On the asteroidal conductivities as inferred from meteorites

Solar wind interactions with planetary bodies without intrinsic magnetic fields depend to a large extent on the electrical conductivities of the objects in question. If the combined (i.e., ionospheric and interior) electrical conductivities are large, as in the case of Venus, the solar wind interaction is strong due to the generation of a large electrical current flow. It is suggested here that a similar interaction may occur at some asteroids, if their interior conductivity can be approximated by the conductivities of carbonaceous or iron-bearing meteorites. This interaction, in turn, can be used as a tool for remote sensing of the asteroidal interior properties in a spacecraft mission to asteroids.     

Fragmentation model dependence of collision cascades

Mass depletion of bodies through successive collisional disruptions (i.e., collision cascade) is one of the most important processes in the studies of the asteroids belt, the Edgeworth-Kuiper belt, debris disks, and planetary formation. The collisional disruption is divided into two types, i.e., catastrophic disruption and cratering. Although some studies of the collision cascades neglected the effect of cratering, it is unclear which type of disruption makes a dominant contribution to the collision cascades. In the present study, we construct a simple outcome model describing both catastrophic disruption and cratering, which has some parameters characterizing the total ejecta mass, the mass of the largest fragment, and the power-law exponent of the size distribution of fragments. Using this simple outcome model with parameters, we examine the model dependence of the mass depletion time in collision cascades for neglect of coalescence of colliding bodies due to high collisional velocities. We find the cratering collisions are much more effective in collision cascades than collisions with catastrophic disruption in a wide region of the model parameters. It is also found that the mass depletion time in collision cascades is mainly governed by the total ejecta mass and almost insensitive to the mass of the largest fragment and the power-law exponent of fragments for a realistic parameter region. The total ejecta mass is usually determined by the ratio of the impact energy divided by the target mass (i.e. Q-value) to its threshold value for catastrophic disruption, as well as in our simple model. We derive a mass depletion time in collision cascades, which is determined by of the high-mass end of collision cascades. The mass depletion time derived with our model would be applicable to debris disks and planetary formation.     

On the orbital isochronism

Formulas for the time of orbital flight in terms of the kinematic parameters are developed, and a kinematic theorem of orbital isochronism is introduced. Through this new theorem, the connection between the celebrated Lambart's theorem and the rather obscure theorem of Hamilton's hodographic isochronism is demonstrated, and the significance and implications of the latter are explored. In the light of Hamilton's hodographic representation, the characteristic features of an isochronal and isoenergetic family of Keplerian trajectories are observed, and a simple geometric method for the construction of such a family is proposed. Finally, Hamilton's time integral is briefly treated, resulting in another set of useful formulas for the time of flight.Presented at the 8th Annual Seminar on Current Problems in Celestial Mechanics, University of Texas, Austin, Texas, January 1970. This work was done partly at the NASA-ERC, Cambridge, Mass., and partly at the Department of Mathematics, MIT under the Senior Research Associateship, NAS-NRC.     

Chaotic behaviour of trajectories for the asteroidal resonances

A systematic study of the main asteroidal resonances of the third and fourth order is performed using mapping techniques. For each resonance one-parameter family of surfaces of section is presented together with a simple energy graph which helps to understand and predict the changes in the surfaces of section within the family. As the truncated Hamiltonian for the planar, elliptic, restricted three-body problem is used for the mapping, the method is expected to fail for high eccentricities. We compared, therefore, the surfaces of section with trajectories calculated by symplectic integrators of the fourth and six order employing the full Hamiltonian. We found a good agreement for small eccentricities but differences for the higher eccentricities (e 0.3).     

A note on the cosmogony of the asteroidal belt

Several arguments based on the orbital dynamics of the asteroids are used to support the idea that the original solar nebula might not necessarily be the Laplacian type of thin disk-like structure. The orbital distribution of the large asteroids is suggested to be a direct result of their accretions of condensed grains in eccentric orbits via the jet stream process.     

