Convection driven by tidal and radiogenic heating in medium size icy satellites

Convection is one of the most important processes responsible for the formation of the surface features on many planetary bodies. Observations of some icy satellites indicate that the satellites’ surfaces are modified due to the internally driven tectonic activity. The tidal heating could be an important source of energy responsible for such internal activity. This suggestion is supported by the correlation of the tidal parameter ψ and tectonic features. Consequently, the tidal and the radiogenic heat sources seem to be of primary importance for the medium size icy satellites. Our research deals with convection in a non-differentiated body. The convection is a results of both uniform radiogenic heating and non-uniform and non-spherically symmetric tidal heating. To investigate the problem a 3D model of convection is developed based on the Navier-Stokes equation, the equation of thermal conductivity, the equation of continuity, and the equation of state. The 3D formulae for the tidal heat generation based on the results of Peale and Cassen [1978. Icarus 36, 245-269] and others are used in the model. To measure the relative importance of radiogenic heating versus tidal heating a dimensionless number C_t is introduced. The systematic investigation of a steady-state convection is performed for different values of the Rayleigh number and for the full range of C_t. The results indicate that for low and moderate value of the Rayleigh number, convection pattern driven by the tidal heating and by the radioactivity in the medium size icy satellites consists of one cell or of two cells. For C_t>0 the critical value of Rayleigh number Ra_cr=0. The one-cell pattern is specific for low Rayleigh numbers but it could be observed for the full range of number C_t. It means that the pattern of convection does not fully follow the pattern of heating. This rather unexpected result could be of great importance for the final stage of convection. All patterns of tidally driven convection are oriented with respect to the direction to the planet. For two-cell patterns the regions of downward motion are situated in the centers of the near and far sides of the satellite, respectively.     

Tidally Induced Volcanism

The dissipation of tidal energy causes the ongoing silicate volcanism on Jupiter's satellite, Io, and cryovolcanism almost certainly has resurfaced parts of Saturn's satellite, Enceladus, at various epochs distributed over the latter's history. The maintenance of tidal dissipation in Io and the occurrence of the same on Enceladus depends crucially on the maintenance of the respective orbital eccentricities by the existence of mean motion resonances with nearby satellites. A formation of the resonances among the Galilean satellites by differential expansion of the satellite orbits from tides raised on Jupiter by the satellites means the onset of the volcanism on Io could be relatively recent. If, on the other hand, the resonances formed by differential migration from resonant interactions of the satellites with the disk of gas and particles from which they formed, Io would have been at least intermittently volcanically active throughout its history. Either means of assembling the Galilean satellite resonances lead to the same constraint on the dissipation function of Jupiter Q _J 10⁶, where the currently high heat flux from Io seems to favor episodic heating as Io's eccentricity periodically increases and decreases. Either of the two models might account for sufficient tidal dissipation in the icy satellite Enceladus to cause at least occasional cryovolcanism over much of its history. However, both models are assumption-dependent and not secure, so uncertainty remains on how tidal dissipation resurfaced Enceladus.     

Saturn's icy satellites: Thermal and structural models

Thermal histories of the small icy Saturnian satellites Mimas, Tethys, Dione, Rhea, and Iapetus are constructed by assuming that they formed as homogeneous ice-silicate mixtures. The models include effects of radiogenic and accretional heating, conductive and subsolidus convective heat transfer, and lithospheric growth. Accretional heating is unlikely to have melted the water ice in the interiors of these bodies and solid state creep of the predominately ice material precludes melting by radiogenic heating. Mimas is so small that its thermal evolution is essentially purely conductive; at present it is a cold, nearly isothermal body. Any subsolidus convection or thermal activity in Mimas would have been confined to a brief period in its early history and would have been due to a warm formation. The four largest satellites are big enough and contain sufficient heat-producing silicates that solid state convection beneath a rigid lithosphere is inevitable independent of initial conditions. Dione and Rhea have convective interiors for most of their thermal histories, while Tethys and Iapetus have mainly conductive thermal histories with early periods of convective 0activity. The thermal histories of the five satellites for the last 4 by are independent of initial conditions; at present they have cold, conductive interiors. The model thermal histories are qualitatively consistent with the appearances of these satellites: Mimas has an ancient heavily cratered surface, Tethys and probably Iapetus have both heavily cratered and more lightly cratered areas, and Dione and Rhea have extensively modified surfaces. Because of their similar sizes and densities, Mimas and Enceladus are expected to have similar surfaces and thermal histories, but instead Enceladus has the most modified surface of all the small icy Saturnian satellites. Our results suggest a heat source for Enceladus, in addition to radiogenic and accretional heating; tidal dissipation is a possibility. Because the water ice in these bodies does not melt, resurfacing must be accomplished by the melting of a low-melting-temperature minor component such as ammonia hydrate.     

