On the bifurcations of a certain family of periodic orbits

In their numerical study of a particular family of periodic orbits in a certain potential, Contopoulos and Zikides (1980; Contopoulos, 1981) have calculated numerically successive values of the energyh at which the family becomes unstable. They showed that the ratio of successive values ofh-h _esc, whereh _esc is the energy of escape, is approximately 9.22. Here we show that the limiting ratio ash→h _esc is exp (π/√2) in this problem, and that the limiting ratio for bifurcations of this type varies from one problem to another.     

A comparison of the thick-target model with stereo data on the height structure of solar hard x-ray bursts

The height structure of a thick-target solar hard X-ray source is predicted for a beam injected vertically downward with a power-law spectrum and dominated by Coulomb collisional energy losses. This structure is characterised by the ratio of hard X-ray flux from an upper part of the source to that from the entire source, and is essentially a function only of the atmospheric column density ΔN (cm^?2) in the upper region. These predictions are compared with the flux ratios at 150 keV and 350 keV observed by two spacecraft for five events in which the solar limb occults part of the source for one spacecraft. In three events the occulting levels h ranged from 0 to 2500 km. For these the theoretical and observed ratios are found to be comparable for values of ΔN in reasonable accord with those found at these altitudes by optical and UV spectroscopic modelling of flare chromospheres. In one event the occultation ratio was observed to rise after the burst peak and it is found that this rise is consistent with an increase in ΔN due to conductively driven chromospheric evaporation. However the energy dependence of the occultation ratio is not consistent with that predicted by the model and it is concluded that non-collisional losses must be significant in beam dynamics. In the other two events, the occultation level h was ? 25 000 km. For these the energy dependence of the occultation ratio is comparable with the model predictions. However the values of ΔN required demand extremely high coronal densities and/or acceleration altitudes. Furthermore, the one observed evolution of the occultation ratio is entirely inconsistent with the model. It is concluded that in these, bremsstrahlung emissions other than that from a beam must be important.     

Anisotropic models of dark halos

An iterative approach is used to construct spherically symmetric equilibrium models with an anisotropic velocity distribution. The potentialities of the method have been tested on models with known distribution functions, the Osipkov-Merritt models. It is shown that models that differ significantly from the Osipkov-Merritt models can be constructed. An N-body model of a dark halo with a density distribution that approximates the results of cosmological simulations (the Navarro-Frenk-White model) has been constructed. The anisotropy profile has been taken to be similar to that yielded by cosmological simulations. The constructed models can serve as direct input data for investigating the dynamics and stability of such systems in N-body simulations.     

Space correction of global models of electron number density in the ionosphere by receiving at one site signals from low-orbit satellites

A technique for updating global models of the electron density N in real time is developed. It employs the ionospheric part of the Doppler shift due to the rate of change of the total electron content determined at one site from measurements of signals from the radio beacons aboard low-orbit satellites. This technique enables corrections of the Chiu model over a region of about 1,000 km in a North-South direction. To study a possibility of predicting N in an East-West direction using the corrected model, we employ the measurements of satellite signals received at three sites spaced up to 2,000 km. For updating global N models, it is determined that spacing between correction sites can be up to about 3,000 to 4,000 km.     

Chaos in N-body systems

Here is a selection of applications of what is now called theory of dynamical systems in galactic dynamics and N-body systems. The study of chaotic motions in potentials used as a model for elliptical galaxies is a first example of these applications. The interest in this problem stems from the fact that there are now many theoretical and observational evidences that the overall potentials of galaxies are indeed non-integrable. There are classes of objects, for example small and intermediate luminosity elliptical galaxies, for which the presence of the famous third integral is not necessary or others in which we observe peculiarities in their photometry or kinematics. We address here some of these issues and their implications in modifying our current understanding of the structure and evolution of galaxies.More in general, there is the natural question of how the systems we see have settled to their present status and what would happen if some external cause perturbs it. This issue is related to the question of the stochasticity involved in the general N-body dynamics, especially when N is very large. An N-body dynamical system is definitely chaotic, as shown by several numerical investigations, at least for N not very large. However, this statement must be reconciled with the picture of non-collisional equilibrium of big systems. The second part of this review presents a survey of numerical experiments and an interpretation of the results obtained using standard chaoticity indicators.     

