A pseudo third order symplectic integrator

The symplectic integrator has been regarded as one of the optimal tools for research on qualitative secular evolution of Hamiltonian systems in solar system dynamics. An integrable and separate Hamiltonian system H = H₀ + Σ_i=1^N ε_iH_i (ε_i ≪ 1) forms a pseudo third order symplectic integrator, whose accuracy is approximately equal to that of the first order corrector of the Wisdom-Holman second order symplectic integrator or that of the Forest-Ruth fourth order symplectic integrator. In addition, the symplectic algorithm with force gradients is also suited to the treatment of the Hamiltonian system H = H₀(q,p) + εH₁(q), with accuracy better than that of the original symplectic integrator but not superior to that of the corresponding pseudo higher order symplectic integrator.     

A model of the terrestrial ionosphere in the altitude interval 50–4000 km I. Atomic ions (H+, He+, N+, O+)

An empirical model of atomic ion densities (H⁺, He⁺, N⁺, O⁺) is presented up to 4000 km altitude as a function of time (diurnal, annual), space (position, altitude) and solar flux (F10.7) — using observations of satellites (AE-B, AE-C, AE-D, AE-E, ISIS-2, OGO-6) and rockets during quiet geophysical conditions (K _p 3). The numerical treatment is based upon harmonic functions for the horizontal pattern and cubic splines for the vertical structure.The ion densities increase with increasing height up to a maximum (depending roughly on the ion mass) and decrease beyond that with increasing altitude. Above 200 km, O⁺ is the main ionic constituent being replaced at approximately 800 km (depending on latitude, local time, etc.) by H⁺. Around polar regions the light ions, H⁺ and He⁺, are depleted (polar wind) and the heavier ones enhanced. During local summer conditions the ion densities increase around polar latitudes and correspondingly decrease during local winter, except He⁺ which reflects the opposite pattern. Diurnal variations are intrinsically coupled to the individual plasma layers: N⁺ and O⁺ peak, in general, during daytime, while the amplitudes and phases of H⁺ and He⁺ change strongly with altitude and latitude. Earth, Moon and Planets Review article.     

High‐mass resolution molecular imaging of organic compounds on the surface of Murchison meteorite

High‐resolution mass spectrometry (HRMS) imaging by desorption electrospray ionization (DESI) coupled with Orbitrap MS using methanol (MeOH) spray was performed on a fragment of the Murchison (CM2) meteorite in this study. Homologues of C_nH_2n–1N₂⁺ (n = 7–9) and C_nH_2nNO⁺ (n = 9–14) were detected on the sample surface by the imaging. A high‐performance liquid chromatography (HPLC)/HRMS analysis of MeOH extracts from the sample surface after DESI/HRMS imaging indicated that the C_nH_2n–1N₂⁺ homologues corresponds to alkylimidazole, and that a few isomers of the C_nH_2nNO⁺ homologues present in the sample. The alkylimidazoles and C_nH_2nNO⁺ homologues displayed different spatial distributions on the surface of the Murchison fragment, indicating chromatographic separation effects during aqueous alteration. Moreover, the distribution pattern of compounds is also different among homologues. This is probably also resulting from the separation of isomers by similar chromatographic effects, or different synthetic pathways. Alkylimidazoles and the C_nH_2nNO⁺ homologues are mainly distributed in the matrix region of the Murchison by mineralogical observations, which is consistent with previous reports. Altered minerals (e.g., Fe‐oxide, Fe‐sulfide, and carbonates) occurred in this region. However, no clear relationship was found between these minerals and the organic compounds detected by DESI/HRMS imaging. Although this result might be due to scale differences between the spatial resolution of DESI/HRMS imaging and the grain size in the matrix of the Murchison, our results would indicate that alkylimidazoles and the C_nH_2nNO⁺ homologues in the Murchison fragment were mainly synthesized by different processes from hydrothermal alteration on the parent body.     

