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harmonic model

harmonic model The representation of a magnetic or gravitational field by a scalar potential V(x,y,z) satisfying Laplace’s equation2V = 0, making V “harmonic”. Because of the spherical geometry of the Earth, both its gravity field and magnetic field are customarily expanded in spherical harmonics, which naturally group the expressions that make up V into monopole, dipole, quadrupole, octopole (etc.) terms, decreasing with radial distance r as 1/r, 1/r2, 1/r3,, 1/r4, etc. The rate at which corresponding field components decrease is larger by one power of r, i.e., these decrease as 1/r2, 1/r3,, 1/r4, 1/r5, etc.

The Earth’s gravity field is dominated by its monopole term, but the axisymmetric terms of higher order m, up to m = 6 (terms whose potential decreases like 1/rm+1) are also needed in accurate calculation of satellite orbits and careful satellite studies give terms up to m ≈ 20. The magnetic field B of the Earth inherently lacks the monopole term and its leading term, which dominates it, is the dipole term. Higher orders can also be fairly important near Earth, while far from Earth additional field sources need to be considered (see empirical models).

Since the Earth’s magnetic field gradually changes with time (“secular variation”), scientists periodically extract from magnetic surveys and observations of each epoch (usually 10 years) an International Geomagnetic Reference Field (IGRF), a harmonic model meant to give (for that epoch) the best available representation of the internal magnetic field and its rates of change, expressed by a given set of spherical harmonic coefficients and their time derivatives. The magnetic fields of other planets have also been represented in this manner, but because of the scarcity of observations, their harmonic models have a much lower accuracy.

heat capacity The thermodynamic quantity dQ/dT , where dQ is an increment in heat energy, and dT is the corresponding increment in temperature. Always specified with some thermodynamic variable held fixed, as heat capacity at constant volume, or heat capacity at constant pressure.

heat flow The study of how bodies generate interior heat and transport this heat to their sur-

faces; a subdiscipline of the field of geophysics. Most planetary bodies begin with substantial amounts of interior heat. Among the sources of such heat are the heat leftover from the formation of the body (accretion), heat produced by differentiation, heat produced by radioactive decay, tidal heating, and solar electromagnetic induction. This heat can melt interior materials, producing magma which can later erupt onto the body’s surface as volcanism. The heat contained in the body’s interior is transported to the surface of the body, where it escapes to space. The three ways in which this energy can be transported through the interior are by radiation (the absorption and reemission of energy by atoms), convection (physical movement of material, with hot material rising and cool material sinking), and conduction (transfer of energy by collisions between atoms). Larger bodies are more efficient at retaining their interior heat, which translates to a longer period of geologic activity. The thermal evolution of a body can be estimated by determining what mechanisms are responsible for its heating, how the body transports that energy to the surface, and how long the body can retain its internal heat.

heat flow density See heat flux.

heat flux The flow of heat energy per unit area and per unit time. It is often called heat flow density or heat flow in geophysics.

Heaviside, Oliver (1850–1925) Physicist and mathematician. Developed the modern vector form of Maxwell’s equations and understanding of the classical electrodynamics (via fundamental physical effects predicted and evaluated by him). He also developed the vector and operational calculi, the ideas and applications of δ- and step-functions, as well as many practical applications of Maxwell’s theory in telephony and electromagnetic waves propagation in the atmosphere (the ionospheric layer, thus long-range radio communications). Heaviside wrote in the telegraph equation and analyzed its technological consequences in 1887, predicted

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Cerenkov radiation in 1888, was the first to introduce the Lorentz force (in 1889, three years before H.A. Lorentz), and he predicted the existence of the Heaviside–Kennelly ionized atmo-

© 2001 by CRC Press LLC

heliosheath

spheric layer in 1902. His highly informal and heuristic approach to mathematics led to general disregard for his results and innovations on the part of formally thinking arbiters of the mathematical and theoretical fashion of his time.

From 1870 to 1874 Heaviside worked as a telegraphist in Newcastle. He began his own experimental research in electricity in 1868, publishing his first paper in 1872. From 1874 Heaviside never had any employment, performing all his research privately at home in London. He lived his later years in need and even in poverty. Heaviside was elected a Fellow of the Royal Society (1891) and honorary member of the American Academy of Arts and Sciences (1899), then awarded Doctor Honoris Causa of the Göttingen University (Germany) in 1905, honorary memberships of the Institute of Electrical Engineers (1908) and the American Institute of Electrical Engineers (1919), and awarded the Faraday Medal of the Institute of Electrical Engineers in 1921.

heavy minerals A generic term used to denote beach sediments that have a specific gravity significantly greater than that of the common quartz and feldspar components of many beach sands. Includes hornblende, garnet, and magnetite, among others, and often appears as dark bands on a beach.

