On the stability of the solar wind

The stability of the solar wind is studied in the case of spherical symmetry and constant temperature. It is shown that the stability problem must be formulated as a mixed initial and boundary-value problem in which are prescribed the perturbation values of velocity and density at an initial time and additionally the velocity perturbation at the base of the corona for all times. The solution is constructed by linear superposition of normal solutions, which contain the time only in an exponential factor. The stability problem becomes a singular eigenvalue problem for the amplitudes of the velocity and pressure perturbations, since additionally to the boundary condition at the base of the corona one must add the condition that the amplitudes behave regularly at the critical point. It is proved that only stable eigenvalues exist.     

The influence of the anomalous cosmic-ray component on the dynamics of the solar wind

It is well known that both the galactic and anomalous cosmic rays show positive intensity gradients in the outer heliosphere which are connected with corresponding pressure gradients. Due to an efficient dynamical coupling between the solar wind plasma and these highly energetic media by means of convected MHD turbulences, there exists a mutual interaction between these media. As one consequence of this scenario the enforced pressure gradients influence the distant solar wind expansion. Here we concentrate in our theoretical study on the interaction of the solar wind only with the anomalous cosmic-ray component. We use the standard two-fluid model in which the cosmic-ray fluid modifies the solar wind flow via the cosmic-ray pressure gradient. Then we derive numerical solutions in the following steps: first we calculate an aspherical pressure distribution for the anomalous cosmic rays, describing their diffusion in an unperturbed radial solar wind. Second, we then consider the perturbation of the solar wind flow due to these induced anomalous cosmic-ray pressure gradients. Within this context we especially take account of the action of a non-spherical geometry of the heliospheric shock which may lead to pronounced upwinddownwind asymmetries in the pressures and thereby in the resulting solar wind flows. As we can show in our model, which fits the available observational data, radial decelerations of the distant solar wind by between 5 to 11% are to be expected, however, the deviations of the bulk solar wind flow from the radialdirections are only slightly pronounced.     

Unique solutions of solar wind models with thermal conductivity

The solutions of a radially symmetrical solar wind model with thermal conductivity are discussed in the neighborhood of the loci of the critical points, called the critical line, and at infinity. It is shown that for constant mass and total energy flux the conditions at infinity are related uniquely to the conditions at the critical line. Furthermore the requirement that the solution is continuous and has the proper behavior at infinity will determine uniquely one specific point on the critical line through which the solution has to pass.Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

Perturbation structures in magnetized relativistic winds

A perturbation formulation is presented for steady three-dimensional compressible structures embedded in a background relativistic magnetohydrodynamic (RMHD) radial outflow with spherical symmetry. The property of the two concurred RMHD fast and slow critical points is examined. Explicit perturbation solutions at large radii are derived analytically in a polytropic background wind. This perturbation approach for compressible structures together with our recent analyses on propagation of Alfvénic perturbations in a RMHD wind forms a useful starting basis for modeling a broad class of complex structures in magnetized relativistic astrophysical outflows.     

Solar-wind model including the effects of rotation,magnetic fields,and anisotropic heat conduction

A time-independent solar-wind model is considered in the case of spherical symmetry and of radial magnetic field at the sun's surface. The energy equation includes besides the usual terms also the heat conduction and magnetic-energy convection (Poynting vector) terms. The dependence of the thermal conductivity on the magnetic field is taken into account. Numerical integrations of the basic equations were performed under the following assumptions: (i) close to the sun the magnetic field is the dominant azimuthal term and solid-body rotation is enforced; (ii) beyond the Alfvénic point the terms quadratic inB are neglected. The model leads to azimuthal velocity at earth between 0.6 and 2.7 km/sec, to radial velocity at earth between 350 and 500 km/sec, and to angular momentum loss of 5×10¹⁸ cm²/sec per unit mass of gas leaving the solar equator. The dependence of the solutions on the reduction of the effective thermal conductivity caused by the micro-structures in the solar wind suggests that the conditions at earth may be largely determined by a transition region in the solar wind, in which the conduction régime changes into an almost adiabatic flow.Presented at the Trieste Colloquium on Mass Loss from Stars, September 12–16, 1968.     

Non-uniformly rotating,self-gravitating,compressible masses with internal meridian circulation

The structure of self-gravitating, inviscid, compressible fluids is investigated assuming a polytropic relation between pressure and density. A class of solutions with non-uniform rotation and internal meridian circulation are presented and the stream lines of the flow calculated using a perturbation technique.     

