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1.
By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L4 and L5 are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L4 and L5, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.  相似文献   

2.
All the Trojan asteroids orbit about the Sun at roughly the same heliocentric distance as Jupiter. Differences in the observed visible reflection spectra range from neutral to red, with no ultra-red objects found so far. Given that the Trojan asteroids are collisionally evolved, a certain degree of variability is expected. Additionally, cosmic radiation and sublimation are important factors in modifying icy surfaces even at those large heliocentric distances. We search for correlations between physical and dynamical properties, we explore relationships between the following four quantities; the normalised visible reflectivity indexes (S′), the absolute magnitudes, the observed albedos and the orbital stability of the Trojans. We present here visible spectroscopic spectra of 25 Trojans. This new data increase by a factor of about 5 the size of the sample of visible spectra of Jupiter Trojans on unstable orbits. The observations were carried out at the ESO-NTT telescope (3.5 m) at La Silla, Chile, the ING-WHT (4.2 m) and NOT (2.5 m) at Roque de los Muchachos observatory, La Palma, Spain. We have found a correlation between the size distribution and the orbital stability. The absolute-magnitude distribution of the Trojans in stable orbits is found to be bimodal, while the one of the unstable orbits is unimodal, with a slope similar to that of the small stable Trojans. This supports the hypothesis that the unstable objects are mainly byproducts of physical collisions. The values of S′ of both the stable and the unstable Trojans are uniformly distributed over a wide range, from 0%/1000 Å to about 15%/1000 Å. The values for the stable Trojans tend to be slightly redder than the unstable ones, but no significant statistical difference is found.  相似文献   

3.
Message derived a method to detect bifurcations of a family of asymmetric periodic solutions from a family of symmetric periodic solutions in the restricted problem of three bodies for the limiting case when the second body has zero mass. This is used to examine several small integer commensurabilities. A total of 21 exterior and 21 interior small integer commensurabilities are examined and bifurcations (two in number) are found to exist only for exterior commensurabilities (q+1):1,q=1, 2,, 7. On investigating other commensurabilities of this form for values ofq up to 50 two bifurcations are still found to exist for each. The eccentricities of the two bifurcation orbits are given for eachq up to 20. For a Sun-Jupiter mass ratio the complete family of asymmetric periodic solutions associated withq=1, 2,..., 5, and the initial segments of the asymmetric family withq=6, 7,..., 12, have been numerically determined. The family associated withq=5 contains some unstable orbits but all orbits in the other four complete families are stable. The five complete families each begin and end on the same symmetric family. The network of asymmetric and symmetric families close to the commensurabilities (q+1):1,q=1, 2,..., 5 is discussed.  相似文献   

4.
The near-Earth objects and their potential threat to our planet   总被引：1，自引：0，他引：1  
The near-Earth object (NEO) population includes both asteroids (NEAs) and comet nuclei (NECs) whose orbits have perihelion distances q<1.3 AU and which can approach or cross that of the Earth. A NEA is defined as a “potentially hazardous asteroid” (PHA) for Earth when its minimum orbit intersection distance (MOID) comes inside 0.05 AU and it has an absolute magnitude H<22 mag (i.e. mean diameter > 140 m). These are big enough to cause, in the case of impact with Earth, destructive effects on a regional scale. Smaller objects can still produce major damage on a local scale, while the largest NEOs could endanger the survival of living species. Therefore, several national and international observational efforts have been started (i) to detect undiscovered NEOs and especially PHAs, (ii) to determine and continuously monitor their orbital properties and hence their impact probability, and (iii) to investigate their physical nature. Further ongoing activities concern the analysis of possible techniques to mitigate the risk of a NEO impact, when an object is confirmed to be on an Earth colliding trajectory. Depending on the timeframe available before the collision, as well as on the object’s physical properties, various methods to deflect a NEO have been proposed and are currently under study from groups of experts on behalf of international organizations and space agencies. This paper will review our current understanding of the NEO population, the scientific aspects and the ongoing space- and ground-based activities to foresee close encounters and to mitigate the effects of possible impacts.  相似文献   

