Accretion of the terrestrial planets. II

Some aspects and consequences of the theory of gravitational accretion of the terrestrial planets are examined. The concept of a “closed feeding zone” is somewhat unrealistic, but provides a lower bound on the accretion time. Safronov's relative velocity relation for planetesimals is not entirely consistent with the feeding zone model. A velocity relation which includes an initial velocity component is suggested. The orbital parameters of the planetesimals and the dimensions of the feeding zone are related to their relative velocities. The assumption of an initial velocity does not seriously change the accretion time.Mercury, Venus, and the Earth have accretion times on the order of 10⁸yr. Mars requires well over 10⁹yr to accrete by the same assumptions. Currently available data do not rule out a late formation of Mars, but the lunar cratering history makes it unlikely. If Mars is as old as the Earth, nongravitational forces or a violation of the feeding zone concept is required. One such possibility is the removal of matter from the zone of Mars by Jupiter's influence. The final sweeping up by Mars after this event would result in the scattering of a considerable mass among the other terrestrial planets. The late postaccretional bombardments infrerred for the Moon and Mercury may have had this source.     

Heat deposition and retention in a solid planet growing by impacts

This paper considers the scaling of impact effects with impactor size and velocity (or planetary radius) and the retention of heat deposited by impacts in a solid planet (i.e., with no convective motions). Some previously used scalings are inconsistent with the general scaling rules of Holsapple and Schmidt (1982), and no study of impact heating has considered the full permissible range of scalings. A sample physical impact model which spans this range is presented. There are three length scales which control impact heat retention: the depth scales of heat deposition and impact stirring and the ratio κ/ν, where κ is the thermal diffusivity and ν is the upward velocity of the accreting surface. These are evaluated in the contexts of the general scaling rules and Safronov's (1972) distribution of impactor sizes. It is found that the effeciency of heat retention (i.e., the fraction of deposited heat which is retained) is independent of the planetary growth rate. It may be low at small planetary radii, but tends to level out around 3000 km radius to values of 40–70%. Combined with an assumed heat deposition effeciency of 20%, this gives melting at a radius between about 2000 and 3000 km in terrestrial planets.     

Voyager U.V. spectrometer observations of He 584 Å dayglow at Jupiter

The intensity of Jupiter's He 584 Å airglow has been measured by the Voyager U.V. spectrometers. The disc-averaged brightness is about 4 Rs and limb darkening is present. The intensity probably varies with longitude, the variation being out of phase with the H Lyman-α intensity bulge. Modelling of resonance scattering of the solar He 584 Å line by Jupiter's atmosphere has shown that the hydrogen and helium emissions can be explained about equally well by at least two self-consistent scenarios involving the structure (temperature and eddy diffusion coefficient) and excitation of the atmosphere. All our evidence points to a dramatic change of conditions in the Jovian atmosphere in the time between Pioneer and Voyager encounters.     

Particle flux associated with stochastic processes

The Lagrange expansion, which may be used to derive the Fokker-Planck equation, is here used to derive the corresponding expression for the flux of particles subject to a stochastic scattering process. The coefficients which occur in this expression are, in general, not the same as the coefficients which occur in the Fokker-Planck equation itself. In the special case that the particle distribution involves only one independent variable, the particle flux is determined by the familiar Fokker-Planck coefficients. Evaluation of particle flux is of special interest in the study of stochastic acceleration.     

The Fokker-Planck equation for charged particles moving in random electromagnetic fields

The Fokker-Planck equation which describes the motion of charged particles in a random electromagnetic field is derived from the Liouville equation by a new method. The size of the perturbing magnetic field, for the Fokker-Planck equation to be valid, is calculated in a regime appropriate for cosmic-ray diffusion.     

Second-order solution of Lindstedt's equation with constant coefficients

The Hori-Lie transformation for a non-conservative system is applied to the Lindstedt's equation with constant coefficients. A second-order solution when the right hand is a quartic polynomial is derived explicity. We made two applications of our solution. We obtained a new form of the trajectorv of a test particle moving in a Schwarzschild field. The radius of the particle is a periodic funciton of the polar angle with a period slightly different from 2π. The deviation is the relativistic precession. We also considered the solution of the coordinates ρ and η in Vinti's problem containing J₃. They are expressed as periodic functions of O'Mathuna's regularization argument.     

