. two bodies formed a binary and the third body escaped) in less than 140 time units, 47% in less than 50 time units and 10% ended in escape in less than 10 time units. Considering three stars with total mass 12M , initially placed at 3, 4 and 5 parsec distances (or three galaxies with mass 2.4×10¹² M , initially placed 30, 40 and 50 kpc apart), the unit of time (approximately the crossing time) becomes 1.5×10⁷ y (3.2×10⁷ y). The average time of disintegration was found to be of the order of 10⁹ y. The average semi-major axis of the binaries left behind after disintegration was 0.7 parsec and the average value of the eccentricity was 0.76. The effect of the masses on the escapes was established and it was found that the bodynot with the smallest mass escaped in 13% of the disintegrated cases. The cases which did not disintegrate in 150 time units were analyzed in detail and the time of their eventual escape was estimated.The numerical results are tabulated regarding escape time, ejection period, total energy, escape energy, terminal velocity, semi-major axis, and eccentricity.The evolution of triple systems is followed from interplays through ejections to escapes and the orbital parameters for the separation of these classes are estimated.     

Orbital evolution with white-dwarf kicks

Recent observations of white dwarfs in globular clusters indicate that these stars may get a velocity kick during their time as giants. This velocity kick could originate naturally if the mass loss while on the asymptotic giant branch is slightly asymmetric. If white dwarfs get a kick comparable to the orbital velocity of the binary, the initial Runge–Lenz vector (eccentricity vector) of the orbit is damped to be replaced by a component pointing toward the cross-product of the initial angular momentum and the force. The final eccentricity may be of the order of unity and if the kick is sufficiently large, the system may be disrupted. These results may have important ramifications for the evolution of binary stars and planetary systems.     

Changes in the gravitational energy of galaxies during collisions

The problem of the change in gravitational energy of a colliding galaxy due to tidal effects is considered. The change in the internal energy, the mass of escaping matter and the change in the mean radius of the test galaxy have been estimated for a relative velocity of 1000 km s^–1 for three distances of closest approach for the following four cases: (a) both galaxies centrally concentrated, (b) both galaxies homogeneous, (c) test galaxy centrally concentrated, field galaxy homogeneous, and (d) test galaxy homogeneous, field galaxy centrally concentrated. The masses and radii of the two galaxies are taken as 10¹¹ M and 10 kpc respectively. For simplicity, the galaxies are assumed to be spherically symmetric and the distribution of mass within a centrally concentrated galaxy is assumed to be that of a polytrope of indexn=4. The results also provide estimates for the minimum relative velocity a galaxy must have in order that it may not be captured by another to form a double system. It has been found that normally a relative velocity of less than about 500 km s^–1 will lead to the formation of a double galaxy by tidal capture. In the case of a head-on collision between two centrally concentrated galaxies even a relative velocity of about 1000 km s^–1 is small enough for tidal capture. The changes in the structure of the galaxies for relative velocities equal to velocity of escape are also indicated. These results show that there is no escape of matter from the test galaxy in cases (b) and (c). In the case (a) the escape of matter can be as high as 4% of the total mass. The head-on collision between galaxies are normally not accompanied by any escape of matter. All the gain in the internal energy of galaxies during such collisions results in increase in their dimensions. The fractional increase in the mean radius of the test galaxy in the head-on collision is 1.5 in the case (a), 3.2 in the case (b) and 0.01 in the case (c). In the case (d) the test galaxy will be disrupted by the tidal forces.     

On the total kinetic energy of our galaxy

The total kinetic energy of the Galaxy is estimated from the potential energy by applying the virial theorem. The limits of the potential energy depend strongly on the value of the local escape velocity. They are estimated to be between −7. 10¹⁶ M_Ȯ km²s⁻² and −1. 10¹⁶ M_Ȯ km²s⁻² (escape velocity approximately between 450 km s⁻¹ and 600 km s⁻¹). The specific kinetic energy of the Galaxy as a whole is most likely about 21 000 km²s⁻², being equally distributed among the subsystems if the local escape velocity is near its lower limit; the higher the local escape velocity is, the higher is the specific kinetic energy of the Galaxy due to the influence of the dark corona. The specific kinetic energy of the dark corona tends to become equal to that of the Galaxy as a whole for very high values of the local escape velocity. For the purpose of estimating the totl potential energy of the Galaxy, inter alia, a new model of the Milky Way is developed, which yields both the potential and the density analytically so that it is suitale for calculating the galactocentric orbits.     

