Separation of the steady-state cosmic ray equation of transport: A generalization

The one-dimensional, steady-state equation of transport for cosmic rays including convection, diffusion and adiabatic deceleration is separated for a spatial diffusion coefficient with an arbitrary momentum dependence and for an arbitrary spatial dependence of the convection velocityV, and applies for planar, cylindrical and spherical geometries. As an application, the previously obtained spherically symmetric steady-state Green's functions, describing the propagation of cosmic rays in interplanetary space, are generalized to the case where the convection velocity is a function of position.     

Maunder convection mode on the Sun and long solar activity minima

A model of velocity field oscillations in the solar convective zone is suggested. The system of convective equations is investigated for a thin rotating spherical envelope when the rotation velocity is depended on the coordinates. It is shown that two different structures of convective cells (longitudinal, or latitudinal) can exist in the envelope depending on gradients values of the rotation velocity and Prandtl number. It is supposed that two different regimes of convection (stationary and autofluctuating) are possible in the envelope when the angular velocity gradients are determined by the convection itself. In the case of autofluctuating regime the alternation of longitudinal and latitudinal structure of convection is realized. If one assumes that on the Sun there exists an autooscillating convection regime, then the periods of the existence of latitudinal convection structure may be associated with long periods of activity minima since according to Cowling's theorem, the action of the axisymmetric magnetic field generation mechanism is impossible under conditions of axisymmetric velocity structures.     

Fourier analysis of the light curves of eclipsing variable stars

The aim of the present paper is to establish the explicit forms of the photometric perturbations, in the frequency-domain, of close binaries, whose components are distorted by axial rotation and mutual tidal action.Following a brief introduction, Section 2 describes the light changes and the photometric perturbations within eclipses in the frequency-domain. In Section 3 the explicit forms of the perturbations for occultation eclipses terminating in totality are given; while in Section 4 analogous results are established for transit eclipses terminating in annular phases. In this latter case the results can be expressed in terms of the photometric perturbations for total eclipses and in terms of some series. To facilitate applications to actual stars these series have been computed and their results are represented in Table I and by the Graphs. Finally, Section 5 gives a discussion of the results.An extension of the photometric perturbations to the case of partial eclipses will be given in a subsequent communication.     

A coupled model of magnetic flux generation and transport in stars

We present a combined model for magnetic field generation and transport in cool stars with outer convection zones. The mean toroidal magnetic field, which is generated by a cyclic thin-layer α Ω dynamo at the bottom of the convection zone is taken to determine the emergence probability of magnetic flux tubes in the photosphere. Following the nonlinear rise of the unstable thin flux tubes, emergence latitudes and tilt angles of bipolar magnetic regions are determined. These quantities are put into a surface flux transport model, which simulates the surface evolution of magnetic flux under the effects of large-scale flows and turbulent diffusion. First results are discussed for the case of the Sun and for more rapidly rotating solar-type stars. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effects of mutual irradiation on the observed radial velocity of the components of close binary systems

The aim of the present paper will be to investigate the effects, on the observed radial velocities of the components of close binary systems, of atmospheric motions caused by mutual irradiation of the two stars. Such motions can (and, in general, will) produce systematic differences between the observed radial velocity and that of the centre of mass of the respective star — differences varying with with the phase and thus giving rise to spurious deformations of the star's radial-velocity curves due to orbital motion. A failure to separate the two could (and in general, will) vitiate the physical elements deduced from these curves —such as the masses or absolute dimensions of the components and of the shape of their orbit; but in order to do so, an investigation of atmospheric motions invoked by irradiation becomes a necessary prerequisite.In the Introduction following this abstract, the problem at issue will be described in general terms, and phenomena outlined which should arise in this connection (together with the observations indicating their presence). In Section 2, general expressions for the radial velocity at any point of stellar surface arising from atmospheric motions will be formulated while Section 3 will isolate such velocities for components of close binary systems as are produced by mutual irradiation of their mates, in terms of hydrodynamical equations of radiative transfer describing the problem. In Sections 4 and 5, the effects of non-rotational motions on the observed radial velocities will be specified, and hydrodynamical equations formulated which specify atmospheric convection caused by irradiation of each component of a close binary by its mate. Linearized versions of such equations will be constructed in Section 6; while Section 7 contains an evaluation of the effects which such gas streams exert on the observed radial velocity of the stars.In the concluding Section 8 applications to practical cases are carried out. It will be shown that no reliable spectroscopic elements of close binary systems (including the masses and absolute dimensions of their components) can be obtained until the effects of atmospheric convection caused by mutual irradiation have been accounted for to permit us to convert the observed radial velocities (influenced as they are by the motion of as in which they originate) to those of the centre of mass of the respective stars.Paper dedicated to Professor Hannes Alfvén on the occasion of his 80th birthday, 30 May 1988.     

