Engineering topics
Publications and source records attributed to Cassen, P..
We present 32 N-body simulations of planetary accretion in the inner Solar System, examining the effect of nebula surface density profile and initial eccentricities of Jupiter and Saturn on the compositions and orbits of the inner planets. Additional information is contained in the original extended abstract.
We develop models of the thermal evolution of the solar nebula designed to test the hypothesis that planetary objects began to form as the nebula cooled from an initially hot state, and compare the results of the models with meteoritic and planetary data.
Direct observations show that the deep Earth contains rare gases of solar composition distinct from those in the atmosphere. We examine the implications of mantle rare gas characteristics on acquisition of rare gases from the solar nebula and subsequent losses due to a large impact. Deep mantle rare gas concentrations and isotopic compositions can be obtained from a model of transport and distribution of mantle rare gases. This model assumes the lower mantle closed early, while the upper mantle is open to subduction from the atmosphere and mass transfer from the lower mantle. Constraints are derived that can be incorporated into models for terrestrial volatile acquisition: (1) Calculated lower-mantle Xe-isotopic ratios indicate that the fraction of radiogenic Xe produced by I-129 and Pu-244 during the first about 10(exp 8) yr was lost, a conclusion also drawn for atmospheric Xe. Thus, either the Earth was made from materials that had lost >99% of rare gases about (0.7-2) x 10(exp 8) yr after the solar system formed, or gases were then lost from the fully formed Earth. (2) Concentrations of 3He and 20Ne in the lower mantle were established after these losses. (3) Neon-isotopic data indicates that mantle Ne has solar composition. The model allows for solar Ar/Ne and Xe/Ne in the lower mantle if a dominant fraction of upper mantle Ar and Xe are subduction-derived. If Earth formed in the presence of the solar nebula, it could have been melted by accretional energy and the blanketing effect of a massive, nebula-derived atmosphere. Gases from this atmosphere would have been sequestered within the molten Earth by dissolution at the surface and downward mixing. It was found that too much Ne would be dissolved in the Earth unless the atmosphere began to escape when the Earth was only partially assembled. Here we consider conditions required to initially dissolve sufficient rare gases to account for the present lower mantle concentrations after subsequent losses at 10(exp 8) yr. It is assumed that equilibration of the atmosphere with a thoroughly molten mantle was rapid, so that initial abundances of gases retained in any mantle layer reflected surface conditions when the layer solidified. For subsequent gas loss of 99.5% and typical solubility coefficients, a total pressure of 100 atm was required for an atmosphere of solar composition. Calculations of the pressure at the base of a primordial atmosphere indicate that this value might be exceeded by an order of magnitude or more for an atmosphere supported by accretional energy. Surface temperatures of about 4000 K would have been produced, probably high enough to melt the deep mantle. Initial distributions of retained rare gases would then be determined by the history of surface pressure and temperature during mantle cooling and solidification, i.e., the coupled cooling of Earth and atmosphere. The Earth's thermal state was determined by its surface temperature and the efficiency of convection in the molten mantle, estimated to be sufficient to maintain an adiabatic gradient. Because the melting curve is steeper than the adiabat, solidification of the mantle proceeded outward from the interior. Incorporation of atmospheric gases in the mantle therefore occurred over a range in surface temperature of a few thousand degrees Kelvin. The thermal state of the atmosphere was controlled by total luminosity of the Earth (energy) released by accreting planetesimals and the cooling Earth), nebular temperature and pressure, and atmospheric opacity. The energy released by accretion declined with time as did nebular pressure. Analytical solutions for an idealized (constant opacity radiative atmosphere show that declining energy sources under constant nebular conditions result in slowly diminishing surface temperature but dramatically increasing surface pressure. For such an atmosphere with declining nebular pressure but constant total luminosity, surface pressure decreases gradually with decreasing temperaure. A decline in accretion luminosity might be compensated by energy released as the mantle cools for about 10(exp 5) year, after which luminosity must decline. The total complement of dissolved rare gases will depend on the particular evolutionary path determined by the declining accretional luminosity, the Earth thermal history, removal of the nebula, and opacity variations of the atmosphere. Models for these coupled evolutionary histories for Earth's acquisition of nebular-derived noble gases are in progress. The later losses required at about 10(exp 8) yr (depleting the interior concentrations of the sequestered solar gases by a factor of > 100) were presumably related to the major impact in which the Moon formed.
