Individual Orbits Research Articles

Far‐ultraviolet reflectance properties of the Moon's permanently shadowed regions

Although of great interest for science and resource utilization, the Moon's permanently shadowed regions (PSRs) near each pole present difficult targets for remote sensing. The Lyman Alpha Mapping Project (LAMP) instrument on the Lunar Reconnaissance Orbiter (LRO) mission is able to map PSRs at far‐ultraviolet (FUV) wavelengths using two faint sources of illumination from the night sky: the all‐sky Ly α glow produced as interplanetary medium (IPM) H atoms scatter the Sun's Ly α emissions, and the much fainter source from UV‐bright stars. The reflected light from these two sources produces only a few hundred events per second in the photon‐counting LAMP instrument, so building maps with useful signal‐to‐noise (SNR) ratios requires the careful accumulation of the observations from thousands of individual LRO orbits. In this paper we present the first FUV albedo maps obtained by LAMP of the Moon's southern and northern polar regions. The results show that (1) most PSR regions are darker at all FUV wavelengths, consistent with their surface soils having much larger porosities than non‐PSR regions (e.g., ∼70% compared to ∼40% or so), and (2) most PSRs are somewhat “redder” (i.e., more reflective at the longer FUV wavelengths) than non‐PSR regions, consistent with the presence of ∼1–2% water frost at the surface.

Secular theory of the orbital evolution of the young stellar disc in the Galactic Centre

We investigate the orbital evolution of a system of N mutually interacting stars on initially circular orbits around the dominating central mass. We include perturbative influence of a distant axisymmetric source and an extended spherical potential. In particular, we focus on the case when the secular evolution of orbital eccentricities is suppressed by the spherical perturbation. By means of standard perturbation methods, we derive semi-analytic formulae for the evolution of normal vectors of the individual orbits. We find its two qualitatively different modes. Either the orbits interact strongly and, under such circumstances, they become dynamically coupled, precessing synchronously in the potential of the axisymmetric perturbation. Or, if their mutual interaction is weaker, the orbits precess independently, interchanging periodically their angular momentum, which leads to oscillations of inclinations. We argue that these processes may have been fundamental for the evolution of the disc of young stars orbiting the supermassive black hole in the centre of the Milky Way.

A halo expansion technique for approximating simulated dark matter haloes

We apply a basis function expansion method to create a time-evolving density/potential approximation of the late growth of simulated N-body dark matter haloes. We demonstrate how the potential of a halo from the Aquarius Project can be accurately represented by a small number of basis functions, and show that the halo expansion (HEX) method provides a way to replay simulations. We explore the level of accuracy of the technique as well as some of its limitations. We find that the number of terms included in the expansion must be large enough to resolve the large-scale distribution and shape of the halo but, beyond this, additional terms result in little further improvement. Particle and subhalo orbits can be integrated in this realistic, time-varying halo potential approximation, at much lower cost than the original simulation, with high fidelity for many individual orbits, and a good match to the distributions of orbital energy and angular momentum. Statistically, the evolution of structural subhalo properties, such as mass, half-mass radius and characteristic circular velocity, are very well reproduced in the halo expansion approximation over several gigayears. We demonstrate an application of the technique by following the evolution of an orbiting subhalo at much higher resolution than can be achieved in the original simulation. Our method represents a significant improvement over commonly used techniques based on static analytical descriptions of the halo potential.

The long-term dynamics of the Jovian irregular satellites

Context. The dynamical region of the Jovian irregular satellites presents an interesting web of resonances that are not yet fully understood. Of particular interest is the influence of the resonances on the stochasticity of the individual orbits of the satellites, as well as on the long-term chaotic diffusion of the different families of satellites. Aims. We make a systematic numerical study of the satellite region to determine the important resonances for the dynamics, to search for the chaotic zones, and to determine their influences on the dynamics of the satellites. We also compare these numerical results to previous analytical works. Methods. Using extensive numerical integrations of the satellites along with an indicator of chaos (MEGNO), we show global and detailed views of the retrograde and prograde regions for various dynamical models of increasing complexity and indicate the appearance of the different types of resonances and the implied chaos. Results. Along with secular and mean motion resonances that shape the dynamical regions of the satellites, we report a number of resonances involving the Great Inequality, and which are present in the system thanks to the wide range of the values of frequencies of the pericenter available for the satellites. The chaotic diffusion of the satellites is also studied and shows the long-term stability of the Ananke and Carme families, in contrast to the Pasiphae family.