Cosmogony of the solar system

In Sections 1–6, we determine an approximate analytical model for the density and temperature distribution in the protoplanetary could. The rotation of the planets is discussed in Section 7 and we conclude that it cannot be determined from simple energy conservation laws.The velocity of the gas of the protoplanetary cloud is found to be smaller by about 5×10³ cm s^–1 in comparison to the Keplerian circular velocity. If the radius of the planetesimals is smaller than a certain limitr ₁, they move together with the gas. Their vertical and horizontal motion for this case is studied in Sections 8 and 9.As the planetesimals grow by accretion their radius becomes larger thanr ₁ and they move in Keplerian orbits. As long as their radius is betweenr ₁ and a certain limitr ₂ their gravitational interaction is negligible. In Section 10, we study the accretion for this case.In Section 11, we determine the change of the relative velocities due to close gravitational encounters. The principal equations governing the late stages of accretion are deduced in Section 12, In Section 13 there are obtained approximate analytical solutions.The effect of gas drag and of collisions is studied in Sections 14 and 15, respectively. Numerical results and conclusions concerning the last and principal stage of accretion are drawn in Section 16.     

Possibility of Convection for Diffusion (Newtonian) Viscosity in the Ice Shell of Europa?

Recent investigations into convection in Europa’s ice shell have been based on non-Newtonian (stress-dependent) or Newtonian (stress-independent) viscosity for water ice. However, despite the wide use of Newtonian convection, experimentally observed water ice flow is non-Newtonian, and analysis of stability against convection of the ice shell using updated flow laws has been only performed for non-Newtonian rheologies. Here we use the flow law proposed for diffusion creep to analyze the possibility of the onset of convection for Newtonian viscosity in relation to the thermal state of Europa. Our findings indicate that for diffusion creep convection might have started, but that significantly lower heat flows (and equivalently higher shell thicknesses) and/or grain sizes are required than for superplastic flow, which is the most probable flow mechanism if the ice shell is convective.     

The Roche Coordinates in three dimensions and their application to problems of double-star astronomy

In a preceding paper (Kopal, 1969; in what follows referred to as Paper I) we introduced a new system of curvilinear coordinates-hereafter referred to as Roche Coordinates — in which spheres of constant radius in spherical polars have been replaced by surfaces of constant potential of a rotating gravitational dipole; while the angular coordinates are orthogonal to the equipotentials. In Paper I we established an explicit form of such a transformation, and related the Roche coordinates with polar coordinates (with which they coalesce in the immediate neighbourhood of each one of the two finite mass-points) in the plane case. The aim of the present investigation will be to generalize the definition of the Roche coordinates to three dimensions.The opening Section 1 of this paper will contain a general outline of the proposed three-dimensional transformation; and in Section 2 details of this transformation will be explicitly worked out correctly to quantities of first order in superficial distortion — an approximation which should prove adequate in regions surrounding the two finite masses; while in Section 3 we shall evaluate (to this degree of accuracy) the metric coefficients of the respective transformation, and its direction cosines, in both polar and curvilinear coordinates. Section 4 will then contain a formulation of the fundamental equations of hydrodynamics in terms of the three-dimensional Roche coordinates; and their advantages for a treatment of certain classes of dynamical problems encountered in doublestar astronomy will be illustrated in the concluding Section 5 by an investigation of the vibrational stability of the Roche model. We shall show that this model is capable of performing free radial oscillations which remain barotropic only if its equilibrium form is spherical (i.e., in the absence of any external mass in the neighbourhood); but not if it is distorted to any extent by rotation or tides.     

Topology of magnetodynamic flows: Kinetic and internal energy of a one-fluid plasma

A mathematical formalism describing the relation between the structure of a one-fluid, collisionless plasma and the topological features of the vector lines of the magnetic, velocity and electric fields is suggested and a rigorous approach to the problem of determining the local features of the plasma from the properties of the velocity field v, observed at a chosen point of space, is developed. The limits to drawing conclusions from the observational data (such as encountered in magnetospheric physics) are outlined in Section 2.Energization of a plasma during a stationary convection is an example of a process characterized by the properties of the tensor $\text{V}\text{= grad}\text{v}$. In particular, if the plasma density and the velocity and pressure components in a volume element of the plasma are known, then the time rates of change of the kinetic and internal energies are described by stretchings of the element and by its shearing in the surface defined by the vectors of magnetic field and perpendicular velocity. Criteria according to which plasma gains or loses kinetic and internal energies are derived.     