Impact and explosion crater ejecta,fragment size,and velocity

A model was developed for the mass distribution of fragments that are ejected at a given velocity for impact and explosion craters. The model is semiempirical in nature and is derived from (1) numerical calculations of cratering and the resultant mass versus ejection velocity, (2) observed ejecta blanket particle size distributions, (3) an empirical relationships between maximum ejecta fragment size and crater diameter, (4) measurements of maximum ejecta size versus ejecta velocity, and (5) an assumption on the functional form for the distribution of fragments ejected at a given velocity. This model implies that for planetary impacts into competent rock, the distribution of fragments ejected at a given velocity is broad; e.g., 68% of the mass of the ejecta at a given velocity contains fragments having a mass less than 0.1 times a mass of the largest fragment moving at that velocity. Using this model, we have calculated the largest fragment that can be ejected from asteroids, the Moon, Mars, and Earth as a function of crater diameter. The model is unfortunately dependent on the size-dependent ejection velocity limit for which only limited data are presently available from photography of high explosive-induced rock ejecta. Upon formation of a 50-km-diameter crater on an atmosphereless planet having the planetary gravity and radius of the Moon, Mars, and Earth, fragments having a maximum mean diameter of ≈30, 22, and 17 m could be launched to escape velocity in the ejecta cloud. In addition, we have calculated the internal energy of ejecta versus ejecta velocity. The internal energy of fragments having velocities exceeding the escape velocity of the moon (～2.4 km/sec) will exceed the energy required for incipient melting for solid silicates and thus, the fragments ejected from Mars and the Earth would be melted.     

A dynamical investigation of the conjecture that Mercury is an escaped satellite of Venus

The possibility that Mercury might once have been satellite of a Venus, suggested by a number of anomalies, is investigated by a series of numerical computer experiments. Tidal interaction between Mercury and Venus would result in the escape of Mercury into a solar orbit. Only two escape orbits are possible, one exterior and one interior to the Venus orbit. For the interior orbit, subsequent encounters are sufficiently distant to avoid recapture or large perturbations. The perihelion distance of Mercury tends to decrease, while the orientation of perihelion librates for the first few thousand revolutions. If dynamical evolution or nonconservative forces were large enough in the early solar system, the present semimajor axes could have resulted. The theoretical minimum quadrupole moment of the inclined rotating Sun would rotate the orbital planes out of coplanarity. Secular perturbations by the other planets would evolve the eccentricity and inclination of Mercury's orbit through a range of possible configurations, including the present orbit. Thus the conjecture that Mercury is an escaped satellite of Venus remains viable, and is rendered more attractive by our failure to disprove it dynamically.     

Planetary dynamics in the system α Centauri: The stability diagrams

The stability of the motion of a hypothetical planet in the binary system ?? Cen A?CB has been investigated. The analysis has been performed within the framework of a planar (restricted and full) three-body problem for the case of prograde orbits. Based on a representative set of initial data, we have obtained the Lyapunov spectra of the motion of a triple system with a single planet. Chaotic domains have been identified in the pericenter distance-eccentricity plane of initial conditions for the planet through a statistical analysis of the data obtained. We have studied the correspondence of these chaotic domains to the domains of initial conditions that lead to the planet??s encounter with one of the binary??s stars or to the escape of the planet from the system. We show that the stability criterion based on the maximum Lyapunov exponent gives a more clear-cut boundary of the instability domains than does the encounterescape criterion at the same integration time. The typical Lyapunov time of chaotic motion is ??500 yr for unstable outer orbits and ??60 yr for unstable inner ones. The domain of chaos expands significantly as the initial orbital eccentricity of the planet increases. The chaos-order boundary has a fractal structure due to the presence of orbital resonances.     

N-body simulations of tidal encounters between stellar systems

N-Body simulations have been performed to study the tidal effects of a primary stellar system on a secondary stellar system of density close to the Roche density. Two hyperbolic, one parabolic and one elliptic encounters have been simulated. The changes in energy, angular momentum, mass distribution, and shape of the secondary system have been determined in each case. The inner region containing about 40 per cent of the mass was found to be practically unchanged and the mass exterior to the tidal radius was found to escape. The intermediate region showed tidal distension. The thickness of this region decreased as we went from hyperbolic encounters to the elliptic encounter keeping the distance of closest approach constant. The numerical results for the fractional change in energy have been compared with the predictions of the available analytic formulae and the usefulness and limitations of the formulae have been discussed.     