NUCLEAR-STRUCTURE EFFECTS ON DOUBLE BETA DECAYS TO 0+ STATES IN 76Ge

Neutrinoless double beta (0νββ) decay of 76 Ge to the ground state and first excited 0+ state in 76 Se is discussed in terms of the associated nuclear matrix elements. The effects arizing from the size of the single-particle model space and the occupancies of the individual orbits are discussed in the framework of the (higher) quasiparticle random-phase approximation with effective, G -matrix-derived nuclear forces. It is found that the orbital occupancies play a role for the size of the nuclear matrix element. Contrary to the ground-state transition the transition to the first excited 0+ state does not depend sensitively on the size of the model space.

Effects of orbital occupancies and spin–orbit partners II: [formula omitted] decays of 76Ge, 82Se and 136Xe to first excited [formula omitted] states

The neutrinoless double beta ( 0 ν β β ) decays of 76Ge, 82Se and 136Xe to the first excited 0 + states ( 0 1 + ) in 76Se, 82Kr and 136Ba are studied in calculations using different sizes of the single-particle model spaces and different occupancies of the individual orbits. This is a continuation of a similar study [J. Suhonen, O. Civitarese, Effects of orbital occupancies and spin–orbit partners on 0 ν β β -decay rates, Nucl. Phys. A 847 (2010) 207–232] of the 0 ν β β decays of 76Ge, 82Se, 128Te, 130Te and 136Xe to the ground states of 76Se, 82Kr, 128Xe, 130Xe and 136Ba. The nuclear matrix elements and half-lives of the decays to the 0 1 + states are calculated by the use of the multiple-commutator model with effective, G-matrix-derived nuclear forces. Interestingly, the basic features of the ground-state and excited-state decays are found to be quite different. The half-lives of the excited-state transitions are suppressed by at least one order of magnitude relative to the ground-state transitions.

Next-order spin–orbit contributions to chaos in compact binaries

This paper is mainly devoted to numerically investigating the effects of the next-order spin–orbit interactions including the 2.5 post-Newtonian order term of the equations of motion and the second post-Newtonian order terms of the spin precession equations on chaos in the conservative Lagrangian dynamics of a spinning compact binary system. It is shown sufficiently through individual orbit simulations, the dependence of the invariant fast Lyapunov indicators on the variations of initial spin angles and the phase space scans for chaos, that the next-order spin–orbit contributions do play an important role in the amplification of chaos.

Development of a 2009 Stable Lights Product using DMSP-OLS data

Since 1994, NGDC has had an active program focused on global mapping of nighttime lights using the data collected by the Defense Meteorological Satellite Program’s Operational Linescan System (DMSP-OLS) sensors. The basic product is a global annual cloud-free composite, which averages the OLS visible band data for one satellite from the cloud-free segments of individual orbits. Over the years, NGDC has developed automatic algorithms for screening the quality of the nighttime visible band observations to remove areas contaminated by sunlight, moonlight, and the presence of clouds. In the Stable Lights product generation, fires and other ephemeral lights are removed based on their high brightness and short duration. Background noise is removed by setting thresholds based on visible band values found in areas known to be free of detectable lights. In 2010, NGDC released the version 4 time series of Stable Lights, spanning the years 1992-2009. These are available online at .