Jupiter's radiation belts

A model for the production and loss of energetic electrons in Jupiter's radiation belt is presented. It is postulated that the electrons originate in the solar wind and are diffused in toward the planet by perturbations which violate the particles' third adiabatic invariant. At large distances, magnetic perturbations, electric fields associated with magnotospheric convection, or interchange instabilities driven by thermal plasma gradients may drive the diffusion. Inside about 10 R_J the diffusion is probably driven by electric fields associated with the upper atmosphere dynamo which is driven by neutral winds in the ionosphere. The diurnal component of the dynamo wind fields produces a dawn-dusk asymmetry in the decimetric radiation from the electrons in the belts, and the lack of obvious measured asymmetries in the decimetric radiation measurements provides estimates of upper limits for these Jovian ionospheric neutral winds. The average diurnal winds are less than or comparable to those on earth, but only modest fluctuating winds are required to drive the energetic electron diffusion referred to above.The winds required to diffuse the energetic particles across the orbit of the satellite lo in a time equal to their drift period are also estimated. If Io is non-conducting, modest winds are required, but if Io is conducting, only small winds are needed. It is concluded that both protons and electrons are diffused in from the solar wind to small distances without serious losses occurring due to the particles being swept up by the satellites.Consideration of proton and electron diffusion in energy shows that once the electrons become relativistic, the ratio of proton to electron energy increases. Thus, if protons and electrons have the same energy in the solar wind, when the electrons reach nMeV, the protons will be nMeV if n ? 1 or n² MeV if n ? 1. If the proton-to-electron energy ratio is initially, e.g., 5, then these figures are 5n and 5n², respectively.     

The evolution of loop structures in flux rings within the solar convection zone

Choudhuri and Gilman (1987) considered certain implications of the hypothesis that the magnetic flux within the Sun is generated at the bottom of the convection zone and then rises through it. Taking flux rings symmetric around the rotation axis and using reasonable values of different parameters, they found that the Coriolis force deflects these flux rings into trajectories parallel to the rotation axis so that they emerge at rather high latitudes. This paper looks into the question of whether the action of the Coriolis force is subdued when the initial configuration of the flux ring has non-axisymmetries in the form of loop structures. The results depend dramatically on whether the flux ring with the loops lies completely within the convection zone or whether the lower parts of it are embedded in the stable layers underneath the convection zone. In the first case, the Coriolis force supresses the non-axisymmetric perturbations so that the flux ring tends to remain symmetric and the trajectories are very similar to those of Choudhuri and Gilman (1987). In the second case, however, the lower parts of the flux ring may remain anchored underneath the bottom of the convection zone, but the upper parts of the loops still tend to move parallel to the rotation axis and emerge at high latitudes. Thus the problem of the magnetic flux not being able to come out at the sunspot latitudes still persists after the non-axisymmetries in the flux rings are taken into account.National Center for Atmospheric Research is sponsored by the National Science Foundation.     

On the nonradiative magnetized accretion flows with convection

In this paper, we explore the radial structure of radiatively inefficient accretion flows (RIAFs) in the presence of an ordered magnetic field and convection. We assume the magnetic field has the toroidal and vertical components. We apply the influences of convection on equations of angular momentum and energy. The convective instability can transport the angular momentum inward or outward. We establish two cases for consideration of the effects of convection parameter on magnetized RIAFs. In the first case, we assume the convection parameter as a free parameter and in the other case we calculate convection parameter through use of mixing length theory. In both cases, the solutions show that a magnetized RIAF is very sensitive to the convection parameter and transport direction of angular momentum due to convection. Moreover, we show that the convection strength strongly depends on magnetic field and viscosity.     

Convection in the Presence of an Inclined Axis of Rotation with Applications to the Sun

Thermal convection in a horizontal fluid layer heated from below and rotating about an arbitrary axis is studied analytically with the attention focused on mean flows and drifts generated by the convection velocity field. Mean flows occur in both horizontal directions when the angle between the rotation vector and the vertical is finite but less than 90°. In the case of a hexagonal convection pattern, a wavelike drift is found in the presence of a horizontal component of rotation. Applications to solar convection are discussed. Considering the simplicity of the model the agreement with observations is surprisingly good.     

CESAM: a free code for stellar evolution calculations

The cesam code is a consistent set of programs and routines which perform calculations of 1D quasi-hydrostatic stellar evolution including microscopic diffusion of chemical species and diffusion of angular momentum. The solution of the quasi-static equilibrium is performed by a collocation method based on piecewise polynomials approximations projected on a B-spline basis; that allows stable and robust calculations, and the exact restitution of the solution, not only at grid points, even for the discontinuous variables. Other advantages are the monitoring by only one parameter of the accuracy and its improvement by super-convergence. An automatic mesh refinement has been designed for adjusting the localisations of grid points according to the changes of unknowns. For standard models, the evolution of the chemical composition is solved by stiffly stable schemes of orders up to four; in the convection zones mixing and evolution of chemical are simultaneous. The solution of the diffusion equation employs the Galerkin finite elements scheme; the mixing of chemicals is then performed by a strong turbulent diffusion. A precise restoration of the atmosphere is allowed for.     