A Geometric Interpretation of Integrable Motions

Integrability, one of the classic issues in galactic dynamics and in general in celestial mechanics, is here revisited in a Riemannian geometric framework, where Newtonian motions are seen as geodesics of suitable -mechanical- manifolds. The existence of constants of motion that entail integrability is associated with the existence of Killing tensor fields on the mechanical manifolds. Such tensor fields correspond to hidden symmetries of non-Noetherian kind. Explicit expressions for Killing tensor fields are given for the N = 2 Toda model, and for a modified Hénon-Heiles model, recovering the already known analytic expressions of the second conserved quantity besides energy for each model respectively.     

The stability of vertical motion in the N-body circular Sitnikov problem

We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N ? 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z ₀ values, where z ₀ lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N ? 1 primary masses.     

Simultaneous storm-time increases of the ionospheric total electron content and the geomagnetic field in the dusk sector

Using the Faraday rotation technique with the ATS-3 satellite, it has been possible to monitor changes in the total electron content (N_T) of the mid-latitude ionosphere during the first day of 20 geomagnetic storms. Our analysis has shown that during the positive phase (ΔN_T > 0) of ionospheric storms the absolute magnitude of the increase in N_T exhibits a very pronounced maximum near sunset. The mean value of ΔN_T at 17:00 LT is more than five times the average ΔN_T value at local noon. This effect is basically independent of the storm commencement time and is usually associated with substantial local enhancements of the total geomagnetic field. The N_T enhancements are discussed in terms of a contraction and draining of the plasmasphere. A model is presented in which the dawn-dusk electric field responsible for the magnetospheric convection slows down the corotational motion of the plasmaspheric ionization in the dusk sector. This braking action causes a ‘pile up’ of the plasma and the magnetic field along the entire dusk sector.     

Interacting ellipsoids: a minimal model for the dynamics of rubble-pile bodies

A simple mechanical model is formulated to study the dynamics of rubble-pile asteroids, formed by the gravitational re-accumulation of fragments after the collisional breakup of a parent body. In this model, a rubble-pile consists of N interacting fragments represented by rigid ellipsoids, and the equations of motion explicitly incorporate the minimal degrees of freedom necessary to describe the attitude and rotational state of each fragment. In spite of its simplicity, our numerical examples indicate that the overall behavior of our model is in line with several known properties of collisional events, like the energy and angular momentum partition during high velocity impacts. Therefore, it may be considered as a well defined minimal model.     

Lithium abundance in sunspots

The observations of lithium were carried out with the TST-2 telescope at CrAO on August 15–20, 2006. A sunspot model was calculated for the dates of observations. The lithium abundance in a sunspot and in the undisturbed photosphere was determined. It is log(N _Li) = 1.35 for the sunspot and log (N _Li) = 1.05 for the undisturbed photosphere.     

Collision with meteoroids as one of the possible mechanisms for the splitting of cometary nuclei

Results are presented of a statistical analysis of dynamic parameters for 114 comets with split nuclei. A list of the objects includes actually split comets, fragments of cometary pairs, lost comets with designation D, and comets with large-scale atmospheric features. Some aspects of the hypothesis that splitting is caused by collisions of cometary nuclei with meteoroid swarms are investigated. To verify the hypothesis, an analysis is conducted of the positions of split comets’ orbits relative to 58 meteor streams from Cook’s catalogue. The calculations give the number (N) of orbital nodes of split comets relative to the plane of each swarm within a distance of 0.001, 0.005, 0.01, 0.05, and 0.1 AU from each swarm. A special algorithm is proposed for determining the degree of redundancy of N by finding the expected value and dispersion for the number of the nodes. The comparison of N with the expected value, together with the consideration of the dispersion, reveals a redundancy of N in 29 cases. Therefore, collisions of comets with meteoroid swarms can be considered as one of the possible causes of comet splitting. A similar testing is conducted for the asteroid belt and Kuiper belt as potential sources of a vast number of sporadic meteoroids. Based on the results of the calculations, the former may be considered as the most effective region of splitting of periodic comets.     

