Solutions of the two-fluid solar wind equations: Adiabatic and conduction dominated solutions

The equations governing the two-fluid spherically symmetric models of the solar wind have been solved numerically for a wide range of base conditions. As predicted from an asymptotic analysis we find a whole domain of solutions which are asymptotically adiabatic with the proton and electron temperatures tending to equality and varying like r ^{- 4/3}. In these 4/3 solutions the electron and proton heat conduction is asymptotically negligible and if it is neglected the resulting equations can be integrated analytically and shown to have the 4/3, 4/3 behaviour.Proceedings of the 14th ESLAB Symposium on Physics of Solar Variations, 16–19 September 1980, Scheveningen, The Netherlands.     

On the collisional theory of the anisotropic solar wind plasma

The collisional equations for the solar wind assuming steady, spherically symmetric flow and including thermal conduction and an anisotropic proton temperature are analysed in the absence of a heat source function. The equations are cast in a form that involves one dimensionless parameter, effectively equal to the inverse Péclet number for protons, and the small quantity , the square-root of the electron-proton mass ratio. Analytic forms for the proton temperature anisotropy and other flow variables are derived by applying the limit 0, and using asymptotic techniques. It is found that the model based purely on Coulomb collisions predicts values for the proton temperature anisotropy in the vicinity of the Earth that are much smaller than those observed, and that increasing the coronal base temperature serves to decrease the predicted anisotropy still further.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Brans-dicke cosmology and the cosmological constant: The spectrum of vacuum solutions

We study the Brans-Dicke vacuum field equations in the presence of a cosmological term A. Considering a Friedmann-Robertson-Walker metric with flat spatial sections (k=0), we provide a qualitative analysis of the solutions and investigate its asymptotic properties. The general solution of the field equations for arbitrary values ofw and A is obtained.Work supported by CNPq (Brazil).     

Formation of emission spectra in movin media

Conclusions It can be seen from the above review that the theory of radiative transfer in moving media in its classical field of applications is a well-developed branch of theoretical astrophysics. Studies i n recent years have clarified important questions such as the asymptotic behavior of the kernel functions and the characteristic lengths of the theory. It has been established that there are two types of radiative coupling, and the influence of nonlocal radiative coupling on the formation of spectral lines and radiation pressure has been investigated. The theory now has at its disposal a large selection of asymptotic, approximate, and numerical methods for solving different applied problems.Despite the competition from numerical methods, the overwhelming majority of calculations of emission spectra in the region of supersonic motions has been made on the basis of the escape-probability method and its generalization to the case of nonlocal radiative coupling. This is explained not only by the simplicity and economy of the method but even more by the fact that the greater accuracy of the calculations that can be achieved by using numerical methods is frequently spurious, since it certainly exceeds the accuracy of the basic assumptions. The real way to increase the reliability in the diagnosis of a radiating gas is, first, to solve simultaneously many-level problems for the group of elements whose lines are observed in the spectrum of the object. Second, where possible, one must solve simultaneously the stationary equations and the heat balance equations.Crimean Astrophysical Observatory. Translated from Astrofizika, Vol. 20, No. 2, pp. 365–417, March–April, 1984.     

A special case of the problem of two fixed centers

This paper deals with the asymptotic case of the problem of two fixed centers. The equations of motion are given in paraboloidal coordinates and then they are solved by the Hamilton-Jacobi method. The surfaces of zero velocity and the qualitative properties of possible motions are also discussed.     

Comparison of solutions to the thirteen-moment and standard transport equations for low speed thermal proton flows

We have compared solutions obtained from the general 13-moment system of transport equations with those obtained from the standard collision-dominated transport equations for conditions corresponding to low speed thermal proton flow in the topside ionosphere in the vicinity of the plasmapause. In general, the solutions obtained from the 13-moment system of equations, which allows for different species temperatures parallel and perpendicular to the geomagnetic field and non-classical heat flows, are different from those obtained from the standard transport equations, which account for isotropic temperatures and classical collision-dominated heat flows. Within the plasmasphere, where the electron density is high, the differences between the 13-moment and standard solutions are typically small. However, outside the plasmasphere where the electron density is lower and in the ionosphere above SAR-arcs, where substantial electron and proton heat flows occur, there can be significant differences between the 13-moment and standard solutions. Generally, the differences are much larger for the protons than for the electrons. Our 13-moment solutions indicate that the proton and electron distributions are anisotropic with the difference between parallel and perpendicular temperatures approaching 4000 K for the protons and 2500 K for the electrons in the ionosphere above SAR-arcs. Also, above SAR-arcs the 13-moment heat flow equations yield proton heat flows as much as a factor of 10 lower and electron heat flows as much as a factor of 2 lower than those predicted by the classical collision-dominated heat flow expressions for the same boundary conditions.     

