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     

A note on the evolution of magnetic helicity in active regions

The evolution of magnetic helicity in a rapidly growing active region is governed mainly by the advection of magnetic fields into the photosphere. Dissipation at the photosphere makes only a minor contribution to helicity evolution. Current helicity dissipation in the three-dimensional domain around the active region can be ignored.     

An observational approach to the early stages of stellar evolution

During the last three decades an observational approach has been applied at the Byurakan Astrophysical Observatory to the problems of the evolution of astronomical bodies and systems. In contradiction to the traditional point of view, assuming that the processes of condensation are dominant in the Universe, this approach makes use of the observed predominance of expansions, ejections, and explosions.In the past, the observational approach has led to the prediction of an expansion of some stellar associations confirmed later by the analysis of observations. It became clear that the stellar associations are very young systems where the star-formation process is still continuing. The new approach has also led to the concept of the activity of galactic nuclei. The observational approach considers as a phenomenon of primary importance the formation of nebulae as a consequence of the activity of dense bodies (nebulae surrounding the novae, planetary nebulae, supernova remnants, cometary nebulae, and the diffuse nebulae in OB-associations).The new approach in application to the early stages of stellar evolution is discussed. The T Tauri-stage is considered as a phase following the more dense protostellar state. The flare stars are regarded as the next phase of evolution. The phenomena of fuors (FU Ori-type brightenings) can be considered as an expression of the same tendency (the transformation of dense matter into a rarefied state).     

Polarization evolution in strong magnetic fields

Extremely strong magnetic fields change the vacuum index of refraction. Although this polarization-dependent effect is small for typical neutron stars, it is large enough to decouple the polarization states of photons travelling within the field. The photon states evolve adiabatically and follow the changing magnetic field direction. The combination of a rotating magnetosphere and a frequency-dependent-state decoupling predicts polarization phase lags between different wavebands, if the emission process takes place well within the light cylinder. This QED effect may allow observations to distinguish between different pulsar-emission mechanisms and to reconstruct the structure of the magnetosphere.     

The evolution of coronal magnetic fields

Slow photospheric motions can produce flow speeds in the corona which are fast enough to violate quasi-static evolution. Therefore, high-speed flows observed in the corona are not necessarily due to a loss of equilibrium or stability. In this letter we present an example where the flow speed increases indefinitely with, height, while the coronal magnetic energy increases quadratically with time.     

Bifurcation of force-free solar magnetic fields: A numerical approach

Numerical calculations of two-dimensional force-free fields as models of solar active regions are presented. For a given toroidal component of the photospheric magnetic field two branches of solutions are numerically obtained which merge at the critical point of maximum allowed toroidal magnetic field. Depending on boundary conditions magnetic islands may or may not form. The results are discussed with respect to their relevance to the flare process.     

Vector magnetic fields, subsurface stresses and evolution of magnetic helicity

Observations of the strength and spatial distribution of vector magnetic fields in active regions have revealed several fundamental properties of the twist of their magnetic fields. First, the handedness of this twist obeys a hemispheric rule: left-handed in the northern hemisphere, right-handed in the southern. Second, the rule is weak; active regions often disobey it. It is statistically valid only in a large ensemble. Third, the rule itself, and the amplitude of the scatter about the rule, are quantitatively consistent with twisting of fields by turbulence as flux tubes buoy up through the convection zone. Fourth, there is considerable spatial variation of twist within active regions. However, relaxation to a linear force-free state, which has been documented amply in laboratory plasmas, is not observed.     

Structure and evolution of magnetic network features

Magnetogram series, obtained with the 512-diode-array magnetograph at KPNO, are used to investigate several properties of magnetic flux tubes in the solar atmosphere. Average size, lifetime and inclination of the flux elements are determined. Further we discuss the question, how magnetic flux appears and disappears on the solar surface. At last it is investigated, if our observational results are consistent with Piddington's flux-rope-fibre theory of solar magnetism.Visiting Astronomer at Kitt Peak National Observatory.     

A direct hydrodynamical approach to the evolution of the density correlations in an expanding flat Friedmann Universe

A direct approach of the dynamical equation for the evolution of the two-point density correlation function is given in an expanding flat Friedmann Universe in the Newtonian approximation. If the third and higher moments are neglected, a wave-like equation of third-order for the two-point density correlation function is found. The exact solution of this equation shows, in the large time limit, the usual Jeans instability t ^4/3. It is suggested that the highern-point correlation function of the density grow liket ^2n/3 in the same approximation. This indicates that every truncation procedure of the hierarchy of the equations is inapplicable at least for large timest.     

Dynamical evolution of the Oort cloud: II. A theoretical approach

We investigate the dynamical evolution of a cloud of comets created by stellar perturbations. We first show the respective advantages of numerical simulations and of studies of more theoretical character. Then we investigate the probability distribution of the velocity changes imparted to comets by passing stars. This distribution is shown to be different from a Maxwellian distribution, mainly because of pronounced tails. The number of fairly large impulses is thus more important than it would be in the case of a Maxwellian distribution. Finally we estimate the probability for a comet to be ejected from the Solar System. About 10% of the cloud population is lost through this mechanism over the age of the Solar System. Taking advantage of the velocity change distribution, we study the random walk of semimajor axes of comets as a function of time. We derive the probability that a comet is lost into interstellar space as a function of its initial semimajor axis.     