Predictions of the electrical conductivity and charging of the aerosols in Titan's atmosphere

The electrical conductivity and electrical charge on the aerosols in atmosphere of Titan are computed for altitudes from 0 to 400 km. Ionization due to both galactic cosmic rays and electron precipitation from the Saturnian magnetosphere is considered. This ionization results in free electrons and the primary ions N₂⁺ and N⁺ which are then rapidly converted into secondary ions such as H₂CN⁺ and NH₄⁺ which in turn form ion clusters such as H₂CN⁺(HCN)_n and NH₄⁺(NH₃)_m. In contrast to the atmospheres of Venus and Earth, we find no species in the Titan atmosphere that lead to the formation of appreciable concentrations of negative ions. Consequently, the predicted conductivity is quite different in that a substantial concentration of electrons exists all the way to the surface of Titan. The ubiquitous aerosols observed in the Titan atmosphere also play an important role in determining the charge distribution in the atmosphere. At altitudes above 100 km and for aerosol concentrations above approximately 10/cc, the recombination of electrons and positive ions is controlled by the recombination on the surface of the aerosols rather than by the gas-kinetic recombination rate. For small aerosol concentrations, the ratio of the number of charges per particle to the radius of the particle is approximately 30, for radii in microns. This value is similar to that obtained by previous investigators for terrestrial noctilucent clouds. Because the aerosol particles are highly charged, coagulation is inhibited, particle sizes are smaller, and their settling rates are reduced. As a consequence, the optical depth of the atmosphere is much higher than it would be if the particles were uncharged.     

Errors in calculated values of electron temperatures in H II regions

This paper considers the classical method to determine the electron temperatures t _3,O, t _2,O and t _2,N from forbidden lines of the ions O⁺⁺, O⁺, and N⁺, and investigates the influence of uncertainties in atomic data on the accuracy of the determined electron temperatures. The uncertainties in atomic data (the Einstein coefficients for spontaneous transitions and electron ionization cross-sections) are estimated as discrepancies between the values computed by various authors. The error in the electron temperature caused by uncertainties in the atomic data is found to increase with the growth in the electron temperature. At a temperature 10000 K, the errors in the electron temperatures t _3,O, t _2,N, and t _2,O do not exceed 1, 3, and 7%, respectively.     

Calculations of the thermochemistry of six reactions leading to ammonia formation in Titan's atmosphere

The thermochemical properties of the six reactions: (1) N₂+hν (solar EUV) → N⁺ + N(⁴S) + e⁻, (2) N⁺ + H₂ → NH⁺ + H, (3) NH⁺ + H₂ → NH⁺₂ + H, (4) NH⁺₂ + H₂ → NH⁺₃ + H, (5) NH⁺₃ + H₂ → NH⁺₄ + H, and (6) NH⁺₄ + e⁻ → NH₃ + H, were theoretically proposed by Atreya in 1986 and were cited in 2003 by Bernard who assumed that this chain reaction would lead to ammonia formation in Titan's atmosphere. The thermochemical properties of these six reactions have been calculated by means of the coupled cluster singles and doubles (CCSD) at the CCSD/cc-pvdz level, and the CCSD/6-311++g(3df,3pd) level, and G2 method. The geometries of the reactants and products of reactions have been optimized, the energies of reactions have been computed. The analysis of the results shows that: (I) The free energies of four reactions among these six reactions are negative. It means that these reactions, namely reactions (1)-(6) except reaction (2), can react spontaneously in Titan's low temperature environment. The converted temperatures of reactions (3) and (5) are 11881.7 and 4596.9 K, respectively. (II) Reaction (2) is an endothermic reaction, its converted temperature is 1797.6 K. When T<1797.6 K, reaction (2) cannot react forward spontaneously. The barrier of reaction (2) is 26.154 kcal mol⁻¹, which is probably too high to allow it to occur in the atmosphere of Titan. The rate for this reaction at 300 K has been calculated, and the value is k=4.16×¹⁰⁻⁷ s⁻¹. (III) The results of the three methods are more or less the same. So it is concluded that this chain reaction cannot be a pathway to lead to ammonia (gas phase) formation in Titan's atmosphere.     

The role of ion-molecule reactions in the conversion of N2O5 to HNO3 in the stratosphere

We have investigated the role of several ion-molecule reactions in the conversion of N₂O₅ to HNO₃. In the proposed conversion, an N₂O₅ molecule would react with an H₂O molecule clustered to an inert ion to produce two HNO₃ molecules. Subsequent clustering of an H₂O molecule to the inert ion would make the reaction catalytic. If such an ion-catalysed conversion of N₂O₅ to HNO₃ occurs, it would probably play a role in the stratospheric chemistry at high latitudes in winter. In this paper we present reaction rate constant measurements made in a flowing afterglow apparatus for hydrated H₃O⁺, H⁺(CH₃CN)_m (m = 1, 2, 3), and several negative ions reacting with N₂O₅. Slow rate constants were found for these ions for hydration levels that are predominant in the stratosphere. With the known stratospheric ion density, these slow rate constants preclude significant N₂O₅ conversion by ion-molecule reactions.     

