Some expansions of the elliptic motion to high eccentricities

Some classic expansions of the elliptic motion — cosmE and sinmE — in powers of the eccentricity are extended to highly eccentric orbits, 0.6627...<e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence (e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions — (a/r), (r/a), (r/a) sinv, ..., — derived straightforward from the expansions ofE, cosE and sinE are also presented.     

Extension of the solution of Kepler's equation to high eccentricities

The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJ _n (ne). The expansion is convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is a positive real number, which is calculated numerically.     

Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque

The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular, the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type, which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

Investigation of equilibria of a satellite subjected to gravitational and aerodynamic torques

Attitude motion of a satellite subjected to gravitational and aerodynamic torques in a circular orbit is investigated. In special case, when the center of pressure of aerodynamic forces is located on one of the principal central axes of inertia of the satellite, all equilibrium orientations are determined. Necessary and (or) sufficient conditions of stability are obtained for each equilibrium orientation. Evolution of domains where stability conditions take place is investigated. All bifurcation values of parameters corresponding to qualitative change of domains of stability are determined.     

Equilibria of a satellite subjected to gravitational and aerodynamic torques with pressure center in a principal plane of inertia

Attitude motion of a satellite subjected to gravitational and aerodynamic torques in a circular orbit is considered. In special case, when the center of pressure of aerodynamic forces is located in one of the principal central planes of inertia of the satellite, all equilibrium orientations are determined. Existence conditions of all equilibria are obtained and evolution of domains with a fixed number of equilibria is investigated in detail. All bifurcation values of the system’s parameters corresponding to the qualitative change of these domains are determined. Sufficient conditions of stability are obtained for each equilibrium orientation using generalized integral of energy.     

Recent changes of water discharge and sediment load in the Zhujiang (Pearl River) Basin, China

The paper is concerned with identifying changes in the time series of water and sediment discharge of the Zhujiang (Pearl River), China. The gradual trend test (Mann–Kendall test), and abrupt change test (Pettitt test), have been employed on annual water discharge and sediment load series (from the 1950s–2004) at nine stations in the main channels and main tributaries of the Zhujiang. Both the Mann–Kendall and Pettitt tests indicate that water discharge at all stations in the Zhujiang Basin showed no significant trend or abrupt shift. Annual water discharges are mainly influenced by precipitation variability, while the construction of reservoirs/dams in the Zhujiang Basin had little influence on water discharge. Sediment load, however, showed significant decreasing trends at some stations in the main channel of the Xijiang and Dongjiang. More stations have seen significantly decreasing trends since the 1990s. The decreasing sediment load in the Zhujiang reflects the impacts of reservoir construction in the basin. In contrast, the Liujiang, the second largest tributary of the Xijiang, has experienced a significant upward shift of sediment load around 1991 likely caused by exacerbated rock desertification in the karst regions. The annual sediment load from the Zhujiang (excluding the delta region) to the estuary has declined from 80.4 × 10⁶ t averaged for the period 1957–1995 to 54.0 × 10⁶ t for the period 1996–2004. More specifically, the sediment load declined steadily since the early 1990s so that in 2004 it was about one-third of the mean level of pre-90s. Water discharge and sediment load of the Zhujiang would be more affected by human activities in the future with the further reservoir developments, especially the completion of the Datengxia hydroelectric project, and an intensification of the afforestation policy in the drainage basin.     

Detection of a Satellite Orbiting The Nucleus of Comet Hale–Bopp (C/1995 O1)

This paper reports on the detection of a satellite around the principal nucleus of comet Hale-Bopp. As shown elsewhere, a successful morphological model for the comet's dust coma necessitates the postulation of overlapping jet activity from a comet pair. The satellite has been detected digitally on images taken with the Hubble Space Telescope's Wide Field Planetary Camera 2 in the planetary mode on five days in May–October 1996. An average satellite-to-primary signal ratio is 0.21 ± 0.03, which implies that the satellite is ∼30 km in diameter, assuming the main nucleus is ∼70 km across. To avoid collision, the separation distance must exceed 50–60 km at all times. The satellite's projected distances on the images vary from 160 to 210 km, or 0.06 to 0.10 arcsec. The satellite was not detected in October 1995, presumably because of its subpixel separation from the primary. The radius of the gravitational sphere of action of the principal nucleus 70 km in diameter is 370–540 km at perihelion, increasing linearly with the Sun's distance: the satellite appears to be in a fairly stable orbit. Its orbital period at ∼180 km is expected to be ∼2–3 days, much shorter than the intervals between the HST observations. If the main nucleus should be no more than 42 km across, Weaver et al.'s upper limit, the satellite's orbit could become unstable, with the object drifting away from the main nucleus after perihelion. Potentially relevant ground-based detections of close companions are reported. Efforts to determine the satellite's orbit and the total mass of the system will get under way in the near future. This revised version was published online in July 2006 with corrections to the Cover Date.     