A former asteroidal planet as the origin of comets

The idea of a missing planet between Mars and Jupiter has been with us since the formulation of the Titius-Bode law. The discovery of the asteroid belt in that location led to speculation about a planetary breakup event. Both ideas remained conjectures until Ovenden's finding in 1972, from which it could be derived that the mass of the missing planet was about 90 Earth masses and that its breakup was astronomically recent. Apparently much of that mass was blown out of the solar system during the disruption of the planet. Because of the action of planetary perturbations, only two types of orbits of surviving fragments could remain at present-asteroid orbits and once-around very-long-period elliptical orbits. Objects in the latter type of orbit are known to exist-the very-long-period comets. A large number of these are on elliptical trajectories with periods of revolution of 5 million years; yet they are known to have made no more than one revolution in an orbit passing close to the Sun. By direct calculation it is possible to predict the distribution of the orbital elements of objects moving on long-period ellipses which might have originated in a breakup event in the asteroid belt 5 million years ago. The comet orbits have the predicted distribution in every case where a measure is possible. Some of the distribution anomalies, such as a bias in the directions of perihelion passage, are statistically strong and would be difficult to explain in any other uncontrived way. In addition, a relative deficiency of orbits with perihelia less than 1 AU indicates that the comets must have had small perihelion distances since their origin, rather than that they have been perturbed into small perihelion orbits from a distant “cloud” of comets by means of stellar encounters. The comet orbital data lead to the conclusion that all comets originated in a breakup event in the asteroid belt (5.5±0.6) × 10⁶ years ago. Asteroid and meteoritic evidence can now be interpreted in a way which not only is supportive but also provides fresh insights into understanding their physical, chemical, and dynamical properties. Particularily noteworthy are the young cosmic-ray exposure ages of meteorites, evidence of a previous high-temperature/pressure environment and of chemical differentiation of the parent body, and compositional similarities among comets, asteroids, and meteorites. Certain “explosion signatures” in asteroid orbital element distributions are likewise indicative. Tektites may also have originated in the same event; but if so, there are important implications regarding the absolute accuracy of certain geological dating methods. Little is known about possible planetary breakup mechanisms of the requisite type, though some speculations are offered. In any case, the asteroid belt is an existing fact; and the arguments presented here that a large planet did disintegrate 5 million years ago must be judged on their merits, even in the absence of a suitable theory of planetary explosions.     

On the collision probability for comets with the Sun

We consider the collision probability for comets with the Sun under the suppositions of different velocity distributions and various initial conditions. We solve the problem applying Laplace's method and using Schiaparelli's hyperboloid of visibility. The probabilities obtained in this manner are given separately for elliptic and hyperbolic orbits.     

On the explosive prevention of collisions of asteroidal and cometary bodies with the earth at their late detection

We consider the questions of an explosive impact on asteroids and comets that approach the Earth in the case of a late forecast of the dangerous situation. Based on models for the destruction of the material of a celestial body in the shock wave produced by a strong self-buried explosion, we estimate the radius of the destroyed region, the ejected mass, and the recoil momentum. We determine the charges needed to completely destroy bodies of various sizes and compositions or to divert bodies from the Earth by the required distance. When comets are dangerous bodies, we compare the efficiencies of the explosive and sublimation methods of changing their orbits. We discuss how to increase the efficiency of the explosive impact on a dangerous body through the use of a high relative velocity of the encounter between this body and a charge-carrying rocket.     

Evidence for the existence of groups of meteorite-producing asteroidal fragments

Abstract— The MORP camera network in western Canada observed 56 events which we associate with meteorites larger than 0.1 kg. An additional 33 Prairie Network (central USA) fireballs with published orbits were previously identified as the sources of meteorites of at least 0.25 kg. A comparison of the MORP orbits with each other and with the PN orbits, using the D′ criterion of orbital similarity, exhibits a surprising number of small values. This suggests there are groups of related objects among the 89 events. We evaluate the probability of small values of D′ arising by chance from a group of random orbits that has the distribution of orbital elements expected for meteorites. There is an excess of small values of D′ among the 89 meteoritic objects over the expectation for random orbits and a marked excess of very small values. Four groups comprising a total of 16 objects account for this excess. These groups exhibit a preference for the larger masses of the population and a very strong concentration of perihelia just slightly inside the Earth's orbit. Although it has been shown by others that gravitational perturbations will disperse Earth-crossing streams in times that are much less than cosmic-ray exposure ages, the properties of the four groups suggest they may be streams of fragments that crossed the Earth's orbit only recently. Such streams may include a considerable fraction of meteorites falling at a given time. Orbital evolution of these streams could alter the sample of meteorites arriving on Earth over time intervals that are less than the accumulation time of the Antarctic collections.     

On the orbital eccentricity of V477 Cygni

Recent photoelectric times of minima support the valuee=0.3 obtained by O'Connell (1970). The lower value obtained by Budding (1974) is ruled out.     

Application of the theory of jet stream to the asteroidal belt

The possibility of incorporating the resonant effect and jet stream formation process into the problems of the Hilda asteroids and Kirkwood gaps is discussed qualitatively. It appears that formation of the precursor jet streams of the resonant asteroids in the main belt would be suppressed due to the collisional perturbation effect of the ambient matter in this region. Together with the biased distribution of near-resonant asteroids, the depletion across the Kirkwood gaps could be understood. Within the context of jet stream theory the existence of Hilda asteroids outside the main belt requires the original limit of the main belt to be not much more extensive than the present value of 3.5 AU. This is suggestive of a cosmogonic origin of the observed outer limit.     