A general cratering-history model and its implications for the lunar highlands

Through analysis of a large number of Monte Carlo and Markov Chain simulations, a model for determining crater accumulation and crater obliteration histories has been derived. The model generally applies to populations of large craters. It predicts that the following relationships hold for subequilibrium-density crater populations: (1) the more negative the production function's exponent, α, (N～D^α) the lower the crater density at which the population size-frequency distribution will significantly depart from its production function; (2) the more negative the production function's exponent, the less obliteration a crater population will sustain after a set number of impacts. Application of the model to the lunar highlands implies (1) the production function for the large craters is highly structured, resembling the observed size-frequency distribution and not the function N～D^?2; (2) even the densely cratered highlands have not attained crater saturation or equilibrium. Direct simulations of the highlands' crater population supports the model's implications.     

The role of the critical ionization velocity phenomena in the production of inner coma cometary plasma

The critical velocity triggering anomalous ionization of the neutral gas by plasma flow is calculated for the model based on the lower hybride instability. It depends strongly on the plasma and gas parameters, denning the instability development of the ionized atoms beam in the counter-streaming plasma.In particular, the possible role of the critical ionization mechanism for Halley's comet is examined. The fulfilment of both Townsend's condition for the selfustained beam plasma discharge and Alfven's condition for the critical velocity mechanism indicates that this mechanism may operate only within 10⁴ km from the cometary nucleus and give an ion production rate close to that observed for Kohoutek's comet.     

Motion in the interiors and atmospheres of Jupiter and Saturn: scale analysis,anelastic equations,barotropic stability criterion

If Jupiter's and Saturn's fluid interiors were inviscid and adiabatic, any steady zonal motion would take the form of differentially rotating cylinders concentric about the planetary axis of rotation. B. A. Smith et al. [Science215, 504–537 (1982)] showed that Saturn's observed zonal wind profile extends a significant distance below cloud base. Further extension into the interior occurs if the values of the eddy viscosity and superadiabaticity are small. We estimate these values using a scaling analysis of deep convection in the presence of differential rotation. The differential rotation inhibits the convection and reduces the effective eddy viscosity. Viscous dissipation of zonal mean kinetic energy is then within the bounds set by the internal heat source. The differential rotation increases the superadiabaticity, but not so much as to eliminate the cylindrical structure of the flow. Very large departures from adiabaticity, necessary for decoupling the atmosphere and interior, do not occur. Using our scaling analysis we develop the anelastic equations that describe motions in Jupiter's and Saturn's interiors. A simple problem is solved, that of an adiabatic fluid with a steady zonal wind varying as a function of cylindrical radius. Low zonal wavenumber perturbations are two dimensional (independent of the axial coordinate) and obey a modified barotropic stability equation. The parameter analogous to β is negative and is three to four times larger than the β for thin atmospheres. Jupiter's and Saturn's observed zonal wind profiles are close to marginal stability according to this deep sphere criterion, but are several times supercritical according to the thin atmosphere criterion.     

Derivation of a conservation equation for mean molecular weight for a two-constituent gas within a three-dimensional,time-dependent model of the thermosphere

Systematic circulation systems within the thermosphere create major departures of composition of both major and minor species from diffusive equilibrium. For example, latitudinal gradients in the mixing ratios of major and minor species in recent empirical models of the Earth's thermosphere are inconsistent with changes of the thermal structure alone or with temporal or spatial changes of the turbopause altitude. A conservation equation describing the time rate of change of mean molecular weight is derived for a two-species gas, in the presence of molecular and turbulent diffusion and general global circulation. The equation is fully three-dimensional and time-dependent and is derived from a combination of the general diffusion equation and the time-dependent continuity equation. In the Earth's thermosphere, the two species are [O] the light species and [N₂,O₂] the heavy species and the approach is valid since the time constants of dissociation of [O₂] and recombination of [O] are long compared with typical dynamical time constants. One of the major effects of allowing a wind-driven departure from diffusive equilibrium is that, at the solstice, the pole to pole exospheric temperature difference is increased by more than 50%, while the prevailing summer to winter meridional wind actually decreases. A conservation equation of this kind has general application to any planetary atmosphere which may be considered to be predominantly comprised of two species. Results for a three-dimensional, time-dependent thermospheric model for solstice conditions are presented for the conditions of solar heating only. The model results are compared with previous model results with composition fixed at pressure levels and with empirical temperature and composition models of MSIS.     