The presence of solar noble gases in the deep interior of the Earth is inferred from the Ne isotopic compositions of MORB (Mid-ocean Ridge Basalts) and OIB (Oceanic Island Basalt); Ar data may also consistent with a solar component in the deep mantle. Models of the transport and distribution of noble gases in the earth's mantle allow for the presence of solar Ar/Ne and Xe/Ne ratios and permit the calculation of lower mantle noble gas concentrations. These mantle data and models also indicate that the Earth suffered early (0.7 to 2 x 10(exp 8) yr) and large (greater than 99 percent) losses of noble gases from the interior, a result previously concluded for atmospheric Xe. We have pursued the suggestion that solar noble gases were incorporated in the forming Earth from a massive, nebula-derived atmosphere which promoted large-scale melting, so that gases from this atmosphere dissolved in the magma ocean and were mixed downward. Models of a primitive atmosphere captured from the solar nebula and supported by accretion luminosity indicate that pressures at the Earth's surface were adequate (and largely more than the required 100 Atm) to dissolve sufficient gases. We have calculated the coupled evolution of the magma ocean and the overlying atmosphere under conditions corresponding to the cessation (or severe attenuation) of the sustaining accretion luminosity, prior to the complete removal of the solar nebula. Such a condition was likely to obtain, for instance, when most of the unaccumulated mass resided in large bodies which were only sporadically accreted. The luminosity supporting the atmosphere is then that provided by the cooling Earth, consideration of which sets a lower limit to the time required to solidify the mantle and terminate the incorporation of atmospheric gases within it. In our initial calculations, we have fixed the nebula temperature at To = 300K, a value likely to be appropriate for nebular temperatures at lAU in the early planet-building epoch. We treated the background (nebula) pressure as an adjustable, time-dependent parameter. Additional information is contained within the original extended abstract.
Observations of T Tauri stars and their circumstellar disks, combined with broadly applied theoretical principles, permit the construction of solar nebular models that are potentially useful for the interpretation of meteoritic data. Quantitative comparisons of model results with meteoritic and planetary data is used to test several propositions regarding the bulk properties of chondrites and chondrule precursor material.
The "moderately volatile" elements are those which condense (or evaporate) in the temperature range 650 - 1350 K, as a mix of material with solar abundances is cooled (or heated) under equilibrium conditions. Their relative abundances in chondritic meteorites are solar (or "cosmic", as defined by tile composition of CI meteorites) to within a factor of several, but vary within that range in a way that correlates remarkably well with condensation temperature, independent of chemical affinity. It has been argued that this correlation reflects a systematically selective process which favored the accretion of refractory material over volatile material from a cooling nebula. Wasson and Chou suggested that condensation and settling of solids contemporaneously with the cooling and removal of nebular gas could produce tile observed abundance patterns, but a quantitative model has been lacking. We show that the abundance patterns of the moderately volatile elements in chondritic meteorites can be produced, in some degree of quantitative detail, by models of the solar nebula that are designed to conform to observations of T Tauri stars and the global conservation laws. For example, even if the local surface density of the nebula is not decreasing, condensation and accretion of solids from radially inflowing gas in a cooling nebula can result in depletions of volatiles, relative to refractories, like those observed. The details of the calculated abundance patterns depend on (but are not especially sensitive to) model parameters, and can exhibit the variations that distinguish the meteorite classes. Thus it appears that nebula characteristics Such as cooling rates, radial flow velocities, and particle accumulation rates can be quantitatively constrained by demanding that they conform to meteoritic data; and the models, in turn, can produce testable hypotheses regarding the time and location of the formation of the chondrite parent bodies and the planets.