On the transfer of radiation at asteroidal surfaces in relation to their orbit deflection — II

The efficiency of absorption of X-rays generated by a nuclear explosion at the surface of an asteroid, estimated earlier, is used to calculate the explosion yield needed to deflect the orbit of an asteroid. Following the work of Ahrens &38; Harris, it is shown that a recoil velocity of 1 cm s⁻¹ is required to deflect an asteroid from a collision course with the Earth, and the necessary yield of explosion energy is estimated. If it is assumed that the scaling law between the energy and the diameter of the resulting crater, obtained from experiments carried out on the Earth, remains valid on the asteroid surface, where gravity is much weaker, an explosion energy of 8 and 800 megaton (Mton) equivalent of TNT would be required for asteroids of diameter 1 and 10 km respectively. If, on the other hand, the crater diameter is proportional to a certain power of the gravity g , the power being determined from a dimension analysis, 130 kton and 12 Mton would be required to endow asteroids of diameters 1 and 10 km with the required velocity, respectively. The result indicates that in order to estimate the required explosion energy, a better understanding of cratering under gravity much weaker than on the Earth would be required.     

On the transfer of radiation at asteroidal surfaces in relation to their orbit deflection – I

One of the methods discussed in deflecting the orbit of an Earth-colliding asteroid is the use of nuclear explosives. In assessing its feasibility, apart from political considerations, it is important to quantify how effective it is in orbit deflection. The transfer of radiation incident at the surface is governed by a non-linear diffusion equation. For low-yield explosions with a slab geometry ( S ₀≃10⁸ kJ μs⁻¹), the temperature at depth x and time t is well approximated by a similarity solution of the form T ( x , t )= T ₀ f (ξ), ξ= x/ ( T ⁿ ₀ t )^1/2, with T ₀ given by ( S ₀/σ)^1/4, where σ is the Stefan–Boltzmann constant, n is an index that specifies the radiation transfer and f (ξ) is the solution of a non-linear differential equation subject to the condition f (0)=1 and lim_ξ→∞ f (ξ)=0. For high-yield explosions ( S ₀≃10¹⁰ kJ μs⁻¹), numerical solutions to the non-linear diffusion equation can be obtained. These solutions have properties similar to the case of low-yield explosions. If the duration of the explosion is d ×10⁻⁸ s, where d is close to 3, the fraction of energy absorbed by the surface is found to be 7, 12 and 23 per cent for S ₀=10⁸, 10⁹ and 10¹⁰ kJ μs⁻¹ respectively.     

On the asteroid belt's orbital and size distribution

For absolute magnitudes greater than the current completeness limit of H-magnitude ∼15 the main asteroid belt's size distribution is imperfectly known. We have acquired good-quality orbital and absolute H-magnitude determinations for a sample of small main-belt asteroids in order to study the orbital and size distribution beyond H=15, down to sub-kilometer sizes (H>18). Based on six observing nights over a 11-night baseline we have detected, measured photometry for, and linked observations of 1087 asteroids which have one-week time baselines or more. The linkages allow the computation of full heliocentric orbits (as opposed to statistical distances determined by some past surveys). Judged by known asteroids in the field the typical uncertainty in the (a/e/i) orbital elements is less than 0.03 AU/0.03/0.5°. The distances to the objects are sufficiently well known that photometric uncertainties (of 0.3 magnitudes or better) dominate the error budget of their derived H-magnitudes. The detected asteroids range from HR=12-22 and provide a set of objects down to sizes below 1 km in diameter. We find an on-sky surface density of 210 asteroids per square degree in the ecliptic with opposition magnitudes brighter than mR=23, with the cumulative number of asteroids increasing by a factor of 10^0.27/mag from mR=18 down to the mR?23.5 limit of our survey. In terms of absolute H magnitudes, we find that beyond H=15 the belt exhibits a constant power-law slope with the number increasing proportional to ¹⁰0.30H from H?15 to 18, after which incompleteness begins in the survey. Examining only the subset of detections inside 2.5 AU, we find weak evidence for a mildly shallower slope for H=15-19.5. We provide the information necessary such that anyone wishing to model the main asteroid belt can compare a detailed model to our detected sample.     

On the Milankovitch orbital elements for perturbed Keplerian motion

We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace–Runge–Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution.     

The influence of the brightness of the asteroidal dust bands on the gegenschein

The position and shape of the Gegenschein’s maximum brightness provide information on the structure of the interplanetary dust cloud. We show that the asteroidal dust bands, extended near the anti-solar point, play an important role in determining both the position of the maximum brightness and the shape of the Gegenschein. After removing the asteroidal dust bands from an observation of the Gegenschein on November 2, 1997, it was found that the maximum brightness point shifted −0.4° in ecliptic latitude, i.e., to the south of the ecliptic plane, at an ecliptic longitude of 180°, in contrast to a latitude value of +0.1° when the dust bands were included. Furthermore, the part of the Gegenschein to the south of the ecliptic plane was brighter than the northern part at the time of observation. Referring to the cloud model of T. Kelsall et al. (1998, Astrophy. J. 508, 44-73), it can be estimated that the ascending node of the symmetry plane of the dust cloud is 57°^−3°_+7° when its inclination is 2.03° ? 0.50°.     

On the orbital eccentricity on an eclipsing binary system

A new method is presented to precisely deduce the orbital eccentricity of an eclipsing binary from observed epochs of its light minima. Application to the system V526 Sagittarii givese=0.2220±0.0016.