Force-free equilibrium at magnetic neutral points

It is shown that, at neutral points of force-free magnetic fields, the electric current density must vanish. This property is independent of whether the neutral points are isolated or (e.g.) fill lines or surfaces. One implication is the fact that in a cold pressureless plasma the formation of neutral current sheets cannot be adiabatically slow. The field-line topology in the neighbourhood of neutral points is discussed. At neutral points of force-free magnetic fields in general three constant- surfaces, defined by the equation ×B=B, with the same value of intersect orthogonally. If, during a time-development, the magnetic field gradient matrix B _i/x _j becomes singular at a neutral point, the field topology can change qualitatively — in general connected with the merger of two or more neutral points into one and/or the splitting up of one neutral point into several others. This can be interpreted as implying the transition from a quasi-static evolution to a dynamical state in which magnetic energy is released.     

On the reconstruction of the nonlinear force-free coronal magnetic field from boundary data

Using a simple model in which the corona is represented by the half-space domain = {z > 0} and the photosphere by the boundary plane = {z = 0}, we discuss some important aspects of the general problem of the reconstruction of the magnetic field B in a small isolated coronal region from the values of the vector B¦ measured by a magnetograph over its whole basis. Assuming B to be force-free in : (i) we derive a series of relations which must be necessarily satisfied by the boundary field B¦ , and then by the magnetograph data if the force-free assumption is actually correct; (ii) we show how to extract directly from the measured B¦ some useful informations about the energy of B in and the topological structure of its field lines; (iii) we present a critical discussion of the two methods which have been proposed so far for computing effectively B in from B¦ .     

A Study of the Lunar Soil in Regions with Temperature Anomalies

Based on a large body of observational data on radio emission from the Moon, we study the dependence of lunar radio temperature variations on illumination conditions. The data were obtained with the RATAN-600 radio telescope with a high sensitivity and resolution, which has not yet been used to construct radio images of the Moon. The harmonic parameters (amplitudes and phase angles) were determined both for the average Moon and for regions with temperature anomalies revealed by the RATAN-600 observations. These parameters allow the physical properties of the lunar soil to be investigated. The distribution of the loss-angle tangent (tan ), one of the characteristics of the lunar material, over the Moon was determined. The loss-angle tangent is related to the content of ilmenite, a rock containing oxygen, iron, and titanium, in the lunar soil. Studies of the ilmenite distribution on the Moon are particularly important in view of the prospects for building a habitable lunar base that needs oxygen. It is relatively easy and cheap to extract oxygen, iron, and titanium from ilmenite.     

The sunlit lunar surface

Directional infrared emission from the sunlit lunar surface is determined for the thermal meridian and as a function of observer elevation and azimuth angles at three Sun elevation angles. A study of selected mare sites at full Moon suggests that brightness temperatures are relatively insensitive to changes in certain surface parameters, such as the photometric function, emissivity, and thermophysical properties of the soil. The observed deviations from predictions for an average surface can be accounted for by changes in surface roughness.Deceased 12 January, 1971.     

The formation of spicules in the course of the chromospheric network magnetic field reconnection

We investigate the physical processes occurring in the supergranule boundary cylinder layer (SBCL). Taking into account the Coriolis force, we obtain an expression for the component of the magnetic field and velocity in the SBCL. Within the framework of linear MHD, we consider the formation and coalescence of magnetic tubes, i.e. spicules, in the course of the reconnection of the SBCL magnetic field. The estimated number of spicules appearing on each supergranule cell is in agreement with observations. This number depends on the solar latitude : (1) if the normal component of the magnetic fieldB _z is assumed to be independent of , then the maximum number of spicules should be at = 71°; (2) ifB _z is assumed to be the component of the dipolar fieldB _z sin , then the maximum number should be at the pole: = 90°. The timescale of the formation and the coalescence of the magnetic tubes is 10–20 min, which is of the order of the observed lifetime of the spicules.     