Excitation class of nebulae—An evolution criterion?

A principally new, quantitative system of the classification of the spectra of planetary nebulae is proposed. Spectral class of excitation class of the nebulap is determined according to the relative intensities of emission lines (N ₁+N ₂) [OIII]/4686 HeII and (N ₁+N ₂) [OIII]/H (Table I, Figure 1). The excitation classes are obtained for 142 planetary nebulae of all classes—low (p=1–3), middle (p=4–8), and high (p=9–12⁺) (Tables II, III, and IV). An empirical relationship between excitation classp and mean radius of nebulae is discovered (Figure 2). This relationship as well as excitation classp, as an independend parameter, admit an evolutionary interpretation. It is shown that after reaching the highest class of excitationp=12⁺ the nebulae decrease their class of excitation with the further increases of sizes. The diagram of this relationship has two nearly-symmetric branches — rising and descending with the apogee onp=12⁺ (Figure 2).     

The production of water-cluster positive ions in the quiet daytime D region

In the quiet daytime D region, the primary positive-ion species is thought to be NO⁺, produced by solar Lyman-alpha ionization of NO. Below the altitude of the mesopause, however, the dominant ambient species observed are water-cluster ions of the general type H⁺(H₂O)_n. No satisfactory reaction scheme for producing these cluster ions from NO⁺ has yet been proposed. Following earlier suggestions, a model calculation has been carried out in which successive hydrations of NO⁺ take place through clustering with N₂ and CO₂, followed by “switching” reactions with H₂O. The third hydrate of NO⁺ is then converted into the water-cluster species H⁺(H₂O)₃, and the other water-cluster species are produced by successive clustering and thermal breakup reactions. Many of the reactions involved have not been measured in the laboratory, but reasonable estimates of their rates can be made on the basis of existing measurements of other species. Since both temperature and water-vapor content are of major importance in the model, calculations were carried out for two temperature profiles and two water-vapor profiles. It is shown that the results are in reasonably good agreement with observations as far as the water-cluster species are concerned. Under low-temperature conditions, the model predicts relatively large concentrations of various clusters of NO⁺, in agreement with some observations but in disagreement with others. The importance of sampling breakup of these weakly bound clusters, and their relevance to the free electron concentrations are discussed.     

Periodic solutions for resonance 4/3: Application to the construction of an intermediary orbit for Hyperion’s motion

The motion of Hyperion is an almost perfect application of second kind and second genius orbit, according to Poincaré’s classification. In order to construct such an orbit, we suppose that Titan’s motion is an elliptical one and that the observed frequencies are such that 4n _H−3n _T+3n _ω=0, where n _H, n _T are the mean motions of Hyperion and Titan, n _ω is the rate of rotation of Hyperion’s pericenter. We admit that the observed motion of Hyperion is a periodic motion such as . Then, .N _H, N _T, k∈ N ⁺. With that hypothesis we show that Hyperion’s orbit tends to a particular periodic solution among the periodic solutions of the Keplerian problem, when Titan’s mass tends to zero. The condition of periodicity allows us to construct this orbit which represents the real motion with a very good approximation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dust and atomic hydrogen nearρ Ophiuchi

We discuss the distribution of dust and atomic hydrogen in and around a small dark cloud near Ophiuchi. Although the obscuration reaches values as high as 8 magnitudes, the values ofN _H derived inside the cloud do not differ markedly from those found at nearby, less obscured points. At low values of the obscuration, dust and hydrogen isophotes show satisfactory agreement, i.e.,N _H increases withN _G. However, we find thatN _H starts decreasing for sufficiently large values ofN _G. In the entire range of optical extinctions investigated our observations are satisfied by an expression of the typeN _H=AN _G exp (-BN _G). Interpretation of this relationship suggests that an appreciable part of the hydrogen has been converted to molecular form inside the cloud. A tentative method for calculating the time elapsed since the conversion process began is proposed.Contribution No. 2 from the Argentine-Carnegie Radio Astronomy Station of the Instituto argentino de Radioastronomía and the Department of Terrestrial Magnetism of the Carnegie Institution of Washington.     