5.
A substantial fraction of the Edgeworth-Kuiper belt objects are presently known to move in resonance with Neptune (the principal commensurabilities are 1/2, 3/5, 2/3, and 3/4). We have found that many of the distant (with orbital semimajor axes a > 50 AU) trans-Neptunian objects (TNOs) also execute resonant motions. Our investigation is based on symplectic integrations of the equations of motion for all multiple-opposition TNOs with a > 50 AU with allowance made for the uncertainties in their initial orbits. Librations near such commensurabilities with Neptune as 4/9, 3/7, 5/12, 2/5, 3/8, 4/27, and others have been found. The largest number of distant TNOs move near the 2/5 resonance with Neptune: 12 objects librate with a probability higher than 0.75. The multiplicity of objects moving in 2/5 resonance and the longterm stability of their librations suggest that this group of resonant objects was formed at early formation stages of the Solar system. For most of the other resonant objects, the librations are temporary. We also show the importance of asymmetric resonances in the large changes in TNO perihelion distances.  相似文献   

6.
We have considered polar ring galaxy candidates, the images of which can be found in the SDSS survey. The sample of 78 galaxies includes the most reliable candidates from the SPRC and PRC catalogs, some of which already have kinematic confirmations. We analyze the distributions of studied objects by the angle between the polar ring and the central disk, and by the optical diameter of the outer ring structures. In the vast majority of cases, the outer structures lie in the plane close to polar (within 10°–20°) which indicates the stability of the corresponding orbits in the gravitational potential of the halo. Moderately inclined outer structures are observed only in about 6% of objects which probably indicates their short lifetime. In such an unstable configuration, the polar ring would often cross the disk of the galaxy, being smaller than it in the diameter. We show that the inner polar structures and outer large-scale polar rings form a single family in the distribution of diameters normalized to the optical size of the galaxy. At the same time, this distribution is bimodal, as the number of objects with d ring = (0.4–0.7) d disk is small. Such a shape of size distribution is most likely due to the fact that the stability of polar orbits in the inner regions of galaxies is maintained by the bulge or the bar, while in the outer regions it is provided by the spheroidal (or triaxial) halo.  相似文献   

7.
We study isofrequency pairing of the circular non-geodesic orbits in the vicinity of the Schwarzschild black hole immersed in external asymptotically uniform magnetic field. The dependence of the isofrequency pairing of non-geodesic orbits from the special quantities, such as radius of the innermost stable circular orbits (ISCO), radius of the circular orbits (r b ) at the limit e(eccentricity)→0, and from the value of the small quantity ξ=(r b ?r ISCO) of the particles moving around the Schwarzschild black hole in the presence of the magnetic field has been found. It is shown that presence of the magnetic field gives rise to the r ISCO and r b of the particles to be slightly shifted from black hole and the contribution of the quantity ξ decreases the amount of isofrequency pairing of non-geodesic orbits due to the reduction of the surface of the region between the separatrix and circular-orbit duals (COD). We study the dependence of surface of region where particles can move and isofrequency pairing of non-geodesic orbits occur around the Schwarzschild black hole from the magnetic field. We find a decrease of nearly (7–10) % for the maximal values of the magnetic field B～106–107 Gauss in the surface of region where isofrequency pairing occurs around central object with compare to the one around Schwarzschild black hole without magnetic field. This result implies that it makes easier to identify signals through the astrophysical observations of compact objects in the presence of magnetic field.  相似文献   

8.
This paper analyzes the distribution of the orbits of near-Earth minor bodies from the data on more than 7500 objects. The distribution of large near-Earth objects (NEOs) with absolute magnitudes of H < 18 is generally consistent with the earlier predictions (Bottke et al., 2002; Stuart, 2003), although we have revealed a previously undetected maximum in the distribution of perihelion distances q near q = 0.5 AU. The study of the orbital distribution for the entire sample of all detected objects has found new significant features. In particular, the distribution of perihelion longitudes seriously deviates from a homogeneous pattern; its variations are roughly 40% of its mean value. These deviations cannot be stochastic, which is confirmed by the Kolmogorov-Smirnov test with a more than 0.9999 probability. These features can be explained by the dynamic behavior of the minor bodies related to secular resonances with Jupiter. For the objects with H < 18, the variations in the perihelion longitude distribution are not so apparent. By extrapolating the orbital characteristics of the NEOs with H < 18, we have obtained longitudinal, latitudinal, and radial distributions of potentially hazardous objects in a heliocentric ecliptic coordinate frame. The differences in the orbital distributions of objects of different size appear not to be a consequence of observational selection, but could indicate different sources of the NEOs.  相似文献   