The physical librations of the moon,including higher harmonic effects

The equations of the physical libration of the Moon are developed using a representation of the Earth-Moon orbit as a Kepler ellipse referred to the lunar equator and expanding the lunar potential in terms of these Kepler elements. TheImproved Lunar Ephemeris is used to calculate solar perturbations, and a linear integration of all effects arising from lunar gravitational harmonics through the fourth degree performed. Aside from unobservable constant offsets of the principal axes, the main effects of the higher harmonics on longitude are: 10 six-yearly (argument), 1.2 three-yearly, 0.5 annual, and 0.1 monthly; on pole direction they are on the order of 0.5 six-yearly and 1.0 monthly. The higher harmonics must hence be taken into account in analyzing ranging data of 10 cm accuracy.Paper dedicated to Prof. Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

Ion cyclotron resonant instability of RH waves propagating at an angle to the interplanetary magnetic field

The anomalous Doppler-shift interaction between positive ions and right-hand (RH) polarized E.M. waves propagating at a small angle to a static magnetic field is investigated. The linear rate of growth of the resulting instability is obtained and compared with the growth rate for the parallel propagation case. For conditions typical of the solar wind at about 1 AU, the rate of growth always decreases with increasing propagation angle. For very large ion pressures (1) and temperature anisotropies (T T 1), the rate of growth may increase with increasing propagation angle.     

Density within the moon and implications for lunar composition

Density models for the Moon, including the effects of temperature and pressure, can satisfy the mass and moment of inertia of the Moon and the presence of a low density crust indicated by the seismic refraction results only if the lunar mantle is chemically or mineralogically inhomogeneous. IfC/MR ² exceeds 0.400, the inferred density of the upper mantle must be greater than that of the lower mantle at similar conditions by at least 0.1 g cm^–3 for any of the temperature profiles proposed for the lunar interior. The average mantle density lies between 3.4 and 3.5 g cm^–3, though the density of the upper mantle may be greater. The suggested density inversion is gravitationally unstable, but the implied deviatoric stresses in the mantle need be no larger than those associated with lunar gravity anomalies. UsingC/MR ³=0.400 and the recent seismic evidence suggesting a thin, high density zone beneath the crust and a partially molten core, successful density models can be found for a range of temperature profiles. Temperature distributions as cool as several inferred from the lunar electrical conductivity profile would be excluded. The density and probable seismic velocity for the bulk of the mantle are consistent with a pyroxenite composition and a 100 MgO/(MgO+FeO) molecular ratio of less than 80.Communication presented at the Lunar Science Institute Conference on Geophysical and Geochemical Exploration of the Moon and Planets, January 10–12, 1973.     

A magnetic dynamo in the moon?

The existence of fossil lunar magnetism has caused speculation that the Moon had, at one time, an internally produced dynamo magnetic field. Quantitative analysis of this idea, constrained by the largest iron lunar core compatible with observations, implies that the Moon would have had to rotate faster than its breakup angular velocity in order to support a dynamo magnetic field.A paper presented at the Lunar Science Institute Conference on Geophysical and Geochemical Exploration of the Moon and Planets, January 10–12, 1973.     

Coronal magnetic-field model with non-spherical source surface

Previous global models of coronal magnetic fields have used a geometrical construction based on a spherical source surface because of requirements for computational speed. As a result they have had difficulty accounting for (a) the tendency of full magnetohydrodynamic (MHD) models to predict non-radial plasma flow out to r 10r and (b) the appreciable magnitude, 3, of B _r , (the radial component of B) consistently observed at r 1 AU. We present a new modelling technique based on a non-spherical source surface, which is taken to be an isogauss of the underlying potential field generated by currents in or below the photosphere. This modification of the source surface significantly improves the agreement between the geometrical construction and the MHD solution while retaining most of the computational ease provided by a spherical source surface. A detailed comparison between the present source-surface model and the MHD solution is made for the internal dipole case. The resulting B field agrees well in magnitude and direction with the coronal B field derived from the full MHD equations. It shows evidence of the slightly equatorward meridional plasma flow that is characteristic of the MHD solution. Moreover, the B field obtained by using our non-spherical source surface agrees well with that observed by spacecraft in the vicinity of the Earth's orbit. Applied to a solar dipole field with a moment of 1 G-r ³ , the present model predicts that B _r at r 1 AU lies in the range of 1–2 and is remarkably insensitive to heliomagnetic latitude. Our method should be applicable also to more general (i.e., more realistic) configurations of the solar magnetic field. Isogauss surfaces for two representative solar rotations, as calculated from expansions of observed photospheric magnetic-field data, are found to show large and significant deviations from sphericity.     