Co-accretion of the Earth-Moon system after the giant impact: reduction of the total angular momentum by lunar impact ejecta

Though the Moon is considered to have been formed by the so-called giant impact, the mass of the Earth immediately after the impact is still controversial. If the Moon was formed during the Earth's accretion, a subsequent accretion of residual heliocentric planetesimals onto the protoearth and the protomoon must have occurred. In this co-accretion stage, a significant amount of lunar-impact-ejecta would be ejected to circumterrestrial orbits, since the mean impact velocity of the planetesimals with the protomoon is much larger than the escape velocity of the protomoon. Orbital calculations of test particles ejected from the protomoon, whose semimajor axis is smaller than that of the present Moon, reveal that most of the particles escaping from the protomoon also escape from the Hill sphere of the protoearth and reduce the planetocentric angular momentum of the primordial Earth-Moon system. Using the results of the ejecta simulations, we investigate the evolution of the mass ratio and the total angular momentum (Earth's spin angular momentum + Moon's orbital angular momentum) of the Earth-Moon system during the co-accretion. We find that the mass of the protomoon is almost constant or rather decreases and the total angular momentum decreases significantly, if the random velocity of planetesimals is as large as the escape velocity of the protoearth. On the other hand, if the random velocity is the half of the escape velocity of the protoearth, the mass ratio is kept to be almost as large as the present value and the decrease of the total angular momentum is not so significant. Comparing with the results of giant impact simulations, we find that the mass of the protoearth immediately after the Moon-forming impact was 0.7-0.8 times the present value if the impactor-to-target mass ratio was 3:7, whereas the giant impact occurred almost in the end of the Earth's accretion if the impactor-to-target mass ratio was 1:9.     

Effect of Orbital Distance on the Atmospheric Escape of Exoplanets

Atmospheric escape is an important sector in the evolution of planetary atmosphere, and its energy is mainly originated from the radiation of the host star at the high energy band. The radiation flux drops dramatically with the increase of orbital distance, there is a large difference of planetary atmospheric escape in different orbits, so it is necessary to study the impact of orbital distance on the atmospheric escape of an exoplanet. We consider the radiation transfer and the photochemical reactions of multiple kinds of particles to study the variation of planetary atmospheric escape with the orbital distance by using a 1-D hydrodynamic model. Due to the large differences of the spectra of host stars in different evolution stages, the Astrophysical Plasma Emission Code (APEC) in the X-Ray Spectral Fitting Package (XSPEC) is used to obtain the spectra of solar-type stars with different ages as the input spectra of the model. The results indicate that the escape rates of the exoplanets in different orbits are different significantly, and the escape mechanism is converted from the drastic hydrodynamic escape into the moderate Jeans escape as the orbital distance increases, the smaller the planetary gravitational potential, the younger the star-planet system, the larger the distance of this conversion. The correlation between the escape rate and the radiation flux decreases for the short-period exoplanets in a younger star-planet system. It is shown that the classical energy-limited escape theory is not suitable for this kind of exoplanets. These results have enriched the studies on the atmospheric escape of exoplanets, especially, extended the studies on the escape mechanism and energy conversion under different orbital distances and stellar ages.     

The dynamical evolution of massive black hole binaries - II. Self-consistent N-body integrations

We use a hybrid N-body program to study the evolution of massive black hole binaries in the centers of galaxies, mainly to understand the factors affecting the binary eccentricity, the response of the galaxy to the binary merger, and the effect of loss-cone depletion on the merger time. The scattering experiments from paper I (Quinlan, 1996)[NewA, 1, 35] showed that the merger time is not sensitive to the eccentricity growth unless a binary forms with at least a moderate eccentricity. We find here that the eccentricity can become large under some conditions if a binary forms in a galaxy with a flat core or with a radial bias in its velocity distribution, especially if the dynamical friction is enhanced by resonances as suggested by Rauch & Tremaine (1996)[NewA, 1, 149]. But the necessary conditions seem unlikely, and our prediction from paper I remains unchanged: in most cases the eccentricity will start and remain small. The ejection of stars caused by the hardening of a binary may explain why large elliptical galaxies have weaker density cusps than smaller galaxies. If so, the central velocity distributions in those galaxies should have strong tangential anisotropies. The wandering of a binary from the center of a galaxy counteracts the effects of loss-cone depletion and helps the binary merge.     

Distribution of lunar impacts by objects in parabolic low inclination orbits

This paper presents a computer investigation extending to the case of parabolic orbits, an earlier investigation conducted by Barricelli and Metcalfe (1969) on lunar impacts by external low eccentricity satellites as a means to interpret the asymmetric distribution of lunar maria. Parabolic orbits can be approximated by two kinds of objects:

1. High eccentricity external satellites may, near periapsis, approach the Moon with orbital velocity and other characteristics closely resembling those of a parabolic orbit.
2. Asteroids and meteoroids approaching the Earth-Moon system with a low velocity may have moved in a nearly parabolic orbit when they reached the lunar distance from the Earth at the time when the impacts which carved the lunar maria took place.