Meridional flow in the solar wind in the presence of latitudinally dependent boundary conditions

The influence of latitudinally dependent boundary conditions on the large radius values of meridional flow in the distant solar wind is examined through a double perturbation expansion of the magnetohydrodynamic equations. A general result is derived for the meridional velocity which allows arbitrary specification of radial velocity, radial magnetic field, and mass flux, as a function of colatitude at some coronal reference surface. Three specific examples are treated, including the model of Pneuman and Kopp (1971). The latter example indicates that there may be flow toward the equator at large radii, as opposed to the pure equatorial divergence of internally generated motion due to a flow which is latitudinally uniform at the reference radius. A solar cycle effect most probably averages the boundary conditions so that only the equatorial divergence from an average spherically symmetric corona is seen in comet-tail observations. This may also explain the off-and-on-again nature of the meridional gradient in the radial velocity of the solar wind as seen in radio scintillation observations.     

A one-dimensional gasdynamical model of magnetospheric convection

The plasma flow in the equatorial plane of the magnetosphere is examined within the framework of a one-dimensional model in which all quantities are supposed to depend only on the distance along the Sun-Earth axis. The following models are considered: (1) the gasdynamical model in which the Ampère force is ignored, (2) the magnetohydrodynamical model in which the normal component of the Ampère force on the magnetopause is taken into account. The flow regime is calculated in the region including two regions: (1) the layer of the return flow where flow velocity is directed from the Sun, (2) the region of convection where the velocity is directed toward the Sun - on the assumption that the form of the magnetopause and the distribution of the solar wind pressure on the magnetopause are known.The following physical mechanisms are taken into account: (1) the appearance of a centrifugal force owing to the magnetopause curvature, the centrifugal force partly compensating for the solar wind pressure; (2) the existence of the critical point which is analogous to the point of transition through the local sound velocity in the Laval nozzle or in the Parker model of the solar corona. The thickness of the layer of the return flow and the velocity of convection in the magnetosphere are calculated; and the following peculiarities are found: (1) in the gasdynamical model the convection regime is only possible with high velocities corresponding to the substorm, (2) in the magnetohydrodynamic model the convection velocity and the thickness of the layer of the return flow are reduced; the reduction being connected to the fact that the pressure of the solar wind is partially compensated for by the jump of the magnetic pressure on the magnetopause.     

The solar wind interaction with Venus through the eyes of the Pioneer Venus Orbiter

Venus has no internal magnetic dynamo and thus its ionosphere and hot oxygen exosphere dominate the interaction with the solar wind. The solar wind at 0.72 AU has a dynamic pressure that ranges from 4.5 nPa (at solar max) to 6.6 nPa (at solar min), and its flow past the planet produces a shock of typical magnetosonic Mach number 5 at the subsolar point. At solar maximum the pressure in the ionospheric plasma is sufficient to hold off the solar wind at an altitude of 400 km above the surface at the subsolar point, and 1000 km above the terminators. The deflection of the solar wind occurs through the formation of a magnetic barrier on the inner edge of the magnetosheath, or shocked solar wind. Under typical solar wind conditions the time scale for diffusion of the magnetic field into the ionosphere is so long that the ionosphere remains field free and the barrier deflects almost all the incoming solar wind. Any neutral atoms of the hot oxygen exosphere that reach the altitude of the magnetosheath are accelerated by the electric field of the flowing magnetized plasma and swept along cycloidal paths in the antisolar direction. This pickup process, while important for the loss of the Venus atmosphere, plays a minor role in the deceleration and deflection of the solar wind. Like at magnetized planets, the Venus shock and magnetosheath generate hot electrons and ions that flow back along magnetic field lines into the solar wind to form a foreshock. A magnetic tail is created by the magnetic flux that is slowed in the interaction and becomes mass-loaded with thermal ions.The structure of the ionosphere is very much dependent on solar activity and the dynamic pressure of the solar wind. At solar maximum under typical solar wind conditions, the ionosphere is unmagnetized except for the presence of thin magnetic flux ropes. The ionospheric plasma flows freely to the nightside forming a well-developed night ionosphere. When the solar wind pressure dominates over the ionospheric pressure the ionosphere becomes completely magnetized, the flow to the nightside diminishes, and the night ionosphere weakens. Even at solar maximum the night ionosphere has a very irregular density structure. The electromagnetic environment of Venus has not been well surveyed. At ELF and VLF frequencies there is noise generated in the foreshock and shock. At low altitude in the night ionosphere noise, presumably generated by lightning, can be detected. This paper reviews the plasma environment at Venus and the physics of the solar wind interaction on the threshold of a new series of Venus exploration missions.     