Околозвездньiе облака поданньiм ультрафиолетовьiх спектров свезл

, ii (2000–3000 Å) i . , i . i (. 2). i i i i + ( 7–10). ii (. 13). ii i i (, 2400 Å) (. 14 15). i i i , iu , i (. 1). i i ii i i . .     

On the uniqueness of the determination of the coronal potential magnetic field from line-of-sight boundary conditions

We consider a simple model in which the coronal magnetic field B is assumed to be potential in the region between the solar surface _o and an exterior source-surface ₁ of arbitrary shape. We prove that the boundary value problem that determines B from the value B _lof its component on ₀ along either (orthoradial direction) or (fixed direction) has at most one solution. On the other hand, we show that a solution can exist only if B _lsatisfies some solubility conditions.     

New approach to determining planetary perturbations in lunar theory

The technique of the general planetary theory has been proposed for constructing a theory of motion of the Moon. This method enables us to elaborate the consistent theory of motion of the principal planets and the Moon, which is of particular importance for determining planetary perturbations in lunar motion. As an initial approximation for lunar motion, an intermediate orbit generalizing the Hill's variational curve has been built. This orbit includes all solar and planetary inequalities independent of eccentricities and inclinations of the Moon, Sun and planets. In calculating this orbit, the motion of the principal planets in quasi-periodic intermediate orbits has been taken into account. This solution was produced with the aid of the Universal Poissonian Processor (UPP) elaborated in the Institute for Theoretical Astronomy (Leningrad).Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

Спектр и поляризация источника синхротронного излучения при релятивистском разлете его компонент

, , , . () , . , .     

Quantum effects in cyclotron plasma absorption

It is shown that X-ray radiation of neutron stars with magnetic fieldsB=10¹¹–10¹³ G near cyclotron resonances=s _B (s=1,2,...) is deeply affected by such quantum effects as electron-positron vacuum polarization (significant at V=3×10²⁸ n _e ^–1 (B/B _C ⁴)1, whereB _C =4.4×10¹³G), the quantizing character of the magnetic field (significant atV=3 x 10²⁸ n _e ^–1 (B/B _c)⁴1 whereB _c =4.4 x 10¹³G), the non-harmonic character of the Landau levels, and the quantum recoil of electrons. The latter two factors shift the resonances by the frequency –s ² _B (B/2B _c )sin², being the angle between the direction of radiation propagation and the magnetic field. IfVV ₀ (for 1,V ₀^–1=(mc ²/2T)^1/2), the normal mode (NM) polarizations, as well as the absorption coefficientk ₁ of the extraordinary NM in the Doppler core of the first resonance (|–| _B cos ), is only slightly affected by varyingb and/orV, whereas for the ordinary NM (at 1)k ₂k ₁ ²[b + (3 + tan²–2V)²]k ₁. For sufficiently largeb and/orV the quantum effects amplify resonant absorption of the ordinary NM at _B , with spin-flip transitions playing a major role atb1+V ². IfVV ₀, the coefficientsk ₁ andk ₂ in the Doppler core of the resonance are of the same order and acquire some peculiar features (shifts, intersections, etc.), with the NM polarizations depending sharply on and being strongly non-orthogonal. AtVV ₀,k ₂=k ₁(cos² +B/2B _C ) and the polarizations are almost linear. Near high resonances (s2), as a rule,k _1,2(1 + b) ^{s–1 2s–3} i.e., absorption increases withb due to replacement of the thermal energy of the transverse motion of electron,T, by the magnetic energy _B . The above effects should be taken into account for an interpretation of observational data on X-ray pulsars (e.g., Her X-1) and other X-ray sources associated with neutron stars.     

