Bifurcations of triple-periodic orbits

We consider families of periodic orbits in potentials symmetric with respect to thex-axis. The characteristics of triple-periodic orbits (i.e. orbits intersecting thex-axis three times) that bifurcate from the central characteristic do not have their maximum or minimum energy (or perturbation) at the point of intersection. We explain theoretically that this happens only for triple-periodic orbits and not for any other type of resonant periodic orbits and verify this fact by numerical calculations.     

Motions in the field of two rotating magnetic dipoles. III: Zero-velocity curves and surfaces

The zero-velocity curves and surfaces created by two parallel or antiparallel magnetic dipoles moving on concentric circles are computed. It is found that for all combinations of values of the mass parameter µ and the ratio of the magnitudes of the magnetic moments of the two dipoles, there exist closed regions inside of which charged particles are permanently trapped.     

Low-speed impacts between rubble piles modeled as collections of polyhedra

We present results of modeling rubble piles as collections of polyhedra. The use of polyhedra allows more realistic (irregular) shapes and interactions (e.g. collisions), particularly for objects of different sizes. Rotational degrees of freedom are included in the modeling, which may be important components of the motion. We solved the equations of rigid-body dynamics, including frictional/inelastic collisions, for collections of up to several hundred elements. As a demonstration of the methods and to compare with previous work by other researchers, we simulated low-speed collisions between km-scale bodies with the same general parameters as those simulated by Leinhardt et al. [Leinhardt, Z.M., Richardson, D.C., Quinn, T., 2000. Icarus 146, 133-151]. High-speed collisions appropriate to present-day asteroid encounters require additional treatment of shock effects and fragmentation and are the subject of future work; here we study regimes appropriate to planetesimal accretion and re-accretion in the aftermath of catastrophic events. Collisions between equal-mass objects at low speeds () were simulated for both head-on and off-center collisions between rubble piles made of a power-law mass spectrum of sub-elements. Very low-speed head-on collisions produce single objects from the coalescence of the impactors. For slightly higher speeds, extensive disruption occurs, but re-accretion produces a single object with most of the total mass. For increasingly higher speeds, the re-accreted object has smaller mass, finally resulting in complete catastrophic disruption with all sub-elements on escape trajectories and only small amounts of mass in re-accreted bodies. Off-center collisions at moderately low speeds produce two re-accreted objects of approximately equal mass, separating at greater than escape speed. At high speed, complete disruption occurs as with the high-speed head-on collisions. Head-on collisions at low to moderate speeds result in objects of mostly oblate shape, while higher speed collisions produce mostly prolate objects, as do off-center collisions at moderate and high speeds. Collisions carried out with the same dissipative coefficients (coefficient of restitution ?n=0.8, zero friction) as used by Leinhardt et al. [Leinhardt, Z.M., Richardson, D.C., Quinn, T., 2000. Icarus 146, 133-151] result in a value for specific energy for disruption , somewhat lower than the value of 2 J/kg found by them, while collisions with a lower coefficient of restitution and friction [?n=0.5, ?t=0, μ=0.5, similar to those used by Michel, et al. [Michel, P., Benz, W., Richardson, D.C., 2004. Planet. Space Sci. 52, 1109-1117] for SPH + N-body calculations] yield .     

Nature of the Martian uplands: Effect on Martian meteorite age distribution and secondary cratering

Abstract— Martian meteorites (MMs) have been launched from an estimated 5–9 sites on Mars within the last 20 Myr. Some 80–89% of these launch sites sampled igneous rock formations from only the last 29% of Martian time. We hypothesize that this imbalance arises not merely from poor statistics, but because the launch processes are dominated by two main phenomena: first, much of the older Martian surface is inefficient in launching rocks during impacts, and second, the volumetrically enormous reservoir of original cumulate crust enhances launch probability for 4.5 Gyr old rocks. There are four lines of evidence for the first point, not all of equal strength. First, impact theory implies that MM launch is favored by surface exposures of near‐surface coherent rock (≤10² m deep), whereas Noachian surfaces generally should have ≥10² m of loose or weakly cemented regolith with high ice content, reducing efficiency of rock launch. Second, similarly, both Mars Exploration Rovers found sedimentary strata, 1–2 orders of magnitude weaker than Martian igneous rocks, favoring low launch efficiency among some fluvial‐derived Hesperian and Noachian rocks. Even if launched, such rocks may be unrecognized as meteorites on Earth. Third, statistics of MM formation age versus cosmic‐ray exposure (CRE) age weakly suggest that older surfaces may need larger, deeper craters to launch rocks. Fourth, in direct confirmation, one of us (N. G. B.) has found that older surfaces need larger craters to produce secondary impact crater fields (cf. Barlow and Block 2004). In a survey of 200 craters, the smallest Noachian, Hesperian, and Amazonian craters with prominent fields of secondaries have diameters of ?45 km, ?19 km, and ?10 km, respectively. Because 40% of Mars is Noachian, and 74% is either Noachian or Hesperian, the subsurface geologic characteristics of the older areas probably affect statistics of recognized MMs and production rates of secondary crater populations, and the MM and secondary crater statistics may give us clues to those properties.     