Convection zone of white-dwarf stars

The boundary convection zones of hot helium white-dwarf stars (WDSs) in the range 17000 KT _e30000 K are studied. Recently, an anisotropic mixing-length theory (AMLT) which determines the mixing-length parameter locally is applied for the convection zones calculation. Comparing with the calculations by using the (MLT), it is found that maximum velocity decreases appreciably, and the other boundary conditions are affected.     

The formation of condensed carbon particles in the atmospheres of white dwarfs

The existence of condensed carbon in the form of liquid droplets and graphite grains is found in white dwarf atmospheres with parametersg=10⁸ cm s^–2, H/He10^–3, andT _eff6000 K on the basis of model atmospheres techniques. It is shown that the condensation layers are dynamically stable and, consequently, that white dwarfs cannot supply the condensed particles to the interstellar medium. Possible observable effects are considered.     

Emden-chandrasekhar axisymmetric,rigidly rotating polytropes

An approximate analytical method of solving the polytropic equilibrium equations, first developed by Seidov and Kuzakhmedov (1978), has been extended and generalized to equilibrium configurations of axisymmetric systems in rigid rotation, with polytropic index,n =n _p + _n , nearn _p =0, 1, and 5. Though the details of the method depend on the value ofn _p , acceptable results are obtained for | _n | 0.5 to describe slowly rotating configurations in the range 0n1.5, 4.5n5. In the limit of rotational equilibrium configurations, when the distorsion may be large enough, a satisfactory approximation holds only in the range 0n, 1n1.5, 4.5n5.     

Satellite theory

In this paper dynamical characteristics of satellites are outlined by classifying the satellites into three categories according to the values of the solar tidal factor (n/n)² which is the disturbing factor due to the sun and the oblateness factor of the primary planetJ ₂/a ². For inner satellites (n/n)² is much smaller thanJ ₂/a ² and there are several pairs among them, for which the mean motions are commensurable to each other, and for some of them secular accelerations in the mean longitudes have been detected. For outer satellites (n/n)² is much larger and the solar perturbations are dominant. For intermediary satellites the motion of the pole of the orbital plane is not so simple as those of the satellites of the other categories.     

The interpretation of flows in molecular clouds

Stellar winds interacting with gas in dense molecular clouds produce flows which may be initially energy or momentum driven. A criterion for this is derived which depends sensitively on the wind velocity. Flows may change from one regime to another depending on the gas distribution about the wind source and these changes are discussed for power law density distributions. In general, the flows observed in CO associated with infrared point sources seem to be in the energy driven regime. By combining CO observations with radio continuum flux measurements, wind parameters are derived for several of these sources. There is some evidence from the derived parameters that high (L _*2×10³ L ) luminosity sources have radiatively-driven winds. Lower luminosity source winds are driven by some agency as yet unknow. We suggest that the widths of infrared lines from wind sources seriously underestimate the wind terminal velocities.     

Dynamical behavior of nuclear reaction systems disturbed by fluid motion in the solar core

Using Euler's equation of motion, the equation for disturbed fluid motion against a hydrostatic equilibrium has been derived, and the nonequilibrium dynamical equation of a P-PI nuclear reaction system driven by He³ has been analysed using developed nonequilibrium theory. We find that the system in the solar core is unstable in the layer extending from about 0.2R to 0.4R if the core is disturbed by fluid motion; this instability may be related to thermal diffusion.     