Electrical conductivity of neutron star cores in the presence of a magnetic field

General theory of electrical conductivity of a multicomponent mixture of degenerate fermions in a magnetic fieldB, developed in the preceding article (this volume), is applied to a matter in neutron star interiors at densities ₀, where ₀ = 2.8×10¹⁴ g cm^–3 is the standard nuclear matter density. A model of free-particle mixture ofn, p, e is used, with account for appearance of ^–-hyperons at > _c , where _c 4₀. The electric resistivities along and acrossB, and , and the Hall resistivity _H are calculated and fitted by simple analytical formulae at _c and > _c for the cases of normal or superfluid neutrons provided other particles are normal. Charge transport alongB is produced by electrons, due to their Coulombic collisions with other charged particles; is independent ofB and almost independent of the neutron superfluidity. Charge transport acrossB at largeB may be essentially determined by other charged particles. If _c , one has = [1 + (B/B ₀)²] for the normal neutrons, and for the superfluid neutrons, while _H = B/B _e for both cases. HereB _e 10⁹ T ₈ ² G,B ₀10¹¹ T ₈ ² G, andT ₈ is temperature in units of 10⁸ K. Accordingly for the normal neutrons atBB ₀, the transverse resistivity suffers an enhancement, _{1/4 1}. When 5₀ andB varies from 0 toBB _p 10¹³ T ₈ ² G, increases by a factor of about 10³–10⁴ and _H changes sign. WhenBB _p , remains constant for the superfluid neutrons, and _H B ² for the normal neutrons, while _H B for any neutron state. Strong dependence of resistivity onB, T, and may affect evolution of magnetic fields in neutron star cores. In particular, the enhancement of at highB may noticeably speed up the Ohmic decay of those electric currents which are perpendicular toB.     

On the Dynamics of Weak Stability Boundary Lunar Transfers

Recent studies demonstrate that lunar and solar gravitational assists can offer a good reduction of total variation of velocity Vneeded in lunar transfer trajectories. In particular the spacecraft, crossing regions of unstable equilibrium in the Earth—Moon—Sun system, can be guided by the Sun towards the lunar orbit with the energy needed to be captured ballistically by the Moon. The dynamics of these transfers, called weak stability boundary (WSB) transfers, will be studied here in some detail. The crucial Earth—Moon—Sun configurations allowing such transfers will be defined. The Sun's gravitational effect and lunar gravitational capture will be analyzed in terms of variations of the Jacobi constants in the Earth—Sun and Earth—Moon systems. Many examples will be presented, supporting the understanding of the dynamical mechanism of WSB transfers and analytical formulas will be obtained in the case of quasi ballistic captures.This revised version was published online in October 2005 with corrections to the Cover Date.     

Bipolar outflows from stars and galaxies as a tornado phenomenon

At an early stage in the lives of stars and galaxies when they are surrounded by discs, vorticity in the disc concentrates into a central vortex, thus converting a Keplerian velocity fieldu _ø r ^–1/2 into an irrotational velocity fieldu _ør ^–1, which implies inward transfer of angular momentum. Centrifugal forces due to spin-up of the inner region and gravity dominant in the outer region then squeeze gas at intermediate layers, increasing pressure gradient in the axial direction sufficiently to drive a wide-angle low-velocity bipolar outflow from the disc. A logarithmic singularity of vorticity at the axis implies strong centrifugal forces which expand plasma to radiusR where pressure gradient balances centrifugal force density of ions; the much weaker centrifugal force density of electrons cannot balance pressure gradient, so that electrons are driven inwards relative to ions until charge separation limits the relative displacement. Now the radial gradient ofu _øcauses ions to rotate at a different rate to electrons, generating an azimuthal current densityj _øwhich is the source of an axial magnetic fieldB _zin the core of the central vortex. Centrifuging carries lines of B to the core wall, where they are wound into helical force-free configuration with B j. An annular channel of radiusR and thickness R into which parallel helical lines ofj andB are compressed constitutes a magnetic vortex tube (MVT). An MVT separates an inner high-velocity highly collimated outflow from the outer low-velocity wide-angle outflow, and is responsible for jets. Magnetic pinches in the MVT may constrict the core flow at HH objects.     