Asymptotic theory of the transfer of polarized radiation with resonance scattering in the doppler core of the line

In the first of the series of papers by Ivanov et al. it was shown that the model problem of the transfer of polarized radiation as a result of resonance scattering from two-level atoms in a homogeneous plane atmosphere in the absence of LTE comes down, in the approximation of complete frequency redistribution, to the solution of an integral matrix equation of the Wiener-Hopf type for a (2 × 2) matrix source function S(τ). In the second paper in this series, devoted to the vector Milne problem, complete asymptotic expansions of the matrix I(z) [which is essentially a Laplace transform of the matrix S(τ)] for the case of a Doppler profile of the coefficient of absorption, and the coefficients of asymptotic expansions of S(τ) (τ » 1) are expressed in terms of coefficients of the expansions of I(z). We show that asymptotic expansions of S(τ) can be found directly from an integral matrix equation of the Wiener-Hopf type for S(τ). We give new recursive equations for the coefficients of these expansions, as well as a new derivation of asymptotic expansions of the matrix I, including its second column, which was considered only briefly by Ivanov et al.     

Massless scalar static spheres

It is known that the requirement of asymptotic flatness places restrictions on spherically-symmetric solutions to field equations. Here it is shown that the most general solution to the static spherically-symmetric massless scalar Einstein equations with zero cosmological constant is asymptotically flat; furthermore, the general solution is derived and shown to be identical to a solution previusly found by M. Wyman.     

On the use of the autonomous Birkhoff equations in Lie series perturbation theory

In this article, we present the Lie transformation algorithm for autonomous Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a symplectic structure of the general form. The Birkhoff equations are derived from the linear first-order Pfaff–Birkhoff variational principle, which is more general than the Hamilton principle. The use of 1-form in formulating the equations of motion in dynamics makes the Birkhoff method more universal and flexible. Birkhoff’s equations have a tensorial character, so their form is independent of the coordinate system used. Two examples of normalization in the restricted three-body problem are given to illustrate the application of the algorithm in perturbation theory. The efficiency of this algorithm for problems of asymptotic integration in dynamics is discussed for the case where there is a need to use non-canonical variables in phase space.     

Line Formation in a Purely Scattering, Optically Thick Atmosphere

A model problem in the theory of line formation in an optically thick, purely scattering, stellar atmosphere is considered. The integral equation of radiation transfer at line frequencies is solved numerically for a two-level atom in the approximation of complete frequency redistribution in scattering. The numerical results are compared with those calculated from equations of the asymptotic theory. On the basis of the asymptotic theory, the positions of intensity maxima in a line are found for different absorption profiles.     

On a common derivation of the averaging method and the two-timescale method

In this paper it is shown that the well-known averaging method (of Krylov, Bogoliubov-Mitropolski) and the two-timescale method, applied to periodic first-order ordinary differential equations, can be derived from one common principle, as two more or less complementary special cases. The uniformity of this treatment includes the proof of asymptotic convergence of both methods, since a single proof can be given under certain hypotheses, which are verifieda posteriori in both cases.     

Diffusion and heat flow equations for the mid-latitude topside ionosphere

For application to the mid-latitude topside ionosphere, we have derived diffusion and heat flow equations for a gas mixture composed of two major ions, electrons and a number of minor ions. These equations were derived by expanding the velocity distribution of each constituent about its 13 lower order velocity moments. As a consequence, each constituent was allowed to have its own temperature and drift velocity. The restriction to mid-latitudes results because we have assumed that the species temperature and drift velocity differences were small. In deriving the diffusion and thermal conduction equations, we have discovered some new transport effects. For the major ions, we have found that: (1) a temperature gradient in either gas causes thermal diffusion in both gases; (2) a temperature gradient in either gas causes heat to flow in both gases; and (3) a relative drift between the major ion gases induces a heat flow in both gases. Similar transport effects have also been found for the minor ions.     