Stochastic coalescence model for terrestrial planetary accretion

A nonequilibrium stochastic coalescence model for terrestrial planetary accretion is developed by using an approximation to the Safronov-Golovin solution for the scalar transport equation with linear kernel. According to this model, formation of comparatively massive objects occurs quite rapidly during the early stages of accretive evolution in a given terrestrial planetesimal population, while during late growth stages, an increasingly substantial fraction of total population mass becomes incorporated into progressively fewer, relatively very large bodies. The model also implies that the (conservative) growth rate of the population's largest member varies directly as its mass, and further suggests that this growth rate may not decline significantly until very nearly final planetary mass is attained.     

Chemistry and evolution of Titan's atmosphere

The chemistry and evolution of Titan's atmosphere is reviewed in the light of the scientific findings from the Voyager mission. It is argued that the present N₂ atmosphere may be Titan's initial atmosphere rather than photochemically derived from an original NH₃ atmosphere. The escape rate of hydrogen from Titan is controlled by photochemical production from hydrocarbons. CH₄ is irreversibly converted to less hydrogen rich hydrocarbons, which over geologic time accumulate on the surface to a layer thickness of ～0.5 km. Magnetospheric electrons interacting with Titan's exosphere may dissociate enough N₂ into hot, escaping N atoms to remove ～0.2 of Titan's present atmosphere over geologic time. The energy dissipation of magnetospheric electrons exceeds solar e.u.v. energy deposition in Titan's atmosphere by an order of magnitude and is the principal driver of nitrogen photochemistry. The environmental conditions in Titan's upper atmosphere are favorable to building up complex molecules, particularly in the north polar cap region.     

Coulomb collisional processes in space plasmas; relaxation of suprathermal particle distributions

Nonequilibrium distributions of space plasmas are often characterized by extended high-energy tails. This paper provides a detailed analysis of the relaxation of such isotropic nonequilibrium plasmas. We consider an energetic charged species dilutely dispersed in a fully ionized plasma, which acts as a heat bath at equilibrium. The minor constituent is referred to as a “test particle” and collisions between test particles are not included. We study the approach to equilibrium with a finite difference method of solution of the Fokker-Planck equation appropriate for collisions between charged particles. The solution of the Fokker-Planck equation is also presented formally as an expansion in the eigenfunctions of the Fokker-Planck operator. The main objective of the paper is the calculation of the energy-dependent relaxation times of the distribution function. A strong energy dependence for these relaxation times is anticipated since, for Coulomb collisions, the Rutherford cross-section varies with relative speed g as g⁻⁴. Analogous results for neutral species are presented for comparison in the following paper.     

A constraint on the distribution of Titan's atmospheric aerosol

The physics of aerosols in Titan's atmosphere is examined, and a simple relatioshi particle size, optical depth, and aerosol replacement rate is derived. The consequences for current models of Titan's aerosol distribution are discussed.     

The Dynamics of Cometary Orbits in Close Encounters with Jupiter. An Analysis of Encounter Durations

The duration of the encounters is analyzed. The encounters studied are divided into low- and high-velocity encounters according to the duration of the comet's presence in the vicinity of Jupiter. Examples of low-velocity (Russel 3, 1931–1941) and high-velocity (Honda–Mrkos–Paidushakova, 1935) encounters are given. The main characteristics of high- and low-velocity encounters are described.     

Spectral evolution of microwaves and hard X-rays in the 1989 March 18 flare and its interpretation

We analyze the time variation of microwave spectra and hard X-ray spectra of 1989 March 18, which are obtained from the Solar Array at the Owens Valley Radio Observatory (OVRO) and the Hard X-Ray Burst Spectrometer (HXRBS) on the Solar Maximum Mission (SMM), respectively. From this observation, it is noted that the hard X-ray spectra gradually soften over 50–200 keV on-and-after the maximum phase while the microwaves at 1–15 GHz show neither a change in spectral shape nor as rapid a decay as hard X-rays. This leads to decoupling of hard X-rays from the microwaves in the decay phase away from their good correlation seen in the initial rise phase. To interpret this observation, we adopt a view that microwave-emitting particles and hard X-ray particles are physically separated in an inhomogeneous magnetic loop, but linked via interactions with the Whistler waves generated during flares. From this viewpoint, it is argued that the observed decoupling of microwaves from hard X-rays may be due to the different ability of each source region to maintain high energy electrons in response to the Whistler waves passing through the entire loop. To demonstrate this possibility, we solve a Fokker-Planck equation that describes evolution of electrons interacting with the Whistler waves, taking into account the variation of Fokker-Planck coefficients with physical quantities of the background medium. The numerical Fokker-Planck solutions are then used to calculate microwave spectra and hard X-ray spectra for agreement with observations. Our model results are as follows: in a stronger field region, the energy loss by electron escape due to scattering by the waves is greatly enhanced resulting in steep particle distributions that reproduce the observed hard X-ray spectra. In a region with weaker fields and lower density, this loss term is reduced allowing high energy electrons to survive longer so that microwaves can be emitted there in excess of hard X-rays during the decay phase of the flare. Our results based on spectral fitting of a flare event are discussed in comparison with previous studies of microwaves and hard X-rays based on either temporal or spatial information.     