Mid-latitude electron content modelling: The role of interhemispheric coupling

A modelling study of the electron content of the mid-latitude ionosphere and protonosphere has been carried out for solstice conditions using the mathematical model of Bailey (1983). In the model calculations coupled time-dependent O⁺, H⁺ continuity and momentum equations and O⁺, H⁺ and electron heat balance equations are solved for a magnetic shell extending over both hemispheres. The inclusion of interhemispheric flow of plasma and of heat balance has enabled us to investigate the role of interhemispheric coupling on the electron content and related shape parameters. The computed results are compared with results from slant path observations of the ATS-6 radio beacon made at Lancaster (U.K.) and Boulder, Colorado (U.S.A.).It has been found that the conjugate photoelectron heating has a major effect on the shape of the daily variation of slant slab thickness (τ) and also on the magnitude of the protonospheric content (N_p). Some of the main features of τ are closely related to the sunrise and sunset times in the conjugate ionosphere. Also it is found that night-time increases in total electron content (N_T) and F2 region peak electron density (N_max) in winter are natural consequences of ionization loss at low altitudes causing an enhanced downward flow of plasma from the protonosphere which is coupled to the summer hemisphere. One other important consequence of the coupled protonosphere is that the effects on N_T of the neutral air wind are not much different in winter from those in summer.     

Ignorable coordinates in the ideal resonance problem

If a dynamical system ofN degrees of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form $$F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 , \mu<< 1.$$ Herey is the momentum-vectory _k withk=1, 2,...,N, andx ₁ is thecritical argument. A first-orderglobal solution,x ₁(t) andy ₁(t), for theactive variables of the problem, has been given in Garfinkelet al. (1971). Sincex _k fork>1 are ignorable coordinates, it follows that $$y_\kappa = const., k > 1.$$ The solution is completed here by the construction of the functionsx _k(t) fork>1, derivable from the new HamiltonianF′(y′) and the generatorS(x, y′) of the von Zeipel canonical transformation used in the cited paper. The solution is subject to thenormality condition, derived in a previous paper fork=1, and extended here to 2≤k≤N. It is shown that the condition is satisfied in the problem of the critical inclination provided it is satisfied fork=1.     

On the relation of dynamics to statistical mechanics

A new conceptual framework for the foundations of statistical mechanics starting from dynamics is presented. It is based on the classification and the study of invariants in terms of the concepts of our formulation of non-equilibrium statistical mechanics. A central role is played by thecollision operator. The asymptotic behaviour of a class of states is determined by the collisional invariants independently of the ergodicity of the system. For this class of states we have an approach to thermodynamical equilibrium. We discuss the existence of classes of states which approach equilibrium. The complex microstructure of the phase space, as expressed by the weak stability concept which was introduced by Moser and others, plays here an essential role. The formalism that we develop is meaningful whenever the “dissipativity condition” for the collision operator is satisfied. Assuming the possibility of a weak coupling approximation, this is in fact true whenever Poincaré's theorem on the nonexistence of uniform invariants holds. In this respect, our formalism applies to few body problems and no transition to the thermodynamic limit is required. Our approach leads naturally to a ‘classical theory of measurement’. In particular a precise meaning can now be given to ‘thermodynamic variables’ or to ‘macrovariables’ corresponding to a measurement in classical dynamics.     

Multiple stars and the number of habitable planets in the Galaxy

The Green Bank Equation for the number of communicative civilizations, N, in our Galaxy is reconsidered in view of recent information on the multiplicity of solar-type stars and the possibilities of forming habitable planets around them. Our results are contingent upon the statistical validity of Abt and Levy's work of stellar multiplicity and upon features of Heppen-heimer's theory of planet formation in binary systems. It is concluded that N may have been overestimated by a factor of at least 50.     