Solutions to bi-Maxwellian transport equations for SAR-arc conditions

We have compared solutions obtained from the bi-Maxwellian based 16-moment transport equations with those obtained from the Maxwellian based 13-moment transport equations for conditions leading to the steady state, subsonic flow of a fully-ionized electron-proton plasma along geomagnetic field lines in the vicinity of the plasmapause. The bi-Maxwellian based equations can account for large temperature anisotropies and the flow of both parallel and perpendicular thermal energy, while the Maxwellian based equations account for small temperature anisotropies and only the total heat flow. Our comparison indicates that for Stable Auroral Red arc (SAR-arc) conditions leading to strong field-aligned heat flows (temperatures of 8000 K and temperature gradients of4K. km⁻¹ at 1500 km), the bi-Maxwellian based equations predict a different thermal structure in the topside ionosphere than the less rigorous Maxwellian based equations. In particular, the bi-Maxwellian based equations predict proton and electron temperature anisotropies with T > T, while the Maxwellian based equations predict the opposite behavior for the same boundary conditions. This difference is related to the way in which the temperature anisotropies and heat flows are treated in the two formulations. For the bi-Maxwellian based equations, the inclusion of separate heat flows for parallel and perpendicular thermal energy allows for the development of a pronounced tail in both the electron and proton distribution functions, which leads to temperature anisotropies with T > T. For the Maxwellian based equations, on the other hand, the tail development is restricted because only the total heat flow is considered. Consequently, as the heat flows down, the presence of an increasing magnetic field acts to produce an anisotropy with T > T, and this process dominates tail formation for the Maxwellian based equations.     

Excitation of non-axially symmetric modes of the Sun's mean magnetic field

The kinematic dynamo equations for the mean magnetic field are solved with an asymptotic method of the WKB type. The excitation conditions and main characteristics of the non-axially symmetric modes for a given distribution of the sources are obtained. Utilization of the helioseismologic data on the Sun's internal rotation permits an explanation, within the framework of dynamo theory, of the excitation of the main non-axially symmetric modes revealed in the Sun's magnetic field sector structure.     

Asymptotic expansions in the perturbed two-body problem with application to systems with variable mass

The main theorems of the theory of averaging are formulated for slowly varying standard systems and we show that it is possible to extend the class of perturbation problems where averaging might be used. The application of the averaging method to the perturbed two-body problem is possible but involves many technical difficulties which in the case of the two-body problem with variable mass are avoided by deriving new and more suitable equations for these perturbation problems. Application of the averaging method to these perturbation problems yields asymptotic approximations which are valid on a long time-scale. It is shown by comparison with results obtained earlier that in the case of the two-body problem with slow decrease of mass the averaging method cannot be applied if the initial conditions are nearly parabolic. In studying the two-body problem with quick decrease of mass it is shown that the new formulation of the perturbation problem can be used to obtain matched asymptotic approximations.     

Excitation of polar motion by earthquake displacement field

We discuss the excitation of polar motion by earthquake displacement field. Instead of the usual static equilibrium equations in the literature, we use an improved set as given in /1/, which guarantee continuity at the core-mantle boundary. We take the parameter values of three earthquakes from /2/.To obviate the singularity at r = 0, we use asymptotic solutions by power series within a small sphere around the centre. Outisde this sphere, the equations are numerically integrated by the Runge-Kutta algorithm. Our equations /1/ gave polar shifts some 3 times larger than Dahlen's equations /2/.     

Some nonlinear effects in the relativistic two-body problem

The test-particle motion in the centrally symmetric gravitational field can be described by the equation in the form appropriate for a nonlinear oscillator — the nonlinear terms being due to the nonrelativistic effects. This enables us to apply to this equation the well-known asymptotic methods of the theory of nonlinear oscillations. Typical nonlinear oscillation phenomena arising from the action of external forces are shown to take place. The form of equations and the main results remain valid in the problem of two bodies of comparable mass in the post-Newtonian approximation.     

Numerical solution of the time-dependent heat energy equations for the magnetosphere

A new solution of the magnetospheric heat equations capable of covering the whole region from 300 km along a field line to the equatorial plane has been achieved by adapting the searching procedure of Murphy (1974). It has been found that the protonospheric heat reservoir is sufficient to maintain T_e >T_n down to the height of the F2-peak electron density all through the night at mid-latitudes. Full solution of the equations has also shown that T_i >T_e in the protonosphere at night and the ions constitute a significant source of heat for the electrons.     

Titan's damp ground: Constraints on Titan surface thermal properties from the temperature evolution of the Huygens GCMS inlet

Abstract— A simple thermal model is developed to determine the temperature history of the inlet tube of the Huygens probe gas chromatograph mass spectrometer (GCMS) after its fortuitous emplacement on the surface of Saturn's moon Titan. The model parameters are adjusted to match the recorded temperature history of a nearby heater, taking into account heat losses by conduction to the rest of the probe and to Titan's cold atmosphere. The model suggests that after impact when forced convective cooling ceased, the inlet temperature rose from ?110 K to an asymptotic value of only ?145 K. This requires that the inlet was embedded in a surface that acted as an effective heat sink, most plausibly interpreted as wet or damp with liquid methane. The data appear inconsistent with a tar or dry, fine‐grained surface, and the inlet was not warm enough to devolatilize methane hydrate.