Interstellar catalysis

Since gas-phase reactions alone cannot account for the observed abundances of H₂ in the typical interstellar cloud, one or more surface reactions are probably involved. Of the three possible candidates, only the catalytic production of H₂ on transition metal grains is supported by laboratory evidence. Using the rate equations developed in a previous paper for this process, the steady-state equilibrium abundances of H, H₂,e ^–, H⁺, H^–, H₂ ⁺, and H₃ ⁺ are calculated for large (r>10 pcs;M10² M ), tenuous (n=10²–10⁴ cm^–3) hydrogen dust clouds under a wide variety of conditions. In addition to the four rate equations involved in the catalytic reactions, 18 gas-phase and one additional surface reaction—the physical adsorption of H-atoms on cold, dielectric surfaces and their subsequent recombination and desorption as H₂ molecules—are included in the calculations. It is found that metal grains can produce as much interstellar H₂ as the best physical adsorption mechanism under optimum conditions if the extinction in the visible is less than 5^m.0. The three critical parameters for efficient catalysis (activation energy of desorption, grain temperature, and the number density of available sites) are examined, and it is shown that catalytic reactions are efficient producers of H₂ under all but the most unfavorable conditions.     

Ion and neutral sources and sinks within Saturn's inner magnetosphere: Cassini results

Using ion-electron fluid parameters derived from Cassini Plasma Spectrometer (CAPS) observations within Saturn's inner magnetosphere as presented in Sittler et al. [2006a. Cassini observations of Saturn's inner plasmasphere: Saturn orbit insertion results. Planet. Space Sci., 54, 1197-1210], one can estimate the ion total flux tube content, N_IONL², for protons, H⁺, and water group ions, W⁺, as a function of radial distance or dipole L shell. In Sittler et al. [2005. Preliminary results on Saturn's inner plasmasphere as observed by Cassini: comparison with Voyager. Geophys. Res. Lett. 32(14), L14S04), it was shown that protons and water group ions dominated the plasmasphere composition. Using the ion-electron fluid parameters as boundary condition for each L shell traversed by the Cassini spacecraft, we self-consistently solve for the ambipolar electric field and the ion distribution along each of those field lines. Temperature anisotropies from Voyager plasma observations are used with (T_⊥/T_∥)_W⁺∼5 and (T_⊥/T_∥)_H⁺∼2. The radio and plasma wave science (RPWS) electron density observations from previous publications are used to indirectly confirm usage of the above temperature anisotropies for water group ions and protons. In the case of electrons we assume they are isotropic due to their short scattering time scales. When the above is done, our calculation show N_IONL² for H⁺ and W⁺ peaking near Dione's L shell with values similar to that found from Voyager plasma observations. We are able to show that water molecules are the dominant source of ions within Saturn's inner magnetosphere. We estimate the ion production rate S_ION∼10²⁷ ions/s as function of dipole L using N_H⁺, N_W⁺ and the time scale for ion loss due to radial transport τ_D and ion-electron recombination τ_REC. The ion production shows localized peaks near the L shells of Tethys, Dione and Rhea, but not Enceladus. We then estimate the neutral production rate, S_W, from our ion production rate, S_ION, and the time scale for loss of neutrals by ionization, τ_ION, and charge exchange, τ_CH. The estimated source rate for water molecules shows a pronounced peak near Enceladus’ L shell L∼4, with a value S_W∼2×10²⁸ mol/s.     

Non-integrability of Hénon-Heiles system

We show that the Hénon-Heiles system with Hamiltonian H=\frac12(y₁²+y₂²)+\frac12(ax₁²+bx₂²)+\frac13dx₂³+cx₁²x₂{H=\frac12(y_1^2+y_2^2)+\frac12(ax_1^2+bx_2^2)+\frac13dx_2^3+cx_1^2x_2} is integrable in Liouvillian sense (i.e., the existence of an additional first integral) if and only if c = 0; or \frac dc=1, a=b; or \frac dc=6, a, b{\frac dc=1, a=b; {\rm or}\, \frac dc=6, a, b} arbitrary; or \frac dc=16, b=16a{\frac dc=16, b=16a}. Therefore, we get a complete classification of the Hénon-Heiles system in sense of integrability and non-integrability.     

The evolution of charged particles in a model of contracting cloud

The evolution of the charged particles are followed during contraction of a model of an interstellar cloud, with initial density number of n = 10 cm^–3. The contraction is followed up to density increase by five orders of magnitude. Special care is given to the details of the negative ions. In addition, we have tested the ambipolar diffusion according to the results of the ion density.The results predict the importance of atomic ions in the diffuse regions. H⁺ and C⁺ are distinctly enhanced in the beginning of contraction but decrease as contraction proceeds. Molecular ions enhance as contraction proceeds and becomes important in dense regions. The most enhanced molecular ions are HCO⁺, O₂ ⁺, C₂H₃ ⁺, H₃O⁺ and SO⁺, H₃ ⁺ is less abundant. The atomic ions (except metalic ions) decrease noticeably as density increases. In general the negative ions are of negligible fractional abundances. It has also been found that the time of ambipolar diffusion is shorter than the dynamical time, hence the magnetic field should be weakened in the central core as the central density increases to n = 10⁴ cm^–3.     

