An archetype hydrogen atmosphere problem

Populations for the first three bound states and the continuum of hydrogen are determined for an isothermal, hydrostatic atmosphere at 20 000 K. The atmosphere is treated as being optically thin in the Balmer and Paschen continua and illuminated by continuum radiation at these wavelengths with prescribed radiation temperatures. The atmosphere is optically thick in the 2-1, 3-1, 3-2 and c-1 transitions. Three stages of approximation are treated:

1. radiative detailed balance in the 2-1, 3-1 and 3-2 transitions,
2. radiative detailed balance in the 3-1 and 3-2 transitions, and
3. all transitions out of detailed balance.

The solution of this problem is non-trivial, and presents sufficient difficulty to have caused failure of at least one rather standard technique. The problem is thus a good archetype against which new methods, or new implementations of old methods may be tested.     

Mechanisms for incorporation of hydrogen in and on terrestrial planetary surfaces

Storage of hydrogen atoms in or on a planetary surface can take place via several different mechanisms. If the hydrogen atom reacts to form a hydroxyl (OH) group or water molecule, an absorption band near 3 μm will be present. Many possible mechanisms for sequestering atomic hydrogen are discussed: internal hydrogen in the form of non-structural OH and H₂O in nominally-anhydrous minerals, structural OH in minerals, structural H₂O in minerals, H₂O in fluid inclusions, and OH and H₂O in glasses; bulk H₂O as either liquid water or ice; and surficial hydrogen that is either physisorbed as H₂O, chemisorbed as an H₂O surface complex, or chemically-bound as an OH group on surface terminal sites and grain boundary regions. Understanding the spectroscopic distinctions among these various phenomena is of critical importance in constraining both the evolution of planetary interiors and the cycling of water on planetary surfaces. Proper interpretation of 3-μm bands in reflectance spectra is shown to depend upon the relative contributions from surficial vs. interior hydrogen, which vary with effective surface area (i.e., the grain size and surface roughness) and the volume sampled by the spectrometer.     

Gravitational N-body problem on the accretion process of terrestrial planets

Numerical integration of the gravitational N-body problem has been carried out for a variety of photoplanetary clusters in the range N = 100 to 200. Particles are assumed to coagulate at collisions irrespective of relative velocity and mass ratio of the particles. It is shown graphically how the dispersed N-bodies accumulate to a single planet through mutual collisions. The velocity distribution and size distribution of bodies are also investigated as functions of time in the accretion process. The root mean square velocity of bodies in a cluster increases with time in an early stage of accretion but decreases with time in a late stage of accretion. Accretion rates of planets are found to be dependent strongly on the initial number density distribution, the initial size distribution, and the initial velocity distribution of bodies. Formation of satellites of about 10% in the planet mass is common to most cases in the present study. A substantial mass of bodies also escapes from the cluster. Many satellites and escapers formed during the accretion process of planets may be source materials of heavy bombardment in the early history of planets.     

The dual origin of the terrestrial atmosphere

The origin of the terrestrial atmosphere is one of the most puzzling enigmas in the planetary sciences. It is suggested here that two sources contributed to its formation, fractionated nebular gases and accreted cometary volatiles. During terrestrial growth, a transient gas envelope was fractionated from nebular composition. This transient atmosphere was mixed with cometary material. The fractionation stage resulted in a high Xe/Kr ratio, with xenon being more isotopically fractionated than krypton. Comets delivered volatiles having low Xe/Kr ratios and solar isotopic compositions. The resulting atmosphere had a near-solar Xe/Kr ratio, almost unfractionated krypton delivered by comets, and fractionated xenon inherited from the fractionation episode. The dual origin therefore provides an elegant solution to the long-standing “missing xenon” paradox. It is demonstrated that such a model could explain the isotopic and elemental abundances of Ne, Ar, Kr, and Xe in the terrestrial atmosphere.     

The terrestrial cratering record: II. The crater production rate

The terrestrial cratering record for the Phanerozoic has a size-frequency distribution of NαD^?2.05 for D > 22.6 km and NαD^?0.24 for D < 11.3 km. This shallowing of the distribution slope at D > 22.6 km reflects the removal of small terrestrial craters by erosion. The number of large craters on the North American and East European cratons provide estimated terrestrial crater production rates for D > 20 km of 0.36 ± 0.1 and 0.33 ± 0.2 × 10^?14 km^?2 year^?1, respectively. These rates are in good agreement with previous estimates and astronomical observations on Apollo bodies. Comparisons with the lunar rate, taking account of the effects of variations in impact velocity, surface gravity, and gravitational cross section, indicate that the lunar and terrestrial rates overlap, if the cratering flux has been constant during the last 3.4 by. If the early (pre 4.0 by) high-flux rate did not decay to a constant value until 3.0 to 2.5 by then the rates differ by a factor of 2 and the Phanerozoic can be interpreted as a period of higher than normal cratering.     