Impact processing of nitrogen on early Mars

An intense impact flux upon a planet having a CO₂ + N₂ atmosphere, such as Mars, provides energy to synthesize nitric oxide, NO, which is likely converted into nitrate minerals. The same impact flux can decompose nitrate minerals if present in the crust. We build a numerical model to study the effects of early impact processes on the evolution of nitrogen in a dominantly CO₂ atmosphere. We model the period of intense post-accretionary bombardment, the roughly 500 Myr period after crustal stabilization that locks in previously accreted volatiles. A best-guess, “fiducial” set of parameters is chosen, with a fixed “veneer” of post-accretionary impactors (δR=950 m thick), assumed to contain carbon at 1 wt% (fg=0.01), with a molar C/N ratio of 18, an initial atmospheric pressure of 1 bar (with CO₂/N₂ = 36), and a power law impactor mass distribution slope b=0.75. This model produces a nitrate reservoir RNO₃?0.5×¹⁰¹⁹ moles, equivalent to ∼30 mbars of N₂, during the intense impact phase. Starting with 1 bar, the atmosphere grows to 2.75 bars. Results of models with variations of parameter values show that RNO₃ responds sluggishly to changes in parameter values. To significantly limit the size of this reservoir, one is required to limit the initial total atmospheric pressure be less than about 0.5 bars, and the impactor volatile content fg to be less than 0.003. The value of fg substantially determines whether the atmosphere grows or not; when fg=0.01, the atmosphere gains about 1.7 bars, while for fg=0.003, the atmosphere gains less than 200 mbars, and for fg=0.001, it loses about 400 mbars. Impact erosion is a minor sink of N, constituting generally less than 10% of the total supply. The loss of impactor volatile plumes can take almost 50% of incoming N and C under fiducial parameters, when atmospheric pressures are low. This nitrogen does not significantly interact with Mars, and hence is not properly delivered. When the initial N is greater than the delivered N, most of the nitrogen ends up as nitrates; when delivered N is larger, most nitrogen ends up in the atmosphere. The reason for this dichotomy seems to be that initial nitrogen is present during the whole bombardment, while delivered N, on average, only experiences half the bombardment. The operating caveat here is that the above results are all conditioned on the assumption that impact processes dominate this period of Mars atmospheric evolution.     

Dynamic status of the wide multiple system α Centauri + Proxima

We study the dynamics of a wide multiple system α Centauri + Proxima. The total energy of the system was estimated according to the available observational data on masses, coordinates, proper motions, and radial velocities of its components. To account for the effect of the observational data errors on the result, we have implemented the Monte Carlo method. From N = 10⁶ statistical tests we show that with the probability of about 90% the motion is hyperbolic, i.e., α Cen AB and Proxima will after a while diverge from each other by a considerable distance. We also perform numerical modeling of dynamic evolution of the wide pair α Cen AB + Proxima in the regular field of the Galaxy. The trajectory of relative motion is constructed. The components diverge from each other by a distance of 20 pc over the time scale of about 200 Myr. The critical parameter for determining the dynamic status of the system is the radial velocity of the C component (Proxima), known with an error of 200 ms^?1. For a reliable determination of the nature of motions in the system, we have to decrease the radial velocity error by at least an order of magnitude.     

A momentum-conserving N-body scheme with individual time steps

We present an N-body code called Taichi for galactic dynamics and controlled numerical experiments. The code includes two high-order hierarchical multipole expansion methods: the Barnes-Hut (BH) tree and the fast multipole method (FMM). For the time integration, the code can use either a conventional adaptive KDK or a Hamiltonian splitting integrator. The combination of FMM and the Hamiltonian splitting integrator leads to a momentum-conserving N-body scheme with individual time steps. We find Taichi performs well in the typical applications in galactic dynamics. In the isolated and interacting galaxies tests, the momentum conserving scheme produces the same result as a conventional BH tree code. But for similar force accuracies, FMM significantly speeds up the simulations compared to the monopole BH tree. In the cold collapse test, we find the inner structure after relaxation can be sensitive to the force accuracies. Taichi is ready to incorporate special treatment of close encounters thanks to the Hamiltonian splitting integrator, suitable for studying dynamics around central massive bodies.     

Production of vibrationally excited N2 by electron impact

Energy transfer from electrons to neutral gases and ions is one of the dominant electron cooling processes in the ionosphere, and the role of vibrationally excited N₂ in this is particularly significant. We report here the results from a new calculation of electron energy transfer rates (Q) for vibrational excitation of N₂, as a function of the electron temperature T_e. The present study was motivated by the development of a new cross-section compilation for vibrational excitation processes in N₂ which supercedes those used in the earlier calculations of the electron energy transfer rates. We show that the energy dependence and magnitude of these cross sections, particularly in the region of the well-known resonance in N₂, significantly affect the calculated values of Q. A detailed comparison between the current and previous calculated electron energy transfer rates is made and coefficients are provided so that these rates for transitions from level 0 to levels 1-10 can be calculated for electron temperatures less than .