Results of calculations performed to determine the conditions necessary for producing the opaque rims on chondrules and CAI rims by high-speed entry into the transient atmosphere of an accreting meteorite parent body are presented. The sensitivity of these results to variations in critical parameters is investigated. The range of entry velocities which can produce such rims is shown to depend on the size, melting temperature, and thermal conductivity of the particles. For particles greater than 2 mm in radius, with thermal conductivities of 20,000 ergs/sm s K or lower, entry velocities of about 3 km/s suffice. For particle sizes less than 1 mm in radius, the range of encounter velocities that can produce rims is narrow or vanishing, regardless of the thermal conductivity, unless the melting temperature in the outer part of the chondrule has been reduced by compositional heterogeneity.
The radial distributions of the effective temperatures of circumstellar disks associated with pre-main sequence (T Tauri) stars are relatively well-constrained by ground-based and spacecraft infrared photometry and radio continuum observations. If the mechanisms by which energy is transported vertically in the disks are understood, these data can be used to constrain models of the thermal structure and evolution of solar nebula. Several studies of the evolution of the solar nebula have included the calculation of the vertical transport of heat by convection. Such calculations rely on a mixing length theory of transport and some assumption regarding the vertical distribution of internal dissipation. In all cases, the results of these calculations indicate that transport by radiation dominates that by convection, even when the nebula is convectively unstable. A simple argument that demonstrates the generality (and limits) of this result, regardless of the details of mixing length theory or the precise distribution of internal heating is presented. It is based on the idea that the radiative gradient in an optically thick nebula generally does not greatly exceed the adiabatic gradient.
The ability of high-speed impacts into a parent body regolith to account for the textures and compositions observed for rims on some chondrules is presently supported by both observational and experimental data. Experimental results obtained with a light-gas gun encompass recovered-grain characteristics ranging from fragmented projectile grains mixed with melted aerogel, demonstrating that rim-like thermal and mechanical alterations of projectiles may be due to a high-velocity encounter with a low density target. Irrespective of conclusions for other types of rims, granular and opaque rim characteristics seem inconsistent with nebular condensation.
One-dimensional, spherically symmetric hydrodynamic calculations of the effect of the impact of a 1-2 earth mass body on the gaseous envelope of Uranus, characterizing the impact as a partition of energy such that a fraction is deposited directly into the envelope, while the rest heats the core. A sharp transition is found between the cases of nearly-complete retention and dispersal, corresponding to energy-deposition values above or below a critical value. Within a realistic parameter range, the angular momentum of the Uranus system may be explained, together with the retention of a gaseous envelope of about 1 earth mass.
Differentially rotating disks of gases and solids occur in several astrophysical systems, in particular in the inner parts of protostellar nebulae, of which our own solar system is thought to be a relic. The objectives of this paper are to: (1) study localized turbulence in circumstances approximating those found in accretion disks using previously existing expertise in performing direct numerical simulation of turbulent, incompressible channel flows with low Reynolds numbers; (2) determine the limitations of such calculations; and (3) extend the type of numerical simulation (e.g., to include density and stratification and compressibility effects and to accommodate higher Reynolds numbers with sub-grid scale modeling) so that the relevant physical effects are realistically captured.
Turbulent convection may have played a major role in determining the structure and evolution of the primordial solar nebula, but current, incomplete models of convection and turbulence give very different results and remain largely untested in the absence of detailed astronomical observations. Numerical simulations provide an 'experimental' database for comparison with these models, and, to this end, direct numerical simulations of turbulent convection were performed with modifications intended to mimic some of the unique physical features of thin accretion disks, such as the primordial solar nebula: internal heating, a gravitational acceleration that is linearly proportional to the distance from midplane of the nebula, and rapid rotation. Peclet numbers in the simulations are comparable to those in solar nebula models; Rossby numbers in the simulations are an order of magnitude larger than those in solar nebula models because of the unrealistically high Prandtl and low, Reynolds numbers required to resolve all scales of the convective flow. We find that, despite the loss of buoyancy at midplane, turbulent motions easily penetrate the midplane region with little loss of intensity, providing efficient transport of heat and turbulent kinetic energy throughout the interior. A simple mixing length model modified to include rotation is found to give convective heat fluxes for the interior flow in rough agreement with the numerical simulations. We discuss these preliminary results with regard to assumptions about heating distributions and convective heat fluxes made in standard solar nebula models. More definitive comparisons with solar nebula modelling will become possible when more realistic effects of shear, density, stratification, and compression are included.