Stability of cloud orbits in the broad-line region of active galactic nuclei

We investigate the global dynamic stability of spherical clouds in the Broad Line Region (BLR) of Active Galactic Nuclei (AGN), exposed to radial radiation pressure, gravity of the central black hole (BH), and centrifugal forces assuming the clouds adapt their size according to the local pressure. We consider both, isotropic and anisotropic light sources. In both cases, stable orbits exist also for very sub-Keplerian rotation for which the radiation pressure contributes substantially to the force budget. We demonstrate that highly eccentric, very sub-Keplerian stable orbits may be found. This gives further support for the model of Marconi et al. 2008, who pointed out that black hole masses might be significantly underestimated if radiation pressure is neglected. That model improved the agreement between black hole masses derived in certain active galaxies based on BLR dynamics, and black hole masses derived by other means in other galaxies by inclusion of a luminosity dependent term. For anisotropic illumination, energy is conserved for averages over long time intervals, only, but not for individual orbits. This leads to Rosetta orbits that are systematically less extended in the direction of maximum radiation force. Initially isotropic relatively low column density systems would therefore turn into a disk when an anisotropic AGN is switched on.

Chaos, Artificial Life and Risk Mathematics

The current work addresses the importance of chaos theory and artificial life as scientific bases for the development of a mathematical research of risk in the systems, within the framework of risk science and risk mathematics. It is argued, in particular, that a combination of chaos theory and artificial life may provide for a theoretical basis for the understanding of systemic situations of risk. It is presented, within the context of category computation theory, a notion of morphic incompressibility and it is addressed how statistical laws emerge in chaotic dynamics, which becomes morphically compressible at the statistical level, while being morphically incompressible at the individual orbits' level. The issue of the systemic origins of chaos is addressed in connection with this dual (orbit) incompressibility/(statistical) compressibility. Through the introduction of a one-dimensional game of life, that combines a coupled map lattice and a cellular automaton, in a model of an artificial ecosystem, it is, then, shown that artificial life can be effectively combined with chaos theory, in order to support the development of a mathematical theory of risk.

Orbit determination for geostationary satellites with the combination of transfer ranging and pseudorange data

Geostationary satellites (GEOs) play a significant role in the regional satellite navigation system. Simulation experiments show that the clock corrections could be mitigated through a single strategy or double differencing strategies for a navigation constellation, but for the mode of individual GEO orbit determination, high precision orbit and clock correction could not be obtained in the orbit determination based on the pseudorange data. A new GEO combined precise orbit determination (POD) strategy is studied in this paper, which combines pseudorange data and C-band transfer ranging data. This strategy overcomes the deficiency of C-band transfer ranging caused by limited stations and tracking time available. With the combination of transfer ranging and pseudorange data, clock corrections between the GEO and the stations can be estimated simultaneously along with orbital parameters, maintaining self-consistency between the satellite ephemeris and clock correction parameters. The error covariance analysis is conducted to illuminate the contributions from the transfer ranging data and the psudoranging data. Using data collected for a Chinese GEO satellite with 3 C-band transfer ranging stations and 4 L-band pseudorange tracking stations, POD experiments indicate that a meter-level accuracy is achievable. The root-mean-square (RMS) of the post-fit C-band ranging data is about 0.203 m, and the RMS of the post-fit pseudorange is 0.408 m. Radial component errors of the POD experiments are independently evaluated with the satellite laser ranging (SLR) data from a station in Beijing, with the residual RMS of 0.076 m. The SLR evaluation also suggests that for 2-h orbital predication, the predicted radial error is about 0.404 m, and the clock correction error is about 1.38 ns. Even for the combination of one C-band transfer ranging station and 4 pseudorange stations, POD is able to achieve a reasonable accuracy with the radial error of 0.280 m and the 2-h predicted radial error of 0.888 m. Clock synchronization between the GEO and tracking stations is achieved with an estimated accuracy of about 1.55 ns, meeting the navigation service requirements.

Effects of orbital occupancies and spin–orbit partners on [formula omitted]-decay rates

Neutrinoless double beta ( 0 ν β β ) decay of the 76Ge, 82Se, 128Te, 130Te and 136Xe nuclei is discussed in terms of the associated nuclear matrix element (NME). The effects of the size of the single-particle model space and the occupancies of individual orbits on the NME are discussed by using the proton–neutron quasiparticle random-phase approximation (pnQRPA) with effective, G-matrix-derived nuclear forces. It is found that only in some cases the orbital occupancies play a decisive role for the size of the NME, whereas the inclusion of all the spin–orbit partners is essential to achieve a NME of reasonable quality. The obtained results are compared with published results of other models.