The structure, ordering and equation of state of ammonia dihydrate (nh3 · 2h2o)

We present the first ab initio simulations of the low-pressure phase of ammonia dihydrate (NH₃ · 2H₂O), ADH I, a likely constituent of many volatile-rich solid bodies in the outer Solar System (e.g., Saturn’s moons). Ordered monoclinic (space group P2₁) and orthorhombic (space group P2₁2₁2₁) variants of the experimentally observed cubic cell (space group P2₁3) may be constructed, with fully ordered water molecule orientations that obey the ice rules. Our calculations show that the most stable structure at 0 K is orthorhombic (P2₁2₁2₁), the monoclinic variants (P2₁) being energetically disfavored. We provisionally call this ordered orthorhombic phase ADH III. The, as-yet-unmeasured, bulk modulus, K₀, is predicted to be 10.67−0.44+0.56 GPa at 0 K. Our results are also combined with literature data to arrive at a revised coefficient of volume thermal expansion, α_v = 2.81 × 10⁻⁷ T^1.39 (from 0-176 K), with the density at 0 K, ρ₀ = 991.7(39) kg m⁻³. We also present a case, based on literature data, that argues for a gradual transformation from a paraelectrically disordered cubic structure (P2₁3) to the proposed antiferroelectrically ordered orthorhombic structure (P2₁2₁2₁) around 130-150 K (cf. ice III → IX), a temperature regime that applies to the surfaces and interiors of many medium-sized (radii ∼500-700 km) icy bodies.     

A dynamical model with a new inversion technique applied to observations of Comet C/2000 WM1 (LINEAR)

Three continuum images of Comet C/2000 WM₁ (LINEAR) obtained on Nov 10, Nov 19, and Dec 03, 2001, are analyzed with the aid of a dynamical model, i.e. with a model that uses the size-dependent motion of dust grains under solar radiation pressure to determine the dust size distribution and its temporal change. The frames are photometrically calibrated in terms of the albedo filling factor product. On Nov 20.2 the Earth transited the orbital plane of the comet and an anti-tail was recognized in the image of Nov 19. For the determination of the particle fluxes describing the contribution of monodisperse particle shells to the cometary brightness the model uses a new regularization method employing Chebyshev polynomials of selected orders in emission time and particle size. It guarantees positiveness of the particle fluxes and imposes a varying degree of smoothness on their dependence on particle size and emission time. The particle emission velocities are still derived by trial and error. The dynamical model is described in detail. Results are presented for several low orders of the Chebyshev polynomials and are compared in order to understand the limitations imposed by the regularization process. The size distributions derived from the different observations do not always agree. This is particularly true for the earliest and most recent synchrones contributing to an image. In the observations of Nov 10 and Dec 03, i.e. excluding the anti-tail image, the integrated mass loss strongly decreases in the most recent time steps of the model although the comet is still approaching the Sun. This is interpreted as an artifact introduced by the overlap of the shells of large particle size emitted shortly before the observation. The model derives an increasing number of small particles released by the comet in the second half of November. This is at least in part considered as real and attributed to particle fragmentation occurring when the comet was at a heliocentric distance of about 1.4 AU.     

Laboratory measurements of the Ka-band (7.5 mm to 9.2 mm) opacity of phosphine (PH3) and ammonia (NH3) under simulated conditions for the Cassini-Saturn encounter

Recently, a model for the centimeter-wavelength opacity of PH₃ under conditions characteristic of the outer planets was developed by Hoffman et al. (2001, PhD thesis), based on centimeter wavelength laboratory measurements. New laboratory measurements have been conducted which show that this model is also accurate at low pressures and temperatures, and at millimeter wavelengths such as will be employed in Cassini Ka-band (9.3 mm) radio occultation studies. The opacity of PH₃ in a hydrogen/helium (H₂/He) atmosphere has been measured at frequencies in the Ka-band region at 32.7 GHz (9.2 mm), 35.6 GHz (8.4 mm), 37.7 GHz (8.0 mm), and 39.9 GHz (7.5 mm) at pressures of 0.5, 1, and 2 bar and at temperatures of 295, 209, and 188 K. Additionally, new high-precision laboratory measurements of the opacity of NH₃ in an H₂/He atmosphere have been conducted under the same temperature and pressure conditions described for PH₃. These new measurements better constrain the NH₃ opacity model supporting use of a Ben-Reuven lineshape model. These measurements will also elucidate the interpretation of millimeter wavelength observations conducted with the NRAO/VLA at 43 GHz (7 mm).     