Temperature lapse rate and methane in Titan's troposphere

We have reanalyzed the Voyager radio occultation data for Titan, examining two alternative approaches to methane condensation. In one approach, methane condensation is facilitated by the presence of nitrogen because nitrogen lowers the condensation level of a methane/nitrogen mixture. The resulting enhancement in methane condensation lowers the upper limit on surface relative humidity of methane obtained from the Voyager occultation data from 0.7 to 0.6. We conclude that in this case the surface relative humidity of methane lies between 0.08 and 0.6, with values close to 0.6 indicated. In the other approach, methane is allowed to become supersaturated and reaches 1.4 times saturation in the troposphere. In this case, surface humidities up to 100% are allowed by the Voyager occultation data, and thus the upper limit must be set by other considerations. We conclude that if supersaturation is included, then the surface relative humidity of methane can be any value greater than 0.08--unless a deep ocean is present, in which case the surface relative humidity is limited to less than 0.85. Again, values close to 0.6 are indicated. Overall, the tropospheric lapse rate on Titan appears to be determined by radiative equilibrium. The lapse rate is everywhere stable against dry convection, but is unstable to moist convection. This finding is consistent with a supersaturated atmosphere in which condensation-and hence moist convection-is inhibited.     

Synchrotron X-ray diagnostics of cutoff shape of nonthermal electron spectrum at young supernova remnants

Synchrotron X-rays can be a useful tool to investigate electron acceleration at young supernova remnants(SNRs).At present,since the magnetic field configuration around the shocks of SNRs is uncertain,it is not clear whether electron acceleration is limited by SNR age,synchrotron cooling,or even escape from the acceleration region.We study whether the acceleration mechanism can be constrained by the cutoff shape of the electron spectrum around the maximum energy.We derive analytical formulae of the cutoff shape in each case where the maximum electron energy is determined by SNR age,synchrotron cooling and escape from the shock.They are related to the energy dependence of the electron diffusion coefficient.Next,we discuss whether information on the cutoff shape can be provided by observations in the near future which will simply give the photon indices and the flux ratios in the soft and hard X-ray bands.We find that if the power-law index of the electron spectrum is independently determined by other observations,then we can constrain the cutoff shape by comparing theoretical predictions of the photon indices and/or the flux ratios with observed data which will be measured by NuSTAR and/or ASTRO-H.Such study is helpful in understanding the acceleration mechanism.In particular,it will supply another independent constraint on the magnetic field strength around the shocks of SNRs.     

Topology of the magnetic field and geomagnetism

We define the topology of an (axisymmetric) magnetic field, a set of qualitative labels characterizing the connectivity of the lines of force. Under the special continuous deformations of theB-field defined by time evolution under the dynamo equation in the convective regime (where the frozen-in behavior dominates the diffusion), this topology is preserved. This theorem should have applications to the study of time-varying magnetic fields in that regime in the case that exact or even approximate solutions are difficult to obtain. A partial generalization to the general case of convection and diffusion is made. As an application, a critique of Hibberd's recent theory of a time-dependent axisymmetric geomagnetic dynamo whose dipole-like field undergoes successive reversals is given.     

Convection in a rapidly rotating system: the smallinclination limit and its planetary applications

An asymptotic method based on a continuous superposition of waves is used to studythe linear stability of convection in a rapidly rotating system. The method gives a uniformrepresentation of the solutions which allows us to impose the boundary conditions and then tominimize the Rayleigh number. This study was done for Prandtl numbers between 0.01 and 100.In the spherical case, for a self-gravitating, internally heated fluid in the small inclination limit,six branches are unveiled. In these branches, infinitesimal amplitude convection takes placepreferentially near the surface of a cylinder coaxial with the axis of rotation in a zone ofthickness ∝ T^−1⧸12, T being the Taylor number. The Rayleigh number ofthree of these flows differs at the most by sixty percent; however, in some intervals of the Prandtlnumber the difference is less than ten percent. Since these flows are located at different radialdistances, this method predicts mixed-modes convection in separate zones at slightlysupercritical values of the Rayleigh number for all Prandtl numbers. A solution exhibitingconvection in separate zones at low supercritical Rayleigh numbers is proposed for the first time.Applications to atmospheres and dynamos of the planets and the starts are discussed.