The evolution of the moon: A finite element approach

Thermal convection has considerable influence on the thermal evolution of terrestrial planets. Previous numerical models of planetary convection have solved the system of partial differential equations by finite difference methods, or have approximated it by parametrized methods. We have evaluated the applicability of a finite element solution of these equations. Our model analyses the thermal history of a self-gravitating spherical planetary body; it includes the effects of viscous dissipation, internal melting, adiabatic gradient, core formation, variable viscosity, decay of radioactive nucleides, and a depth dependent initial temperature profile. Reflecting current interest, physical parameters corresponding to the Moon were selected for the model.Although no initial basalt ocean is assumed for the Moon, partial melting is observed very early in its history; this is presumably related to the formation of the basalt maria. The convection pattern appears to be dominated by an L-2 mode. The present-day lithospheric thickness in the model is 600 km, with core-mantle temperatures close to 1600 K. Surface heat flux is 25.3 mW m^–2, higher than the steady state-value by about 16%.The finite element method is clearly applicable to the problem of planetary evolution, but much faster solution algorithms will be necessary if a sufficient number of models are to be examined by this method.Notation coefficient of thermal expansion - _ij Kronecker delta - absolute or dynamic viscosity - perturbation in temperature - thermal diffusivity - kinematic viscosity - density - stress tensor - B.P. before present - c specific heat at constant pressure or volume (Boussinesq approximation) - d depth of convection - E ^* activation energy for creep - g gravity - Ga billions of years - H(t) heat generation per unit mass per unit time at timet - k Boltzmann's constant - K mean thermal conductivity - Ma millions of years - p pressure - q heat flux - q _ss steady-state heat flux - Ra Rayleigh number - S volumetric heat sources, includes radioactivity and viscous dissipation - t time - T temperature - u verocity vector - V ^* activation volume for creep     

Libration of Laplace's argument in the Galilean satellites theory

In the Galilean satellites motion, the Laplace argument ₁3₂+2₃ librates around the value . The amplitude of libration is very small so that the classical theories have not been set up to take into account large librations. On the other hand large librations have to be considered when we describe possible scenarii of capture into resonance by tidal effects. The aim of this paper is to present a new way of applying Hamiltonian perturbation methods to the problem of the Galilean satellites in such a way that the theory is valid for large librations. Preliminary results from such a theory are discussed.     

Spatial dependence of the pitch-angle and associated spatial diffusion coefficients for cosmic rays in interplanetary space

The spatial dependence of the pitch-angle and associated spatial diffusion coefficients for cosmic ray particles in interplanetary space is calculated in the WKB approximation. The model considers only Alfven waves of solar origin to be responsible for scattering of moderate energy particles. After developing the general theory results are presented for the asymptotic case corresponding to radial distancesr greater than about 1 to 2 AU. The radial diffusion coefficient _r increases with energyE like _r E , wherev2/3. The radial mean free path turns out to increase proportional tor ³ at medium and low heliographic latitudes. This behaviour is consistent with a very small radial cosmic ray gradient and the existence of a free boundary for particle diffusion. At equal radial distances the high latitude mean free path is not only much smaller than the one calculated at the lower latitudes but in addition increases only weakly with distance. Some conceivable dynamical implications for the outer solar system are indicated.     

Approximate expressions for Gyrosynchrotron radiation in Transverse Propagation

Two sets of accurate approximate expressions for the gyrosynchrotron radiation in the transverse propagation case are presented for the first time. They contain emissivity _–/BNand absorptivity _– B/Nfor e-mode, effective temperature T _effand frequency of peak brightness _p. The expressions are designed for the range 2 to 7 of electron energy spectral index and for the ranges from 2 to 10 and 10 to 100 of harmonic numbers s(=/_B). Their statistical error is, respectively, ±18% and ±29% for _–/BNand _– B/Nfor 10/_B100, ±128% and and ±170% for 2/_B10.     

Rate-dependent strength of porous ice–silica mixtures and its implications for the shape of small to middle-sized icy satellites