The molecular composition of Comet C/2007 W1 (Boattini): Evidence of a peculiar outgassing and a rich chemistry

We measured the chemical composition of Comet C/2007 W1 (Boattini) using the long-slit echelle grating spectrograph at Keck-2 (NIRSPEC) on 2008 July 9 and 10. We sampled 11 volatile species (H₂O, OH^∗, C₂H₆, CH₃OH, H₂CO, CH₄, HCN, C₂H₂, NH₃, NH₂, and CO), and retrieved three important cosmogonic indicators: the ortho-para ratios of H₂O and CH₄, and an upper-limit for the D/H ratio in water. The abundance ratios of almost all trace volatiles (relative to water) are among the highest ever observed in a comet. The comet also revealed a complex outgassing pattern, with some volatiles (the polar species H₂O and CH₃OH) presenting very asymmetric spatial profiles (extended in the anti-sunward hemisphere), while others (e.g., C₂H₆ and HCN) showed particularly symmetric profiles. We present emission profiles measured along the Sun-comet line for all observed volatiles, and discuss different production scenarios needed to explain them. We interpret the emission profiles in terms of release from two distinct moieties of ice, the first being clumps of mixed ice and dust released from the nucleus into the sunward hemisphere. The second moiety considered is very small grains of nearly pure polar ice (water and methanol, without dark material or apolar volatiles). Such grains would sublimate only very slowly, and could be swept into the anti-sunward hemisphere by radiation pressure and solar-actuated non-gravitational jet forces, thus providing an extended source in the anti-sunward hemisphere.     

Correction to: Observation of a Flare and Filament Eruption in Lyman- α $\alpha $ on 8 September 2011 by the PRoject for OnBoard Autonomy/Large Yield Radiometer (PROBA2/LYRA)

In most of the solar cycles, activity in the northern and southern hemispheres peaks at different times. One hemisphere peaks well before the other, and at least one of the hemispheric maxima frequently does not coincide with the whole sphere maximum. Prediction of the maximum of a hemisphere and the corresponding north–south asymmetry of a solar cycle may help to understand the mechanisms of the solar cycle, the solar-terrestrial relationship, and solar-activity influences on space weather. Here we analysed the sunspot-group data from the Greenwich Photoheliographic Results (GPR) during 1874?–?1976 and Debrecen Photoheliographic Data (DPD) during 1977?–?2017 and studied the cycle-to-cycle variations in the values of 13-month smoothed monthly mean sunspot-group area in the whole sphere (WSGA), northern hemisphere (NSGA), and southern hemisphere (SSGA) at the epochs of maxima of Sunspot Cycles 12?–?24 and at the epochs of maxima of WSGA, NSGA, and SSGA Cycles 12?–?24 (note that solar-cycle variation of a parameter is expressed as a cycle of that parameter). The cosine fits to the values of WSGA, NSGA, and SSGA at the maxima of sunspot, WSGA, NSGA, and SSGA Cycles 12?–?24, and to the values of the corresponding north–south asymmetry, suggest the existence of a ≈132-year periodicity in the activity of the northern hemisphere, a 54?–?66-year periodicity in the activity of the southern hemisphere, and a 50?–?66 year periodicity in the north–south asymmetry in activity at all the aforementioned epochs. By extrapolating the best-fit cosine curves we predicted the amplitudes and the corresponding north–south asymmetry of the 25th WSGA, NSGA, and SSGA cycles. We find that on average Solar Cycle 25 in sunspot-group area would be to some extent smaller than Solar Cycle 24 in sunspot-group area. However, by inputting the predicted amplitudes of the 25th WSGA, NSGA, and SSGA cycles relationship between sunspot-group area and sunspot number we find that the amplitude (\(130\pm 12\)) of Sunspot Cycle 25 would be slightly larger than that of reasonably small Sunspot Cycle 24. Still it confirms that the beginning of the upcoming Gleissberg cycle would take place around Solar Cycle 25. We also find that except at the maximum of NSGA Cycle 25 where the strength of activity in the northern hemisphere would be dominant, the strength of activity in the southern hemisphere would be dominant at the maximum epochs of the 25th sunspot, WSGA, and SSGA cycles.