9.
As follows from dynamical studies, in the course of evolution, most near-Earth objects reach orbits with small perihelion distances. Changes of the asteroids in the vicinity of the Sun should play a key role in forming the physical properties, size distribution, and dynamical features of the near-Earth objects. Only seven of the discovered asteroids are currently moving along orbits with perihelion distances q < 0.1 AU. However, due to the Kozai–Lidov secular perturbations, the asteroids, having recently passed near the Sun, could by now have moved to orbits farther from the Sun. In this study, we found asteroids that have been recently orbiting with perihelion distances q < 0.1 AU. Asteroids may be on such orbits for hundreds to tens of thousands of years. To carry out astrophysical observations of such objects is a high priority.  相似文献   

10.
In October 2009, a new set of optical observations of Apophis, a potentially hazardous asteroid, was published. These data have significantly expanded the interval of observations and their total number. In the article we compare the results of refinement of Apophis’ orbit made at the Jet Propulsion Laboratory (JPL, United States), the University of Pisa (Italy), and the Institute of Applied Astronomy (IAA) of the Russian Academy of Sciences with consideration for new observations. New orbits lead to a significant decrease in the probability of Apophis’ collision with the Earth in 2036. As a result of processing a large number of observations of asteroids approaching the Earth and main belt asteroids less than 40 km in size, with a large number of optical and, in many cases, radar observations in different oppositions, one of the authors revealed that additional acceleration affects their motion. This acceleration can be represented by the transversal component A 2 in the orbital coordinate system. The presence of this acceleration can be interpreted as the Yarkovsky effect. The statistical properties of distribution of A 2 for asteroids, for which it is determined quite reliably, evidence in favor of this interpretation. The value of additional acceleration for bodies the size of Apophis falls in the range ±10−13 AU/day2. In this paper we have calculated the probability of Apophis colliding with the Earth in 2036 at different values of the transversal component of additional acceleration A 2. For the resulting points, a plot of the probability of the collision against the A 2 value has been constructed. At A 2 = −8.748 × 10−14 AU/day2 (and zero values of the radial A 1 and normal A 3 components) the nominal solution for Apophis’ orbit on April 13, 2029, is only 90 m from the middle of a “keyhole” 600 m in width, which leads to a collision of Apophis with the Earth in 2036. Since the scattering ellipse in the target plane in 2029 significantly overlaps the keyhole, the probability of collision at the given additional acceleration value is 0.0022. This result has been verified by the Monte Carlo method. Tests of 10000 random sets of orbital elements, which were found taking into account their correlation, have shown that 22 cases have resulted in virtual asteroids colliding with Earth in 2036. A plot of the probability of the collision against the value of A 2 has been constructed.  相似文献   

11.
The classical Öpik theory provides an estimate of the collision probability between two bodies on bound, heliocentric or planetocentric orbits under restrictive assumptions of: (i) constant eccentricity and inclination, and (ii) uniform circulation of the longitude of node and argument of pericenter. These assumptions are violated whenever either of the orbits has a large inclination with respect to the local Laplace plane or large eccentricity, and their motion is perturbed by an exterior (tidal) gravitational field of a planet or the Sun. In this situation, known as the Lidov–Kozai regime, the eccentricity and inclination values exhibit large and correlated oscillations. At the same time, the longitude of node and the argument of pericenter may have strongly nonlinear time evolution, with the latter being even bound to a small interval of values. Here we develop a new Öpik-type collision probability theory which is valid even for highly inclined and/or eccentric orbits of the projectile. We assume that the orbit of the target is circular and in the local Laplace plane. Such a generalized setting is necessary, as an example, to correctly estimate the terrestrial impact fluxes of sporadic micrometeoroids on high-inclination orbits (notably those from the toroidal source and the associated helion and anti-helion arcs).  相似文献   