The investigation gives, therefore, not only additional information relevant to the interpretation of the distribution of lunar maria by the satellite impacts hypothesis (in this case high eccentricity ones), but also information about the alternative hypothesis (Wood, 1973) that asteroid impacts rather than satellite impacts were involved.     

Impact erosion of planetary atmospheres

The impact of a large extraterrestrial body onto a planet deposits considerable energy in the atmosphere. If the radius of the impactor is much larger than an atmospheric scale height and its velocity much larger than the planetary escape velocity, some of the planetary atmosphere may be driven off into space. The process is analyzed theoretically in this paper. The amount of gas that escapes is equal to the amount of gas intercepted by the impacting body multiplied by a factor not very different from unity. Escape occurs only if the velocity of the impacting body exceeds the planetary escape velocity. At large impact velocities the enhancement factor, which is the factor multiplying the amount of atmosphere intercepted by the impacting body, approaches a constant value approximately equal to 10¹²/V_e², where V_e is the escape velocity (in cm/sec). The enhancement factor is independent of atmospheric mass or surface pressure. Ablation of the impacting body and the planetary surface adds to the mass of gas that must be accelerated into space if escape is to occur. As a result, impact erosion of the atmosphere does not occur from a planet with an escape velocity in excess of 10 km/sec.     

Circinus X-1: survivor of a highly asymmetric supernova

We have analysed the kinematical parameters of Cir X-1 to constrain the nature of its companion star, the eccentricity of the binary and the pre-supernova parameter space. We argue that the companion is most likely to be a low-mass (≲2.0 M_⊙) unevolved star and that the eccentricity of the orbit is 0.94±0.04. We have evaluated the dynamical effects of the supernova explosion and we find it must have been asymmetric. On average , we find that a kick of ∼740 km s⁻¹ is needed to account for the recently measured radial velocity of +430 km s⁻¹ (Johnston, Fender & Wu) for this extreme system. The corresponding minimum kick velocity is ∼500 km s⁻¹. This is the largest kick needed to explain the motion of any observed binary system. If Cir X-1 is associated with the supernova remnant G321.9-0.3 then we find a limiting minimum age of this remnant of ∼60 000 yr. Furthermore, we predict that the companion star has lost ∼10 per cent of its mass as a result of stripping and ablation from the impact of the supernova shell shortly after the explosion.     

Stability of Binary Asteroids

The stability and final outcome of a strongly interacting binary asteroid system is considered. We discuss the implications of the system transferring energy and angular momentum between rotational and translational motion while conserving the total system energy and angular momentum. Using these results we can develop a set of sufficient conditions for stability against escape and impact. These allow us to delineate several classes of final outcomes for a binary asteroid system, each of which may have implications for asteroid observations. The effects of energy dissipation on an asteroid binary system are also considered and are shown to be able to change the stability of the system against escape and impact. An example computation for the near-Earth asteroid binary 1996 FG3 is given along with a series of numerical explorations of an evolving binary system consisting of an ellipsoid and a sphere of equal mass.     

Analytical study of the powered Swing-By maneuver for elliptical systems and analysis of its efficiency

Analytical equations describing the velocity and energy variation of a spacecraft in a Powered Swing-By maneuver in an elliptic system are presented. The spacecraft motion is limited to the orbital plane of the primaries. In addition to gravity, the spacecraft suffers the effect of an impulsive maneuver applied when it passes by the periapsis of its orbit around the secondary body of the system. This impulsive maneuver is defined by its magnitude \(\delta V\) and the angle that defines the direction of the impulse with respect to the velocity of the spacecraft (\(\alpha\)). The maneuver occurs in a system of main bodies that are in elliptical orbits, where the velocity of the secondary body varies according to its position in the orbit following the rules of an elliptical orbit. The equations are dependent on this velocity. The study is done using the “patched-conics approximation”, which is a method of simplifying the calculations of the trajectory of a spacecraft traveling around more than one celestial body. Solutions for the velocity and energy variations as a function of the parameters that define the maneuver are presented. An analysis of the efficiency of the powered Swing-By maneuver is also made, comparing it with the pure gravity Swing-by maneuver with the addition of an impulse applied outside the sphere of influence of the secondary body. After a general study, the techniques developed here are applied to the systems Sun-Mercury and Sun-Mars, which are real and important systems with large eccentricity. This problem is highly nonlinear and the dynamics very complex, but very reach in applications.