Lunar physical librations and laser ranging

The analysis of lunar laser ranging data requires very accurate calculations of the lunar physical librations. Libration terms are given which arise from the additive and planetary terms in the lunar theory. The large size of the recently discovered terms due to third degree gravitational harmonics will allow some of these harmonics to be measured, in addition to and, by laser ranging to the Moon. Combining the laser ranging determinations of = 630.6 ± 0.5 × 10^–6 and = 226.4 ± 3.0 × 10^–6 with lunar orbiter measurements ofC ₂₀ andC ₂₂ givesC/MR ²=0.395 _-0.010 ^+0.006 . Numerical integration promises to be an effective method of calculating librations. Comparison of numerical integrations with analytic series indicates that the calculation of the series due to third and fourth degree harmonics is not yet as accurate as the more extensively developed second degree terms.Communication presented at the Conference on Lunar Dynamics and Observational Coordinate Systems, held January 15–17, 1973, at the Lunar Science Institute, Houston, Tex., U.S.A.     

Doubly averaged effect of the Moon and Sun on a high altitude Earth satellite orbit

Infinite series expansions are obtained for the doubly averaged effects of the Moon and Sun on a high altitude Earth satellite, and the results used to interpret numerically integrated examples. New in this paper are: (1) both sublunar and translunar satellites are considered; (2) analytic expansions include all powers in the satellite and perturbing body semi-major axes; (3) the fact that retrograde orbits have more benign eccentricity behavior than direct orbits should be exploited for high altitude satellite systems; and (4) near circular orbits can be maintained with small expenditures of fuel in the face of an exponential driving force one forI _ab, whereI _b=180°–I _a andI _a is somewhat less than 39.2° for sublunar orbits and somewhat greater than 39.2° for translunar orbits.Nomenclature a semi-major axis - A _lk coefficient defined in Equation (11) - B _lk coefficient defined in Equation (24) - C _km coefficient defined in Equation (25) - D, E, F coefficients in Equations (38), (39) - e eccentricity - H _k expression defined in Equation (34) - expression defined in Equation (35) - I inclination of satellite orbit on lunar (or solar) ring plane - J ₂ coefficient of second harmonic of Earth's gravitational potential (1082.637×10^–6 R _E ² ) - K _k, L_k, M_k expressions in Section 4 - expressions in Section 4 - p=a(1–e ²) semi-latus rectum - P _l Legendre polynomial of degreel - q argument of Legendre polynomial - radial distance of satellite - R _E Earth equatorial radius (6378.16 km) - R, S, W perturbing accelerations in the radial, tangential and orbit normal directions - syn synchronous orbit radius (42 164.2 km=6.6107R _E) - t time - T satellite orbital period - T orbital period of perturbing body (Moon) - T _e period of long periodic oscillations ine for |I|<I _a - T _s synodic period - U gravitational potential of lunar (or solar) ring - x, y, z Cartesian coordinates of a satellite with (x, y) being the ring plane - coefficient defined in Equation (20) - average change in orbital element over one orbit (=a, e, I, , ) - ₁,₂₃ unit vectors in thex, y, z coordinate directions - _r , _s , _w unit vectors in the radial, tangential and orbit normal directions - =+ angle along the orbital plane from the ascending node on the ring plane to the true position of the satellite - angle around the ring - gravitational constant times mass of Earth (3.986 013×10⁵ km s^–2) - gravitational constant times mass of Moon (or Sun) - _m gravitational constant times mass of Moon (/81.301) - _s gravitational constant time mass of Sun (332 946 ) - ratio of the circumference of a circle to its diameter - radius of lunar (or solar) ring - _m radius of lunar ring (60.2665R _E) - _s radius of solar ring (23455R _E) - true anomaly - argument of perigee - ₀ initial value of - _i critical value of in quadranti(i=1, 2, 3, 4) - longitude of ascending node on ring plane This work was sponsored by the Department of the Air Force.