The interplay of chaos between the terrestrial and giant planets

We report on some simple experiments on the nature of chaos in our planetary system. We make the following interesting observations. First, we look at the system of Sun + four Jovian planets as an isolated five-body system interacting only via Newtonian gravity. We find that if we measure the Lyapunov time of this system across thousands of initial conditions all within observational uncertainty, then the value of the Lyapunov time seems relatively smooth across some regions of initial condition space, while in other regions it fluctuates wildly on scales as small as we can reliably measure using numerical methods. This probably indicates a fractal structure of Lyapunov exponents measured across initial condition space. Then, we add the four inner terrestrial planets and several post-Newtonian corrections such as general relativity into the model. In this more realistic Sun + eight-planet system, we find that the above structure of chaos for the outer planets becomes uniformly chaotic for almost all planets and almost all initial conditions, with a Lyapunov time-scale of about 5–20 Myr. This seems to indicate that the addition of the inner planets adds more chaos to the system. Finally, we show that if we instead remove the outer planets and look at the isolated five-body system of the Sun + four terrestrial planets, then the terrestrial planets alone show no evidence of chaos at all, over a large range of initial conditions inside the observational error volume. We thus conclude that the uniformity of chaos in the outer planets comes not from the inner planets themselves, but from the interplay between the outer and inner ones. Interestingly, however, there exist rare and isolated initial conditions for which one individual outer planetary orbit may appear integrable over a 200-Myr time-scale, while all the other planets simultaneously appear chaotic.

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus' equilateral equilibrium points is investigated with respect to the inclination of the asteroid's orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L4 and L5, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 billion years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 < a < 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids' orbit was varied in the range 0 < i < 60 and the semimajor axes were fixed. It turned out that only four 'windows' of stable orbits survive: these are the orbits for the initial inclinations 0 < i < 7, 9 < i < 13, 31 < i < 36 and 38 < i < 50. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.

The orbital evolution induced by baryonic condensation in triaxial haloes

Using spectral methods, we analyse the orbital structure of prolate/triaxial dark matter (DM) halos in N-body simulations to understand the processes that drive the evolution of shapes of DM halos and elliptical galaxies in which central masses are grown. A longstanding issue is whether the change in the shapes of DM halos is the result of chaotic scattering of box orbits, or whether they change shape adiabatically in response to the evolving galactic potential. We use orbital frequencies to classify orbits, to quantify orbital shapes, and to identify resonant orbits and chaotic orbits. The frequency-based method overcomes the limitations of Lyapunov exponents which are sensitive to numerical discreteness effects. Regardless of the distribution of the baryonic component, the shape of a DM halo changes primarily due to changes in the shapes of individual orbits within a given family. Orbits with small pericentric radii are more likely to change both their orbital type and shape than orbits with large pericentric radii. Whether the evolution is regular (and reversible) or chaotic (and irreversible), depends primarily on the radial distribution of the baryonic component. The growth of an extended baryonic component of any shape results in a regular rather than chaotic change in orbital populations. In contrast the growth of a massive and compact central component results in chaotic scattering of a significant fraction of both box and long-axis tube orbits. The growth of a disk causes a significant fraction of halo particles to become trapped by major global orbital resonances. Despite the fact that shape of a DM halo is always quite oblate following the growth of a central baryonic component, a significant fraction of its orbit population has characteristics of its triaxial or prolate progenitor (ABRIDGED).