     

Analysis of OMEGA/Mars Express data hyperspectral data using a Multiple-Endmember Linear Spectral Unmixing Model (MELSUM): Methodology and first results

The mineralogical composition of the Martian surface is investigated by a Multiple-Endmember Linear Spectral Unmixing Model (MELSUM) of the Observatoire pour la Minéralogie, l’Eau, les Glaces et l’Activité (OMEGA) imaging spectrometer onboard Mars Express. OMEGA has fully covered the surface of the red planet at medium to low resolution (2–4 km per pixel). Several areas have been imaged at a resolution up to 300 m per pixel. One difficulty in the data processing is to extract the mineralogical composition, since rocks are mixtures of several components. MELSUM is an algorithm that selects the best linear combination of spectra among the families of minerals available in a reference library. The best fit of the observed spectrum on each pixel is calculated by the same unmixing equation used in the classical Spectral Mixture Analysis (SMA). This study shows the importance of the choice of the input library, which contains in our case 24 laboratory spectra (endmembers) of minerals that cover the diversity of the mineral families that may be found on the Martian surface. The analysis is restricted to the 1.0–2.5 μm wavelength range. Grain size variations and atmospheric scattering by aerosols induce changes in overall albedo level and continuum slopes. Synthetic flat and pure slope spectra have therefore been included in the input mineral spectral endmembers library in order to take these effects into account. The selection process for the endmembers is a systematic exploration of whole set of combinations of four components plus the straight line spectra. When negative coefficients occur, the results are discarded. This strategy is successfully tested on the terrestrial Cuprite site (Nevada, USA), for which extensive ground observations exist. It is then applied to different areas on Mars including Syrtis Major, Aram Chaos and Olympia Undae near the North Polar Cap. MELSUM on Syrtis Major reveals a region dominated by mafic minerals, with the oldest crustal regions composed of a mixing between low-calcium pyroxenes (LCPs) (orthopyroxenes (OPx)) and high-calcium pyroxenes (HCPs) (clinopyroxenes (CPx)). The Syrtis volcanic edifice appears depleted in LCP (OPx) and enriched in HCP (CPx), which is consistent with materials produced with a lower partial fusion degree at an age younger to the surrounding crust. Strong olivine signatures are found between the two calderas Nili Patera and Meroe Patera and in Nili Fossae. A strong signature of iron oxides is found within Aram Chaos, with a spatial distribution also consistent with thermal emission spectrometer (TES). Gypsum is unambiguously detected in the northern polar region, in agreement with the study of Langevin et al. [2005. Sulfates in the north polar region of Mars detected by OMEGA/Mars Express. Science 307(5715), 1584–1586]. Our results show that the linear spectral unmixing provides good first order results in a variety of mineralogical contexts, and can therefore confidently be used on a wider scale to analyze the complete archive of OMEGA data.     

Radiative decay of the 4d5(6S)5p z5,7Po states in Tc ii: comparison along the homologous and isoelectronic sequences. Application to astrophysics

Using three independent theoretical approaches (CA, HFR + CP, AUTOSTRUCTURE), oscillator strengths have been calculated for a set of Tc  ii transitions of astrophysical interest and the reliability of their absolute scale has been assessed. The examination of the spectra emitted by some Ap stars has allowed the identification of Tc  ii transitions in HD 125248. This Tc  ii detection should however await confirmation from spectral synthesis relying on dedicated model atmospheres. New partition functions are also provided for Tc  i , Tc  ii and Tc  iii for temperatures ranging between 4000 and 13 000 K.     

No compelling evidence of distributed production of CO in Comet C/1995 O1 (Hale-Bopp) from millimeter interferometric data and a re-analysis of near-IR lines

Based on long-slit infrared spectroscopic observations, it has been suggested that half of the carbon monoxide present in the atmosphere of Comet C/1995 O1 (Hale-Bopp) close to perihelion was released by a distributed source in the coma, whose nature (dust or gas) remains unidentified. We re-assess the origin of CO in Hale-Bopp’s coma from millimeter interferometric data and a re-analysis of the IR lines.Simultaneous observations of the CO J(1–0) (115 GHz) and J(2–1) (230 GHz) lines were undertaken with the IRAM Plateau de Bure interferometer in single-dish and interferometric modes. The diversity of angular resolutions (from 1700 to 42,000 km diameter at the comet) is suitable to study the radial distribution of CO and detect the extended source observed in the infrared. We used excitation and radiative transfer models to simulate the single-dish and interferometric data. Various CO density distributions were considered, including 3D time-dependent hydrodynamical simulations which reproduce temporal variations caused by the presence of a CO rotating jet. The CO J(1–0) and J(2–1) observations can be consistently explained by a nuclear production of CO. Composite 50:50 nuclear/extended productions with characteristic scale lengths of CO parent L_p > 1500 km are rejected.Based on similar radiation transfer calculations, we show that the CO v = 1–0 ro-vibrational lines observed in Comet Hale-Bopp at heliocentric distances less than 1.5 AU are severely optically thick. The broad extent of the CO brightness distribution in the infrared is mainly due to optical depth effects entering in the emitted radiation. Additional factors can be found in the complex structure of the CO coma, and non-ideal slit positioning caused by the anisotropy of dust IR emission.We conclude that both CO millimeter and infrared lines do not provide compelling evidence for a distributed source of CO in Hale-Bopp’s atmosphere.