Equilibrium points and zero velocity surfaces in the restricted four-body problem with solar wind drag

We have analyzed the motion of an infinitesimal mass in the restricted four-body problem with solar wind drag. It is assumed that the forces which govern the motion are mutual gravitational attractions of the primaries, radiation pressure force and solar wind drag. We have derived the equations of motion and found the Jacobi integral, zero velocity surfaces, and particular solutions of the system. It is found that three collinear points are real when the radiation factor 0<β<0.1 whereas only one real point is obtained when 0.125<β<0.2. The stability property of the system is examined with the help of Poincaré surface of section (PSS) and Lyapunov characteristic exponents (LCEs). It is found that in presence of drag forces LCE is negative for a specific initial condition, hence the corresponding trajectory is regular whereas regular islands in the PSS are expanded.     

A theoretically derived energy coupling function for the magnetosphere

The energy coupling function between the solar wind and the magnetosphere can be obtained for two extreme situations, in which the magnetospheric geometry is determined primarily by either (i) the interplanetary magnetic field, or (ii) the solar wind pressure. In this paper, we obtained an expression for the energy coupling function by assuming a simple interpermeation of the interplanetary and geomagnetic fields. Two important quantities in this case are the potential difference between the two neutral points and the amount of open flux. From these two overall quantities, the voltage and the current of the magnetospheric dynamo are calculated. The dynamo power output represents the rate at which energy is transferred from the solar wind to the magnetosphere. The derived functional dependence on the interplanetary conditions provides a theoretical basis for the energy coupling function previously deduced from observations.     

Interplanetary magnetic field structure and solar wind parameters as inferred from solar magnetic field observations and by using a numerical 2-D MHD model

An attempt is made to infer parameters of the solar corona and the solar wind by means of a numerical, self-consistent MHD simulation. Boundary conditions for the magnetic field are given from the observations of the large-scale magnetic field at the Sun. A two-region, planar (the ecliptic plane is assumed) model for the solar wind flow is considered. Region I of transonic flow is assumed to cover the distances from the solar surface up to 10R _S (R _S is the radius of the Sun). Region II of supersonic, super-Alfvénic flow extends between 10R _S and the Earth's orbit. Treatment for region I is that for a mixed initial-boundary value problem. The solution procedure is similar to that discussed by Endler (1971) and Steinolfson, Suess, and Wu (1982): a steady-state solution is sought as a relaxation to the dynamic equilibrium of an initial state. To obtain a solution to the initial value problem in region II with the initial distribution of dependent variables at 10R _S (deduced from the solution for region I), a numerical scheme similar to that used by Pizzo (1978, 1982) is applied. Solar rotation is taken into account for region II; hence, the interaction between fast and slow solar wind streams is self-consistently treated. As a test example for the proposed formulation and numerical technique, a solution for the problem similar to that discussed by Steinolfson, Suess, and Wu (1982) is obtained. To demonstrate the applicability of our scheme to experimental data, solar magnetic field observations at Stanford University for Carrington rotation 1682 are used to prescribe boundary conditions for the magnetic field at the solar surface. The steady-state solution appropriate for the given boundary conditions was obtained for region I and then traced to the Earth's orbit through region II. We compare the calculated and spacecraft-observed solar wind velocity, radial magnetic field, and number density and find that general trends during the solar rotation are reproduced fairly well although the magnitudes of the density in comparison are vastly different.     

Hysteresis effects and diagnostics of the shock formation in low angular momentum axisymmetric accretion in the Kerr metric

The secular evolution of the purely general relativistic low angular momentum accretion flow around a spinning black hole is shown to exhibit hysteresis effects. This confirms that a stationary shock is an integral part of such an accretion disc in the Kerr metric. The equations describing the space gradient of the dynamical flow velocity of the accreting matter have been shown to be equivalent to a first order autonomous dynamical systems. Fixed point analysis ensures that such flow must be multi-transonic for certain astrophysically relevant initial boundary conditions. Contrary to the existing consensus in the literature, the critical points and the sonic points are proved not to be isomorphic in general, they can form in a completely different length scales. Physically acceptable global transonic solutions must produce odd number of critical points. Homoclinic orbits for the flow possessing multiple critical points select the critical point with the higher entropy accretion rate, confirming that the entropy accretion rate is the degeneracy removing agent in the system. However, heteroclinic orbits are also observed for some special situation, where both the saddle type critical points of the flow configuration possesses identical entropy accretion rate. Topologies with heteroclinic orbits are thus the only allowed non-removable degenerate solutions for accretion flow with multiple critical points, and are shown to be structurally unstable. Depending on suitable initial boundary conditions, a homoclinic trajectory can be combined with a standard non-homoclinic orbit through an energy preserving Rankine-Hugoniot type of stationary shock, and multi-critical accretion flow then becomes truly multi-transonic. An effective Lyapunov index has been proposed to analytically confirm why certain class of transonic flow cannot accommodate shock solutions even if it produces multiple critical points.     