12.
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13.
This paper analyzes the capture of comets into Halley-type and Jupiter-family orbits from the nearparabolic flux of the Oort cloud. Two types of capture into Halley-type orbits are found. The first type is the evolution of near-parabolic orbits into short-period orbits (with heliocentric orbital periods P < 200 years) as a result of close encounters with giant planets. This process is followed by a very slow drift of cometary orbits into the inner part of the Solar System. Only those comets may pass from short-period orbits into Halley-type and Jupiter-family orbits, which move in orbits with perihelion distances q < 13 au. In the second type of capture, the perihelion distances of cometary orbits become rather small (< 1.5 au) during the first stage of dynamic evolution under the action of perturbations from the Galaxy, and then their semimajor axes decrease as a result of diffusion. The capture takes place, on average, in 500 revolutions of the comet about the Sun, whereas in the first case, the comet is captured, on average, after 12500 revolutions. The region of initial orbital perihelion distances q > 4 au is found to be at least as important a source of Halley-type comets as the region of perihelion distances q < 4 au. More than half of the Halley-type comets are captured from the nearly parabolic flux with q > 4 au. The analysis of the dynamic evolution of objects moving in short-period orbits shows that the distribution of Centaurs orbits agrees well with the observed distribution corrected for observational selection effects. Hence, the hypothesis associating the origin of Centaurs with the Edgeworth-Kuiper belt and the trans-Neptunian region exclusively should be rejected.  相似文献   

14.
We have integrated the orbits of the 76 scattered disk objects (SDOs), discovered through the end of 2002, plus 399 clones for 5 Gyr to study their dynamical evolution and the probability of falling in one of the following end states: reaching Jupiter's influence zone, hyperbolic ejection, or transfer to the Oort cloud. We find that nearly 50% of the SDOs are transferred to the Oort cloud (i.e., they reach heliocentric distances greater than 20,000 AU in a barycentric elliptical orbit), from which about 60% have their perihelia beyond Neptune's orbit (31 AU<q<36 AU) at the moment of reaching the Oort cloud. This shows that Neptune acts as a dynamical barrier, scattering most of the bodies to near-parabolic orbits before they can approach or cross Neptune's orbit in non-resonant orbits (that may allow their transfer to the planetary region as Centaurs via close encounters with Neptune). Consequently, Neptune's dynamical barrier greatly favors insertion in the Oort cloud at the expense of the other end states mentioned above. We found that the current rate of SDOs with radii R>1 km incorporated into the Oort cloud is about 5 yr−1, which might be a non-negligible fraction of comet losses from the Oort cloud (probably around or even above 10%). Therefore, we conclude that the Oort cloud may have experienced and may be even experiencing a significant renovation of its population, and that the trans-neptunian belt—via the scattered disk—may be the main feeding source.  相似文献   

15.
The second species periodic solutions of the restricted three body problem are investigated in the limiting case of μ=0. These orbits, called consecutive collision orbits by Hénon and generating orbits by Perko, form an infinite number of continuous one-parameter families and are the true limit, for μ→0, of second species periodic solutions for μ>0. By combining a periodicity condition with an analytic relation, for criticality, isolated members of several families are obtained which possess the unique property that the stability indexk jumps from ±∞ to ?∞ at that particular orbit. These orbits are of great interest since, for small μ>0, ‘neighboring’ orbits will then have a finite (but small) region of stability.  相似文献   

16.
We compare the properties of orbits in five different rotationally symmetric models of the regular Galactic force field. Our goal is to estimate the possible range of the discrepancies in stellar orbital elements attributable to the choice of the model for the Galactic potential. For disk objects, the discrepancies in eccentricities e attributable to the choice of the model do not exceed 0.05. The relative distances h from the symmetry plane can differ by more than a factor of 2. Since there are resonance regions, the types of orbits (box-shaped or tube-shaped) cannot be reliably identified. The situation is even worse for halo objects: the discrepancies in h can exceed 100%, and identifying the type of orbits is also highly problematic.  相似文献   