GPS-Based On-Board Orbit Determination of a Satellite Using Extended H ∞ Filtering Algorithms

In this paper the extended H filtering algorithms for the design of the GPS-based on-board autonomous navigation system for a low earth orbit (LEO) satellite are introduced. The dynamic process models for the estimation of position, velocity and acceleration from the GPS measurements are established. The nominal orbit of the small LEO satellite is determined by using the 7th–8th order Runge—Kutta algorithms. Three filtering approaches are applied to smooth the orbit solutions, respectively, based upon the simulated GPS pseudo range observables using the Satellite Navigation Tool Box. The simulation shows that the observed orbit errors obtained by using the extended H filtering algorithms can be reduced to a lower level than the observed orbit errors in the sense of RMS within 12 h of tracking time by using the H filtering algorithms and the extended Kalman filtering algorithms under the appropriately designed parameters. Based upon the position errors predicted by the three filtering algorithms after the last observation, we find that the extended H filtering algorithm provides the least position errors of the user satellite.This revised version was published online in October 2005 with corrections to the Cover Date.     

Ion composition and chemistry in the coma of Comet 1P/Halley—A comparison between Giotto's Ion Mass Spectrometer and our ion-chemical network

In order to understand the cometary plasma environment it is important to track the closely linked chemical reactions that dominate ion evolution. We used a coupled MHD ion-chemistry model to analyze previously unpublished Giotto High Intensity Ion Mass Spectrometer (HIS-IMS) data. In this way we study the major species, but we also try to match some minor species like the CH_x and the NH_x groups. Crucial for this match is the model used for the electrons since they are important for ion-electron recombination. To further improve our results we included an enhanced density of supersonic electrons in the ion pile-up region which increases the local electron impact ionization. In this paper we discuss the results for the following important ions: C⁺, CH⁺, CH⁺₂, CH⁺₃, N⁺, NH⁺, NH⁺₂, NH⁺₃, NH⁺₄, O⁺, OH⁺, H₂O⁺, H₃O⁺, CO⁺, HCO⁺, H₃CO⁺, and CH₃OH⁺₂. We also address the inner shock which is very distinctive in our MHD model as well as in the IMS data. It is located just inside the contact surface at approximately 4550 km. Comparisons of the ion bulk flow directions and velocities from our MHD model with the data measured by the HIS-IMS give indication for a solar wind magnetic field direction different from the standard Parker angle at Halley's position. Our ion-chemical network model results are in a good agreement with the experimental data. In order to achieve the presented results we included an additional short lived inner source for the C⁺, CH⁺, and CH⁺₂ ions. Furthermore we performed our simulations with two different production rates to better match the measurements which is an indication for a change and/or an asymmetric pattern (e.g. jets) in the production rate during Giotto's fly-by at Halley's comet.     

Measured ion-molecule reaction rates for modelling Titan's atmosphere

Rate coefficients for several two- and three-body ion-molecule reactions involving hydrocarbons have been determined at thermal energies and above using drift tube-mass spectrometer techniques. The measured rates for clustering and breakup reactions involving CH₅⁺ and C₂H₅⁺ ions in methane are found to be strongly temperature dependent in the range from 80 to 240 K. The equilibrium constants determined for these reactions differ somewhat from those of Hiraoka and Kebarle. Rate coefficients for two-body reactions of CH₅⁺, C₂H₅⁺, N⁺, H⁺ and D⁺ ions with methane and/or ethane have been measured. The results indicate that the product yields of several of the fast ion-molecule reactions depend strongly on ion energy (temperature), and therefore previous room-temperature results may be of limited value for model calculations of Titan's atmosphere.     

Symmetry in hierarchical cosmology

An idealized model of a hierarchy of clusters is considered, and the number-count asymmetry measure in two different directions,R ₁ |N ⁺-N ^-|/(N ⁺+N ^-), is evaluated, for values ofl _I /c _I =(distance between cluster centres)/(cluster diameter). Providedl _I /c _I 10, theory predictsR _I 0.1, in agreement with the symmetry of high-redshift radio sources.