The application of rotational stability criteria to a specific model of star formation leads to the conclusion that the growth of stellar angular momentum is limited by its transfer to the disk. Excess accreted angular momentum can be transferred by torques connected with spiral density waves induced by even a slight protostellar triaxiality. In addition, viscous damping of the density waves is likely to cause the excess angular momentum to be deposited within a small region close to the protostar. Thus, it would be appropriate to treat that part of the growing protostellar disk beyond the outer Lindblad resonance as an accretion disk with a torque applied to its inner edge. It is noted that this situation is directly relevant to certain models of the evolution of the protosun and solar nebula.
Solutions that describe the collapse of a molecular cloud core that is initially in unstable equilibrium, embedded within an envelope of uniform density, and rotating at the same rate as the envelope are given. Hydrodynamic equations, including self gravity, are deduced to a set of ordinary differential equations, which are solved by the method of matched asymptotic expansions. Results of these calculations are: (1) the range of stellar masses derived seems to correspond to realistic ranges of observed stellar masses and interstellar cloud parameters, (2) the proper measure of dissipation rate is the ratio of accretion time to viscous diffusion time, and (3) the pressure distribution on the surface of an accreting protostar is nonuniform in a way that favors the channeling of a stellar wind into a bipolar flow directed along the rotation axis.
The status of theoretical work on protostellar disks is reviewed. Accretion disk theory and its application to models of the solar nebula and protostellar disks are discussed. A unified view of the process of star formation is presented, starting from the evolution of molecular clouds, and leading naturally to the formation of protostellar disks. The models used to describe this process are idealized, but are believed to provide good prototypes that well represent the essential hydromagnetic phenomena involved in star and disk formation. Several possible evolutionary paths and final configurations are qualitatively discussed, showing how the outcomes depend on the relative efficiencies of various angular momentum transport processes.
A generalized model which accounts for the effects of initially uniform and slow rotation is defined for the spherical collapse of a singular isothermal sphere such as protosolar and binary nebulae. An initial unstable equilibrium state is described for a sound speed of 0.35 km/sec and a rotation rate of 10 to the -14th/sec for the molecular cloud surrounding the accreting core. The total angular momentum and mass of the inner cloud is set equal to solar system values. The evolution of the collapse is traced by applying a perturbation analysis to the similarity solution for a nonrotating condition, and matched asymptotic expansions solve the hydrodynamic equations. The model is concluded a valid tool for studying star and nebular disk formation.
Evaluation of earlier observations indicated that layered rims on coarse-grained Allende CAI's were possibly the result of partial melting by ablation/drag-heating and reaction of CAI exteriors with a gas or gases of non-solar composition. Bunch and Chang reported the common occurrence of thin, fine-grained, matrix-like bands that at least partially surround rims of CAI's. Although material in these bands in general appears to be similar to matrix, SEM observations show them to be dissimilar in volatile element content, mineral composition, and grain morphology. Moreover, they appear to be related in time of formation with rim development and Na-metasomatism of CAI's. Observations indicate a short-lived but intense heating episode followed by radid cooling as the mechanism responsible for these CAI features.
Protostellar cloud collapse and solar nebula formation models indicate that the size of the nebula produced will be larger in terms of both gas centrifugal balance R(CF) and collapse time diffusion length R(V). From this, it can be deduced that low mass nebulas are produced if (R(V)/R(CF))-squared is much greater than unity, while nebulas result for values lower than approximately unity. The total angular momentum value distinguishes most current models of the solar nebula. Analytic expressions for the surface density, nebular mass flux and photospheric temperature distributions during the formation stage are presented for simple modes illustrating and general properties of growing protostellar disks.