17.
The calculation of collision probability is the foundation of collision detection and avoidance maneuver for space objects. Now an assumption of linear relative motion is usually applied in the calculation of collision probability and then the complex 3-dimensional problem can be reduced to a 2-dimensional integral of probability density function over the area of circle. However, if the relative velocity value is very small, the term of linear relative motion is not valid. So it is necessary to consider the calculation of collision probability for nonlinear relative motions. The method used to calculate collision probability for nonlinear relative motion is studied, and test cases are designed to justify the validity of this method. It is applicable to collision probability problems involving relative velocity and error covariance varying with time. The results indicate that it is necessary to calculate collision probability with this nonlinear method under certain circumstances. For example, for elliptical relative motions in Satellite Formation Flying, when the relative velocity is below 100 m/s, the relative error between the linear method and the nonlinear method exceeds 5%; for the problem of conjunction analysis of two satellites with circular orbits, when the relative velocity is below 10 m/s, the relative error is also larger than 1%. Some significant conclusions are obtained for the collision detection system of our country.  相似文献   

18.
In the restricted circular three-body problem, two massive bodies travel on circular orbits about their mutual center of mass and gravitationally perturb the motion of a massless particle. The triangular Lagrange points, L4 and L5, form equilateral triangles with the two massive bodies and lie in their orbital plane. Provided the primary is at least 27 times as massive as the secondary, orbits near L4 and L5 can remain close to these locations indefinitely. More than 2200 cataloged asteroids librate about the L4 and L5 points of the Sun-Jupiter system, and five bodies have been discovered around the L4 point of the Sun-Neptune system. Small satellites have also been found librating about the L4 and L5 points of two of Saturn's moons. However, no objects have been discovered around the Earth-Moon L4 and L5 points. Using numerical integrations, we show that orbits near the Earth-Moon L4 and L5 points can survive for over a billion years even when solar perturbations are included, but the further addition of the far smaller perturbations from other planets destabilize these orbits within several million years. Thus, the lack of observed objects in these regions cannot be used as a constraint on Solar System formation, nor on the tidal evolution of the Moon's orbit.  相似文献   

19.
The present paper deals with the periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are oblate bodies. We have illustrated the periodic orbits for different values of μ, h, σ1 and σ2 (h is energy constant, μ is mass ratio of the two primaries, σ1 and σ2 are oblateness factors). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by (Karimov and Sokolsky, 1989). We have applied the predictor-corrector algorithm to construct the periodic orbits in an attempt to unveil the effect of oblateness of the primaries by taking the fixed values of parameters μ, h, σ1 and σ2.  相似文献   

20.
The steady-state distribution of orbits of Apollo-Amor objects is calculated for a variety of possible sources. These include asteroids near the inner edge of the belt, cometary orbits similar to Encke, and hypothetical extinct cometary orbits with perihelia larger than that of Encke. In all but one case, the steady-state distributions are similar for all these sources, and predict Amor/Apollo ratios of 1.5 to 3. These ratios are lower than those predicted by work in which the effects of the ν6 secular resonance were not considered. These results are in general agreement with observation, although the higher (～3) Amor/Apollo ratios found for many of the sources may turn out to be unacceptably high. The absolute number of Apollo-Amors observed is found to require an injection rate of ～15 objects/(106 years). This rate is easily achieved if the present existence of Encke is assumed to be a reasonably probable event, and if Encke becomes a ～1-km-diameter Apollo object following exhaustion of its volatile material; best estimates of the injection rate from the asteroid belt [～1.5/(106 years)] are too low. Hence a dominant cometary component is suggested. The predicted number of Apollo objects in small (q < 1.0 AU, a < 1.4 AU orbits is in agreement with observation. Predicted lunar and terrestrial cratering rates agree approximately with observation. An unexplained difference between the lunar and terrestrial results is probably caused by uncertainties in the scaling laws or crater counts used. This discrepancy precludes an exact test of these calculations using cratering data.  相似文献   

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