Statistical approach to the theory of circumstellar discs

The equations for the viscous motion of a mixture of gas and dust in a gravitational field are derived from the statistics of particle orbits and radiative processes in a general form which gives the Navier-Stokes equation as a special case. Diffusion, partially elastic collisions and — for larger bodies — the gravitational encounters are included. The results are applied to the evolution of circumstellar discs.     

Large seasonal variations in Triton's atmosphere

Triton's seasons differ materially from those of Pluto owing to four important differences in the governing physics: First, the obliquity of Triton is significantly less than Pluto's obliquity. Second, Triton's inclined orbit precesses rapidly about Neptune so that a complicated seasonal variation in the latitude of the Sun occurs for Triton. Third, Neptune's orbit is much more circular than Pluto's orbit so that the sunlight intercepted by Triton's disk does not vary seasonally. Finally, Triton's atmosphere cannot be saturated at the lower latitudes so that the mass of the atmosphere is controlled by the temperature of the high-latitude ices or liquids (polar caps), as for CO₂ on Mars. The consequences of Triton's entire surface being covered with volatile substances have been examined. It is found that the circularity of Neptune's orbit then implies that Triton would have hardly any seasonal variation at all in surface temperature or atmospheric bulk, in spite of the complicated precessional effects of Triton's orbit. The only seasonal effect would be the migration of surface ices and liquids. This scenario is ruled out because it implies a column CH₄ abundance much higher than that observed and because it quickly depletes the lower latitudes of volatiles. It is concluded that Triton's most volatile surface substances are probably relegated to latitudes higher than 35° and probably form polar caps. The temperature of the polar caps should be nearly equal, even during midwinter/midsummer when the insolation of the summer pole is greatest. If the summer pole completely sublimates during one of the “major” summers, Triton's atmosphere may begin to freeze out over the winter caps. It is therefore expected that Triton's atmosphere undergoes large and complex seasonal variations. Triton is currently approaching a “maximum southern summer”, and over the remainder of this century, a dramatic increase in CH₄ abundance above the current upper limit of 1 m-Am may be witnessed.     

Late stages of planetary accretion

We derive first-order differential equations for the late stages of planetary accretion (planetesimal mass >10¹³ g). The effect of gravitational encounters, energy exchange, collisions, and gas drag has been included. Two simple models are discussed, namely, (i) when all planetesimals have the same mass and (ii) when there is one large planetesimal and numerous small planetesmals. Gravitational two-body encounters are modeled according to Chandrasekhar's classical theory from stellar dynamics. It is shown that the velocity increase due to mutual encounters can be modeled according to the simple theory of random flights. We find analytical equations for the average velocity decrease due to collisions. Gas drag, if present, is modeled in averaged form up to the first order in the eccentricities and inclinations of the planetesimals. Characteristic time scales for the formation of the terrestrial planets are found for the most favorable models to be of order 10⁸ year. The calculated mass of rock and ice of the giant planets is too low as compared to the observed one. This difficulty of our model could be overcome by assuming a several times larger surface density, an enlarged accretion cross section, and gas accretion during the final stages of accretion of the solid cores of the giant planets. Analytical and numerical results are presebted, the evolutionary tracks showing satisfactory agreement with observations for some models.     

The Fokfer-Planck equapions for magnetic monopoles and charged particles

We investigate the Fokker-Planck equations (non-relativistic case) for the interaction between magnetic monopoles and charged particles. We find that, the collision integral in this case is also logarithmically divergent, showing that the main effect is still produced by far encounters. We give, for thermal equilibrium, expressions for the three relaxation times. Compared with the well-known expressions for charged particles, these are generally amplified by a factor (light velocity/thermal velocity)², showing that, under ordinary conditions, the interaction between monopoles and charged particles is negligible but that, under certain astrophysical conditions, it is not. In the latter case, the MHD equations must be modified accordingly and the modified equations are given.