Correlation-induced single-flux-quantum penetration in quantum rings

Conventionally, the states of a two-dimensional quantum ring in a high magnetic field have a well-defined spatial structure. But Coulomb repulsion between individual orbits causes oscillations in the size of this structure each time a magnetic flux-quantum enters or leaves the system. This effect has now been measured experimentally in semiconducting quantum rings. The confined electronic states of mesoscopic structures in a magnetic field are arranged in Landau levels consisting of spatially discrete eigenstates. These Landau orbits are the quantum mechanical analogue of classical cyclotron orbits. Here we present magnetoconductance oscillations in semiconductor rings, which visualize the spatial discreteness of the Landau orbits in high magnetic fields (typically B>2 T). We will show that these oscillations are caused by the flux-quantized, discrete electronic size of the ring leading to a corresponding modulation of its two-point conductance. The oscillation period is given by the number of flux quanta penetrating the conducting area of the structure. These high-field oscillations are distinctively different from the well-known Aharonov–Bohm effect1, where, most generally, the penetration of individual flux quanta h/e through a nanostructure causes periodic crossings of field-dependent energy levels, which give rise to magneto-quantum oscillations in its conductance2,3,4.

A pressure dependence for bound and unbound clusters

Hierarchical structure in gas and young stars produces clusters in high-density regions where the individual stellar orbits rapidly mix. For a fixed density at the onset of gas collapse (e.g. determined by changes in the ionization equilibrium and grain properties), the efficiency of star formation is automatically high in the high-density regions of giant molecular clouds. Thus, bound cluster formation follows somewhat trivially from hierarchical structure. The density where the efficiency is high enough to produce a bound cluster depends on the dispersion of the density probability distribution function (pdf), decreasing for higher dispersions and making bound cluster formation more likely. Similarly, the mass fraction of star formation in the form of bound clusters increases with the pdf dispersion. Because this dispersion is related to the turbulent Mach number, and also to the interstellar medium pressure and star-formation rate per unit volume, it follows that high-pressure or highly active regions tend to produce bound clusters, while low-pressure and inactive regions tend to produce stars in unbound associations.

Asymptotic behavior of individual orbits of discrete systems

We consider the asymptotic behavior of bounded solutions of the difference equations of the form x(n + 1) = Bx(n) + y(n) in a Banach space X, where n = 1,2,..., B is a linear continuous operator in X, and (y(n)) is a sequence in X converging to 0 as n → ∞. An obtained result with an elementary proof says that if σ(B) ∩ {|z| = 1} ⊂ {1}, then every bounded solution x(n) has the property that lim n →∞ (x(n + 1) - x(n)) = 0. This result extends a theorem due to Katznelson-Tzafriri. Moreover, the techniques of the proof are furthered to study the individual stability of solutions of the discrete system. A discussion on further extensions is also given.

0νββnuclear matrix elements and the occupancy of individual orbits

The measured occupancies of valence orbits in 76Ge and 76Se are used as a guideline for modification of the effective mean field energies that results in better description of these quantities. With them, in combination with the self-consistent renormalized quasiparticle random phase approximation (SRQRPA) method that ensures conservation of the mean particle number in the correlated ground state, we show that the resulting 0νββ nuclear matrix element for the 76Ge-->76Se transition is reduced by ~25% compared to the previous QRPA value, and therefore the difference between the present approach and the interacting shell model predictions becomes correspondingly smaller. Analogous modification of the mean field energies for the A=82 system also results in a reduction of 0νββ matrix element for the 82Se-->82Kr transition, making it also closer to the shell model prediction.

The warped young stellar disc in the Galactic centre

Aims. Within the central parsec of the Galaxy, several dozen young stars orbiting a central supermassive black hole are observed. A subset of these stars forms a coherently rotating disc. Other observations reveal a massive molecular torus that lies at a radius ∼ 1.5 pc from the centre. In this paper we consider the gravitational influence of the molecular torus upon the stars of the stellar disc. Methods. We derive an analytical formula for the rate of precession of individual stellar orbits and we show that it is highly sensitive to the orbital semi-major axis and inclination with respect to the plane of the torus, as well as to the mass of the torus. Results. Assuming that both the stellar disc and the molecular torus are stable on the timescale ≥6 Myr, we constrain the mass of the torus and its inclination with respect to the young stellar disc. We further suggest that all young stars observed in the Galactic centre may have a common origin in a single coherently rotating structure with an opening angle ≤5°, which was partially destroyed (warped) during its lifetime by the gravitational influence of the molecular torus.