Strongly-cooled ionizing plasma flows with application to venus

On the solar wind's penetration into an atmosphere of hydrogen or helium, symmetric charge exchange interactions give energy and momentum losses as the dominant source terms in the flow equations. One-dimensional, supersonic to subsonic solutions are available if the cooling is strong enough. In a model with transverse field and adiabatic (non-thermal) ions, a range of weakly-shocked solutions with upstream mach number less than 2.5 are discovered. As in the case of detonation waves, the shock strength is independent of downstream boundary conditions. The solutions may apply in the solar wind flow into the Venusian atmosphere.     

Solar wind expansion with a heat-loss function

The problem of steady expansion of the solar wind is investigated allowing for heating and cooling mechanism in the medium. Single fluid equations are employed, neglecting viscosity and magnetic fields. It is found that the flow characteristics are modified in the regions near the Sun though the location of the critical point and the flow speeds beyond it are unaffected. It is surmised that the solar wind must originate at higher levels if it is subject to a heat-loss due to bremsstrahlung.     

On convection in the sun

Convective motions driven by a superadiabatic temperature gradient in a viscous thermally conductive medium are considered. Approximate linearized equations governing the perturbation are derived under the following conditions: (i) The ratio of the excess temperature gradient over the adiabatic gradient is small compared with the gradient itself, (ii) The perturbation is of low-frequency type, (iii) The rotation is slow. Only the convective mode is described by these equations (as in the Boussinesq approximation), and the equations are valid for compressible configurations with any ratio between the scale heights of the equilibrium and perturbed quantities. Results of a numerical calculation of unstable perturbations for configurations with a large density stratification are given. They show that under conditions appropriate for the solar convection zone an extremely strong instability is expected to occur if the mixing length is assumed to be equal to 1.5 times the pressure scale height. The horizontal scale of the instability is intermediate between those of granulation and supergranulation. The larger the mixing length, the smaller the growth rate of the instability, and the larger its horizontal scale. Therefore it seems possible to adjust the mixing length to obtain the characteristics corresponding to those of the solar supergranulation. The possible origin of the granulation as an instability in a subsurface zone, where a local increase in the density scale height takes place, is also discussed. To achieve agreement with observations, it seems necessary to assume that the ratio of the mixing length to pressure scale height is an increasing function of the pressure.     

Long-term evaluation of three-dimensional heliocentric solar sail trajectories with arbitrary fixed sail setting

The solar radiation effects upon the orbital behaviour of an arbitrarily shaped spacecraft (or a solar sail in particular) in a general fixed orientation with respect to the local coordinate frame are investigated. Through introduction of a quasi-angle in the osculating plane, the motion of the orbital plane becomes uncoupled from the in-plane perturbations. Exact solutions in the form of conic sections and logarithmic spirals can readily be formulated for certain specific initial conditions. An effective out-of-plane spiral transfer trajectory is obtained by reversing the force component normal to the orbital plane at specified positions in the orbit. By choosing the appropriate control angles for the sail orientation, any point in space can be reached eventually. In the case of general initial conditions, the long-term orbital behaviour is assessed asymptotically by means of the two-variable expansion procedure. An implicit expression for the eccentricity is derived and explicit results are established by an iteration scheme. The other orbital elements can be expressed in terms of the eccentricity and their asymptotic series for near-circular initial orbits are also obtained. While equations for the higher-order contributions as well as the periodic parts of their solutions can be formulated readily, their secular terms are determined only for a circular initial orbit.     

The effect of finite electrical conductivity on the angular-momentum loss of the sun due to the solar wind

In general the solutions to the azimuthal equations of motion of the solar wind possess a singularity at the radial Alfvénic critical point if the fluid is non-viscous and has an infinite electrical conductivity. This singularity can only be removed by a proper choice of boundary conditions. If a plasma with a large but finite conductivity is considered, the singularity is removed in the mathematical sense. However, it is shown that for this latter case the actual solution does not differ significantly from the former model and that the boundary condition is still determined from the behavior of the solution at or near the Alfvénic critical point.Kitt Peak National Observatory Contribution No. 411.Operated by The Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

A new shock locus for similarity solutions in one-dimensional unsteady gas dynamics

This paper deals with shock conditions for the progressing wave (or similarity) solutions of one-dimensional, unsteady gas dynamics. These solutions have hitherto been used to deal with the flow behind shocks moving into stationary atmospheres. By generalising the shock conditions to the case of moving atmospheres, it is shown that the progressing wave solutions can be used to describe a certain class of flows, and a new shock locus can be constructed in the phase plane of the solutions. It is hoped that such solutions will be of use in describing the unsteady flow behind shocks propagating into the ambient solar wind.