Porosity is one of the most important physical properties in the rheology of small icy satellites composed of ice–silicate mixtures. Deformation experiments involving ice and 1 μm silica bead mixtures were conducted to clarify the effect of porosity on the flow law of ice–silica mixtures. Mixtures with silica mass contents of 0, 30, and 50 wt.% were used for the experiments, and the porosity was changed from 0% to 25% in each mixture. The temperature ranged from −10 to −20 °C, and the strain rate was changed from 1.2 × 10⁻⁶ to 4.2 × 10⁻⁴ s⁻¹. As a result, it was found that the ice–silica mixtures deformed plastically, and that the relationship between the maximum stress, σ_max, on the stress–strain curve and the applied strain rate, , could be described by the following flow law: . The mixture became softer as the porosity or silica mass content increased, and the stress exponent n and activation energy Q were independent of porosity, depending only on the silica mass content. Furthermore, the parameter A₀ could be written as A₀ = B(1 − ?)^−α, where ? is the porosity. The constants B and α also depended only on the silica mass content, and they increased with the increase in this content. The Maxwell relaxation time was calculated in order to estimate the conditions for topographic relaxation of icy satellites, and it was found that topographic relaxation occurred at temperatures higher than 160 K in the case of icy satellites with mean radii of 200 km.     

Convective instability of thin flux tubes

The stability of magnetic flux tubes embedded vertically in a convection zone is investigated. For thin tubes, the dominant instability is of the convective type, i.e. it is driven by buoyancy forces associated with displacements along the tube. The stability is determined by = 8P/B ²; if _c the tube is convectively stable, otherwise it is unstable, where the critical value _c depends on the stratification of the convection zone. For a solar convection zone model, _c = 1.83, corresponding to a magnetic field strength of 1350 G at the surface of the Sun. It is concluded that the flux tubes making up the small scale field of the Sun are probably hydrodynamically stable.In tubes with > _c, the instability is expected to transform the tube either into a state of vanishing surface field strength (in the case of an upward flow), or one with a field strength higher than the original value (if the instability sets in as a downward flow). Following Parker, we suggest that this effect is related to the concentrated nature of the observed solar fields.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The Energy Equation in the Lower Solar Convection Zone

As a consequence of the Taylor–Proudman balance, a balance between the pressure, Coriolis and buoyancy forces in the radial and latitudinal momentum equations (that is expected to be amply satisfied in the lower solar convection zone), the superadiabatic gradient is determined by the rotation law and by an unspecified function of r, say, S(r), where r is the radial coordinate. If the rotation law and S(r) are known, then the solution of the energy equation, performed in this paper in the framework of the ML formalism, leads to a knowledge of the Reynolds stresses, convective fluxes, and meridional motions. The ML-formalism is an extension of the mixing length theory to rotating convection zones, and the calculations also involve the azimuthal momentum equation, from which an expression for the meridional motions in terms of the Reynolds stresses can be derived. The meridional motions are expanded as U _r(r,)=P ₂(cos)₂(r)/r ²+P ₄(cos)₄(r)/r ² +..., and a corresponding equation for U (r,). Here is the polar angle, is the density, and P ₂(cos), P ₄(cos) are Legendre polynomials. A good approximation to the meridional motion is obtained by setting ₄(r)=–H₂(r) with H–1.6, a constant. The value of ₂(r) is negative, i.e., the P ₂ flow rises at the equator and sinks at the poles. For the value of H obtained in the numerical calculations, the meridional motions have a narrow countercell at the poles, and the convective flux has a relative maximum at the poles, a minimum at mid latitudes and a larger maximum at the equator. Both results are in agreement with the observations.     

Parameters of Dust Particles in the Martian Atmosphere

A critical analysis of the methods and results of estimating the optical thickness of the dust component in the Martian atmosphere ₀, the particle size r ₀, and the imaginary part of the refractive index n _ihas shown the following. (1) Observational data on the brightness distribution over the Martian disk as well as the phase dependences of diffusely reflected light and the azimuthal dependences of diffusely transmitted light are most appropriate to use only for verifying the reliability of the aerosol parameters determined by other methods. (2) If the morning and evening fogs in the atmosphere are disregarded, the Bouguer–Lambert–Beer method used to analyze the solar-brightness attenuation measured on the planetary surface yields overestimated extraatmospheric solar intensity I ₀and atmospheric optical depth ₀. At the Viking 1landing site, I ₀and ₀could be overestimated by a factor of 1.7 and by 0.35, respectively. (3) The aerosol size determined by analyzing measurements of the azimuthal dependences for the Martian sky brightness at low elevations of the Sun most likely corresponds to the fog particles. (4) If overestimated values of I ₀were used to standardize the observations of the solar radiation transmitted by the Martian atmosphere, then n _iwere also overestimated; using overestimated ₀also affected the reliability of the latter. (5) The problem of reliability of the available ₀and r ₀estimates for periods of high atmospheric transparency is yet to be solved. For the highest activity of the dust storm in 1971, it was found that 4.5 r ₀ 7.5 m for the lognormal particle size distribution with ²= 0.2 and the optical thickness of a dust cloud ₀ 15. (6) The spectral values of the apparent albedo of Mars measured in October 1971 at a phase angle of 42° in the spectral range 0.250 0.717 allowed the imaginary part of the refractive index to be estimated in terms of a model of a dust cloud composed of spherical particles with the lognormal size distribution with r ₀= 4.5 m and ²= 0.2.     