Fermion zero-mode influence on neutron-star magnetic field evolution

The quantum phenomenon of spectral flow which has been observed in laboratory superfluids, such as ³He-B, controls the drift velocity of proton type II superconductor vortices in the liquid core of a neutron star and so determines the rate at which magnetic flux can be expelled from the core to the crust. In the earliest and most active phases of the anomalous X-ray pulsars and soft-gamma repeaters, the rates are low and consistent with a large fraction of the active crustal flux not linking the core. If normal neutrons are present in an appreciable core matter-density interval, the spectral flow force limits flux expulsion in cases of rapid spin-down, such as in the Crab pulsar or in the propeller phase of binary systems.     

Structure and evolution of supermassive rotating magnetic polytropes

The structure of rotating magnetic polytropes is considered in Roche approximation. Investigation of the influence of poloidal as well as toroidal magnetic fields on the conditions of the beginning of matter outflow due to rotational instability is carried out. The influence of the turbulent convection and twisting of magnetic force lines on the time of smoothing of differential rotation is considered. The estimate of the magneto-turbulence energy generated by differential rotation is presented. Both maximum possible energy output and duration of the quasi-statical evolution phase up to the appearance of hydrodynamic instability due to the effects of General Relativity are calculated for supermassive magnetic polytropes of index three with uniform or differential rotation. The radiusmass relation is obtained for supermassive differentially-rotating magnetic polytropes referring to the longest part of the quasi-statistical evolution stage; some consequences are pointed out, including the period-luminosity relation.The evolution of the considered models of supermassive rotating magnetic polytropes with different character of rotation and different geometry of a magnetic field is discussed.The results obtained are summarized in the last section.Receipt delayed by postal strike in Great Britain.     

A multiple shooting approach for the numerical treatment of stellar structure and evolution

We present a new numerical method for solving the system of partial differential equations describing the structure and evolution of a spherically symmetric star. As usual, we employ the transversal method of lines in order to split the equations into a coupled spatial and temporal part. The novel features of the algorithm are the following: (a) Instead of using the Lagrangian picture we formulate the system of partial differential equations in the Eulerian picture. (b) We reformulate the equations of stellar structure as a multipoint boundary-value problem. By means of this reformulation the rather clumsy iterative matching procedure of stellar atmosphere and interior is avoided. (c) The multipoint boundary-value problem is solved by the multiple shooting method. This approach not only ensures a high accuracy of the stellar models calculated at each time step but also allows the free boundaries inside the star due to different energy transport mechanisms to be located exactly. (d) The time derivatives involved in the stellar-structure equations are discretized implicitly to second order accuracy. Moreover, at each time step, the chemical abundances are determined by using a sophisticated update procedure. In this way, a high accuracy is achieved with respect to the integration in time. The algorithm has turned out to be exceedingly reliable and numerically accurate. This is shown by the evolution of a 1 M_⊙ star up to the hydrogen-shell burning phase. In this example, the virial theorem, the law of mass conservation, and the law of energy conservation is fulfilled to a hitherto unattainable degree of accuracy. Since the multiple shooting method, which is at the heart of our approach, is a perfect example of a parallel algorithm, the computational speed of the algorithm might be substantially improved provided easy-to-program, high-performance parallel computers with sufficiently many processors become available in the near future.     

Energetic particle losses and trapping boundaries as deduced from calculations with a realistic magnetic field model

An interpretation of the stable trapping boundaries of energetic electrons and protons during quiet periods is given basing on a realistic magnetospheric magnetic field model. Particle losses are explained in terms of an ionospheric and drift loss cone filling due to a non-adiabatic pitch-angle scattering in the nightside magnetotail current sheet. The proposed mechanism is shown to provide a good agreement of the observed and calculated positions of the energetic particle trapping boundaries, as well as their energy dependence. The obtained results can be applied as a tool for investigating the magnetospheric magnetic field structure using the particle data of low-altitude satellites.     

The evolution of the magnetic fields of neutron stars

Observational evidence, and theoretical models of the magnetic field evolution of neutron stars is discussed. Observational data indicates that the magnetic field of a neutron star decays significantly only if it has been a member of a close interacting binary. Theoretically, the magnetic field evolution has been related to the processing of a neutron star in a binary system through the spin evolution of the neutron star, and also through the accretion of matter on the neutron star surface. I describe two specific models, one in which magnetic flux is expelled from the superconducting core during spin-down, via a copuling between Abrikosov fluxoids and Onsager-Feynman vortices; and another in which the compression and heating of the stellar crust by the accreted mass drastically reduces the ohmic decay time scale of a magnetic field configuration confined entirely to the crust. General remarks about the behaviour of the crustal field under ohmic diffusion are also made.     

Observations of magnetic field evolution in a solar flare

I±V and I±Q profiles of nine spectral lines of Fei, Feii, and Hi in the 2B flare of 16 June 1989 have been analyzed. Two bright flare knots outside and inside of a spot were investigated. To measure the true magnetic field strength in the flare, two different methods were applied. In addition to these data, the magnetic field and thermodynamic conditions were determined using the non-LTE program for line profile synthesis. According to the measurements, the magnetic field in both flare knots changed in synchronism and non-monotonically, and reached its peak (nearly 1.6 kG for non-spot areas and approximately 4.0 kG for sunspot locations) at the time of flare peak. For the flare knot outside the spot, a background field component was also detected; the magnetic field in this component was found to have mixed polarity and remained practically unchanged during the flare. The non-LTE calculations show that the unique local magnetic field peak existed near the temperature minimum zone in the flare peak too. The observed perturbations do not exclude such phenomena as a magnetic field transient in flare.