First measurement of helium on Mars: implications for the problem of radiogenic gases on the terrestrial planets

108 +/- 11 photons of the martian He 584-angstroms airglow detected by the Extreme Ultraviolet Explorer satellite during a 2-day exposure (January 22-23, 1993) correspond to the effective disk average intensity of 43 +/- 10 Rayleigh. Radiative transfer calculations, using a model atmosphere appropriate to the conditions of the observation and having an exospheric temperature of 210 +/- 20 K, result in a He mixing ratio of 1.1 +/- 0.4 ppm in the lower atmosphere. Nonthermal escape of helium is due to electron impact ionization and pickup of He+ by the solar wind, to collisions with hot oxygen atoms, and to charge exchange with molecular species with corresponding column loss rates of 1.4 x 10(5), 3 x 10(4), and 7 x 10(3) cm-2 sec-1, respectively. The lifetime of helium on Mars is 5 x 10(4) years. The He outgassing rate, coupled with the 40Ar atmospheric abundance and with the K:U:Th ratio measured in the surface rocks, is used as input to a single two-reservoir degassing model which is applied to Mars and then to Venus. A similar model with known abundances of K, U, and Th is applied to Earth. The models for Earth and Mars presume loss of all argon accumulated in the atmospheres during the first billion years by large-scale meteorite and planetesimal impacts. The models show that the degassing coefficients for all three planets may be approximated by function delta = delta (0)(t(0)/t)1/2 with delta (0) = 0/1, 0.04, and 0.0125 Byr-1 for Earth, Venus, and Mars, respectively. After a R2 correction this means that outgassing processes on Venus and Mars are weaker than on Earth by factors of 3 and 30, respectively. Mass ratios of U and Th are almost the same for all three planets, while potassium is depleted by a factor of 2 in Venus and Mars. Mass ratios of helium and argon are close to 5 x 10(-9) and 2 x 10(-8) g/g in the interiors of all three planets. The implications of these results are discussed.     

The three-body problem

Book review

The three-body problemby Christian Marchal, Elsevier, Amsterdam, 1990, (ISBN 0444874402)     

The Trojan problem

There has been a renewed interest in the Trojan problem in recent years. Significant progress has been made in discovering and understanding dynamical features of motion of Jupiter's Trojan asteroids. The dynamics of hypothetical Trojan-type asteroids of other major planets has also been the subject of several recent investigations. This paper offers an overview on the current status of researches on real and hypothetical Trojan asteroids of the major planets. Results of analytical and numerical works are surveyed. Questions of dynamical properties, long-term evolution of orbits, stability regions around the triangular Lagrangian points are discussed among other problems of the Trojans.     

Dating terrestrial impact events

Abstract— Systematic examination of dating results from various craters indicates that about 90% of the rocks affected by an impact preserve their pre-shock ages because shock and post-shock conditions are not sufficient to disturb isotopic dating systems. In the other 10% of target lithologies, various geochronometers show significant shock-induced effects. Major problems in dating impactites are caused by their non-equlibrated character. They often display complex textures, where differently shocked and unshocked phases interfinger on the sub-mm scale. Due to this, dating on whole rock samples or insufficiently pure mineral fractions often yielded ambiguous results that set broad age limits but are not sufficient to answer reliably questions such as a possible periodicity in cratering on Earth, or correlation of impact events with mass extinctions. Dating results from shock recovery experiments indicate that post-shock annealing plays the most important role in resetting isotopic clocks. Therefore, the major criterion for sample selection in and around craters is the post-shock thermal regime. Based on their different thermal evolution, the following geological impact formations can be distinguished: (1) the coherent impact melt layer, (2) allochthonous breccia deposits, (3) the crater basement, and (4) distant ejecta deposits. Samples of the coherent impact melt layer are the most suitable candidates for dating. Excellent ages of high precision can be obtained by internal Rb-Sr, and Sm-Nd isochrons, U-Pb analyses on newly crystallized accessory minerals, and K-Ar (³⁹Ar-⁴⁰Ar) dating of clast-free melt rocks. Fission track counting on glassy material has yielded correct ages, and paleomagnetic measurements have been successfully applied to post-Triassic craters. In the ideal case of a fast-cooling impact melt layer, all these different techniques should give identical ages. Allochthonous breccias contain shocked, unshocked, and/or glassy components in various proportions; and, hence, each of these ejecta deposits has its own individual thermal history, making sample evaluation difficult Glassy melt particles in suevitic breccias are well suited for fission track and Ar-Ar dating. Weakly shocked material may yield reliable Ar-Ar and fission track ages, if formation temperatures were high, and cooling rates moderate. In contrast, highly shocked but rapidly cooled lithologies show only disturbed and not reset isotopic systems. For ejecta deposits and the crater wall of young craters, dating with cosmogenic nuclides is a new and powerful technique. Crater basement lithologies have a high potential in impact dating, although it has not been exploited so far. A prerequisite for resetting of isotopic clocks in these lithologies is the presence of an overlaying impact melt layer, which causes thermal metamorphism. Fission track and K-Ar techniques are most promising, because both systems are easily reset at low temperatures. Good candidates for impact dating are long-term annealed rocks, even if shock metamorphic overprint is very weak. In addition, Ar-Ar dating dating of pseudotachylites appears promising. In large impact structures, where high temperatures persist for long times, polymict “footwall” breccias beneath the melt sheet are also appropriate for dating, using the isochron approach and U-Pb on accessory minerals. Distant ejecta material have undergone very fast cooling, and the ejecta deposits have ambient formation temperatures. Among this material, tektites and impact melt glass are ideal objects for Ar-Ar and fission track impact dating. Dating on other material from distant ejecta deposits, such as U-Pb analyses on zircons, offers new possibilities. Efforts to correlate distant ejecta with distinct craters critically depend on proper error assignment to a specific age. This aspect is illustrated on the K/T boundary example.     