A study of massive classical polytropes in general relativity

A detailed study of classical polytropes in general relativity has been presented for _O ((dP/dE)_O) 1.0 and _O((P/E _O)_O. The behaviour of various structural parameters with _O/_O, _O and _O are the values ofP/E and dP/dE at the centre) has been studied. The most important result of this study is the fact the qualitative behaviour of all the structural parameters depends only on the value of µ_O for the various assigned _O values. The maximum value of surface red shift occurs when µ_O=0.6 and for _O=1.0 it equals 0.618. These structures are gravitationally bound for µ_O0.8 and most so for µ_O=0.4. The maximum value of binding coefficient comes out to be 0.181 when _O=1.0. These structures have been used to model neutron stars. The maximum mass of neutron star based upon such a model comes out to be 2.55M (for µ_O=0.4 and _O=1.0) and maximum size comes out to be 15.0 km (for µ_O=0.2 and _O=1.0). It is also seen that the structures are pulsationally stable for µ0.6.     

Radial gradients and anisotropies due to galactic cosmic rays

Numerical calculations have been made of the radial gradients and the anisotropyvector atr=1 AU due to galactic cosmic-ray protons and helium nuclei. The model used assumes transport by convection and anisotropic diffusion, and includes the energy losses due to adiabatic deceleration. The present calculations are for the 1964–65 solar minimum. An important constraint applied ineach case was that the model reproduces the electron modulation known from deductions of the galactic spectrum and observations of the near-Earth spectrum; and also reproduces the near-Earth proton and helium nuclei spectra. The diffusion coefficients have been based upon those deduced from magnetic-field power spectra.The principal aim has been to provide estimates of radial gradients and anisotropies, particularly at kinetic energiesT100 MeV/nucleon, by the complete solution of realistic models. Typical values for protons, obtained with a galactic differential number density (total energy)^–2.5, atT50 MeV are: radial gradient, 25%/AU; radial anisotropy, –0.2%; azimuthal anisotropy, 0.2%. These values change markedly when the galactic spectrum is cut-off or greatly enhanced atT<150 MeV, but the intensity spectrum near Earth remains substantially unchanged.It has been shown that it is possible to obtain negative radial gradients and positive radial anisotropies atT50 MeV for galactic particles and thus to mimic solar sources. The radial gradient for 1964–65 reported by Anderson (1968) and by Krimigis and Venkatesan (1969) are shown to be consistent with the diffusion coefficient deduced from the magnetic-field power spectrum; those reported by O'Gallagher are higher than expected and that for 20T30 MeV protons appears to be inconsistent. More precise data on conditions throughout the solar cavity are required if more definitive gradients and anisotropies are to be determined.     

A Systematic Study of the Stability of Symmetric Periodic Orbits in the Planar, Circular, Restricted Three-Body Problem

We investigate symmetric periodic orbits in the framework of the planar, circular, restricted, three-body problem. Having fixed the mass of the primary equal to that of Jupiter, we determine the linear stability of a number of periodic orbits for different values of the eccentricity. A systematic study of internal resonances, with frequency p/q with 2p 9, 1 q 5 and 4/3 p/q 5, offers an overall picture of the stability character of inner orbits. For each resonance we compute the stability of the two possible periodic orbits. A similar analysis is performed for some external periodic orbits.Furthermore, we let the mass of the primary vary and we study the linear stability of the main resonances as a function of the eccentricity and of the mass of the primary. These results lead to interesting conclusions about the stability of exosolar planetary systems. In particular, we study the stability of Earth-like planets in the planetary systems HD168746, GI86, 47UMa,b and HD10697.