The relationship between the amplitudinal characteristics of the high-frequency spectral components of wind-generated waves and the wind velocity over the sea

The relationship between the RMS amplitudes of the wind wave spectral components and the wind speed has been studied at ten frequencies in the band of 0.65–23 Hz. To measure the parameters of the high-frequenci waves, a resistance elevation wave gauge was operated, which was deployed in the Black See on an oceanographic platform near Katsively. The correlation between the wave amplitudes and the wind velocity at high frequencies of 5–23 Hz, corresponding to gravitation-capillary ripples, was found to reach a value of 0.8. At lower frequencies of 0.65–4.3 Hz, corresponding to short gravity waves, it dropped to 0.5–0.7. The response of spectral components to the wind speed variations in the gravity-capillary range is higher than in the range of short gravity waves. The results obtained differ from Phillips' idea about a saturated range for the frequency form of the spectrum of high-frequency gravity waves, since a linear dependence of the spectral amplitudes on the wind speed is established at a wind of force 1–8.Translated by Mikhail M. Trufanov.     

Nonlinear inertial oscillations of a multilayer eddy: An analytical solution

Nonlinear axisymmetric oscillations of a warm baroclinic eddy are considered within the framework of an reduced-gravity model of the dynamics of a multilayer ocean. A class of exact analytical solutions describing pure inertial oscillations of an eddy formation is found. The thicknesses of layers in the eddy vary according to a quadratic law, and the horizontal projections of the velocity in the layers depend linearly on the radial coordinate. Owing to a complicated structure of the eddy, weak limitations on the vertical distribution of density, and an explicit form of the solution, the latter can be treated as a generalization of the exact analytical solutions of this form that were previously obtained for homogeneous and baroclinic eddies in the ocean.     

Differential rotation of the solar electron corona

Autocorrelation analyses of K-coronameter observations made at Haleakala and Mauna Loa, Hawaii, during 1964–1967 have established average yearly rotation rates of coronal features as a function of latitude and height above the limb. At low latitudes the corona was found to rotate at the same rate as sunspots but at higher latitudes was consistently faster than the underlying photosphere. There were differences as large as 3–4% in the rate at specific latitudes from year to year and between the two hemispheres. In 1967 a nearly constant rotation was found for heights ranging from 1.125 to 2.0 R ₀. For 1966 there was a more complicated pattern of height dependence, with the rate generally decreasing with height at low latitudes and increasing at high latitudes.At Hawaii Institute of Geophysics.     

A morphological classification of e and SO galaxies in terms of equilibrium isopleths

A theory of galactic morphology results from the consideration of isoplethic surfaces in states of geometric equilibrium within the theory of general relativity. In the linearized case, one solves ?²ψ+ρ²ψ=0 on the surface of an oblate spheroid with unit semimajor axis and eccentricity ε. The surface defined by $$x^2 + y^2 + z^2 /(1 - \varepsilon ^2 ) = {\text{exp (}}\psi {\text{)}}$$ are then equilibrium isopleths. The resulting morphological system contains two continuous parameters (ε and the norm of ψ) and an integer-valued parametern. The system includes the ellipticals as special cases, and the various available profiles are given for a variety of values of the parameters. The forms are sufficiently varied as to represent NGC 3115, 128, 7332, and IC 3973. These forms cease to be enigmatic for they fit into an orderly sequence progressing out of the ellipticals and have equilibrium interpretations as previously obtained only for the ellipticals.     

About the first integrals of the generalized problem of translatory-rotary motion of rigid bodies

The existence of ten first integrals for the classical problem of the motion of a system of material points, mutually attracting according to Newtonian law, is well known.The existence of the analogous ten first integrals for the more complicated problem of the motion of a system of absolutely rigid bodies, whose elementary particles mutually attract according to the Newtonian law, was established by the author (Duboshin, 1958, 1963, 1968).In his later papers (Duboshin, 1969, 1970), the problem of the motion of a system of material points, attracting each other according to a more general law, was considered and, in particular, it was shown under what conditions the ten first integrals, analogous to the classical integrals, may exist for this problem.In the present paper, the generalized problem of translatory-rotatory motion of rigid bodies, whose elementary particles acting upon each other according to arbitrary laws of forces along the straight line joining them, is discussed.The author has shown that the first integrals for this general problem, analogous to the integrals of the problem of the translatory-rotatory motion of rigid bodies, whose elementary particles acting according to the Newtonian law, exist under certain well known conditions.That is, it has been established that if the third axiom of dynamics (action = reaction) is satisfied, then the integrals of the motion of centre of inertia and the integrals of the moment of momentum exist for this generalized problem.If the third axiom is not satisfied, then the above mentioned integrals do not exist.The third axiom is a necessary but not a sufficient condition for the existence of the tenth integral-the energy integral. The tenth integral always exists if the elementary particles of the bodies acting with a force, depend only on the mutual distances between them. In this case the force function exists for the problem and the energy integral can be expressed in a well known form.The tenth integral may exist for some more general case, without expressing the principle of conservation of energy, but permitting calculation of the kinetic energy, if the configuration of a system is given.The problem, in which the elementary particles acting according to the generalized Veber's law (Tisserand, 1896) has been cited as an example of this more general case.