The stellar three-body problem

Two applications of von Zeipel's method to the stellar three-body problem eliminate the short period terms and establish two new integrals of the motion beyond the classical integrals. The remaining time averaged problem with only the second order Hamiltonian has one additional integral and can be solved. The motion with the third order averaged Hamiltonian included is more complex, in that there may be additional resonances, and the additional integral does not exist in all cases.     

The accelerated Kepler problem

The accelerated Kepler problem (AKP) is obtained by adding a constant acceleration to the classical two-body Kepler problem. This setting models the dynamics of a jet-sustaining accretion disk and its content of forming planets as the disk loses linear momentum through the asymmetric jet-counterjet system it powers. The dynamics of the accelerated Kepler problem is analyzed using physical as well as parabolic coordinates. The latter naturally separate the problem’s Hamiltonian into two unidimensional Hamiltonians. In particular, we identify the origin of the secular resonance in the AKP and determine analytically the radius of stability boundary of initially circular orbits that are of particular interest to the problem of radial migration in binary systems as well as to the truncation of accretion disks through stellar jet acceleration.     

The solar neutrino problem

The observed capture rate for solar neutrinos in the ³⁷Cl detector is lower than the predicted capture rate. This discrepancy between theory and observation is known as the ‘solar neutrino problem’. I review the basic elements in this problem: the detector efficiency, the theory of stellar (solar) evolution, the nuclear physics of energy generation, and the uncertainties in the predictions. I also answer the questions of: So What? and What Next?

     

The rectilinear three-body problem

The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities, escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches or a definite time. In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this periodic orbit. We have studied the phase space structure via Poincaré sections. Using these sections and symbolic dynamics, we study the fine structure of the region of initial conditions, in particular the chaotic scattering region.     

The formation of the atmospheres of the terrestrial planets by impact

It is generally supposed that the atmospheres of the terrestrial planets were formed by secondary degassing processes. We propose, instead, that they are of primary origin, forming as an immediate and necessary consequence of the final stages of planetary accretion. Once the planetary embryo reached a critical size, the impacting material began to vaporize. The atmosphere, so created, then decelerated other impacting material, thus limiting the rate of atmospheric growth. We show that, given reasonable assumptions concerning the chemical composition of the impacting material, an acceptable model for the early atmosphere of the Earth, and the present atmospheres of Venus and Mars results.A discussion of the noble gas data for the terrestrial atmosphere indicates that these can be readily reconciled with an impact origin.     

The origin and evolution of the atmospheres of the terrestrial planets

The chemical nature of the Earth's atmosphere is determined by its interaction with the biosphere, hydrosphere and lithosphere. Detailed balance is maintained over long time periods by a complex series of cyclical processes. The chemical differences between the atmosphere of the Earth, on the one hand, and the atmospheres of Venus and Mars, on the other, can be understood in terms of the greater complexity of the terrestrial interactions. When this has been taken into account, the origin of all three planetary atmospheres can be explained as resulting from degassing. Despite the similarity of the atmospheres of Venus and Mars, it seems necessary to invoke different mechanisms for the low amount of water vapour on each.