Hebe Sixth asteroid to be discovered, in 1847. Orbit: semimajor axis 2.4246AU, eccentricity 0.2021, inclination to the ecliptic 14◦.76835, and period 3.78 years.

hedgehog configuration A configuration frequently encountered in various condensed matter phase transitions where molecules point outward away from a pointlike topological defect. It is also used to describe monopoles produced during cosmic phase transitions. See also monopole.

Helene Moon of Saturn, also designated SXII. It was discovered by Laques and Lecacheux in 1980. Its orbit has an eccentricity of 0.005, an inclination of 0◦, a semimajor axis of 3.77 ×105 km, and it orbits Saturn at the leading Lagrange point in Dione’s orbit. Its size is 18 × 16 × 15 km, but its mass is not known.

Its geometric albedo is 0.7, and it orbits Saturn once every 2.737 Earth days.

heliacal rising The first visibility of an astronomical object (star) in the predawn sky, after months of being invisible by virtue of being up in day.

helicity In plasma physics, the sense and amount of twist of magnetic fields characterized by several different parameters. The density of magnetic helicity is Hm = A·B, where

A is the magnetic vector potential of the magnetic field, B. Hm determines the number of linkages of magnetic field lines. The density of current helicity, Hc, is B · ×B which varies in a way similar to Hm and describes the linkage of electric currents. In hydrodynamics, one of two quadratic invariants (the other is energy) occurring in the theory of three-dimensional incompressible Navier–Stokes turbulence. The helicity within a volume V is defined as

HNS = ω · Vd3x ,

where V is the velocity, ω =curl V is the vorticity, and the integral is taken over V. In a dissipation-free fluid, for suitable boundary conditions, HNS is conserved. In fully developed three-dimensional turbulence with viscous dissipation, the helicity, as well as the energy, cascades from large eddies down to smaller eddies where dissipation can occur. See cross helicity, hydromagnetic turbulence, magnetic helicity.

heliocentric Centered on the sun, as in the Copernican model of the solar system.

heliopause The boundary between the heliosphere and local interstellar medium. See heliosphere, solar wind.

helioseismology The science of studying wave oscillations in the sun. Temperature, composition, and motions deep in the sun influence the oscillation periods. As a result, helioseismology yields insights into conditions in the solar interior.

heliosheath Shocked solar wind plasma, bounded on the inside by the heliospheric ter-

© 2001 by CRC Press LLC

Helios mission

mination shock, and on the outside by the heliopause. See heliosphere, solar wind.

Helios mission German–US satellite mission to study the inner heliosphere. The instrumentation includes plasma, field, particle, and dust instruments. Two identical satellites, Helios 1 and 2, were launched into highly elliptical orbits with a perihelion at 0.3 AU and an aphelion at 0.98 AU. The combination of the two satellites allowed the study of radial and azimuthal variations; the mission lasted from 1974 to 1986 (Helios 1) and 1976 to 1980 (Helios 2).

heliosphere The cavity or bubble in the local interstellar medium due to the presence of the solar wind. The size of the heliosphere is not yet established, but typical length scales must be of order one to several hundred astronomical units. Heliospheric plasma and magnetic field are of solar origin, although galactic cosmic rays and neutral interstellar atoms do penetrate into the heliosphere.

The heliosphere is thought to comprise two large regions; the interior region is the hypersonic solar wind, separated from the exterior shocked-plasma (heliosheath) region by the heliospheric termination shock. The boundary between the heliosheath and local interstellar medium is called the heliopause.

If the flow of the local interstellar medium is supersonic with respect to the heliosphere, a termination shock will be formed in the interstellar gas as it is deflected by the heliosphere. Because of substantial temporal variations in the solar wind (and, possibly, in the local interstellar medium) it is likely that the termination shock and heliopause are never static, but undergo some sort of irregular inward and outward motions.

heliospheric current sheet (HCS) The current sheet that separates magnetic field lines of opposite polarity which fill the northern and southern halves of interplanetary space (“the heliosphere”). The sun’s rotation, combined with the stretching action of the solar wind, gives the HCS the appearance of a sheet with spiral waves spreading from its middle. The spiral structures are responsible for the interplanetary sectors ob-

served near the Earth’s orbit. See interplanetary magnetic sector.

heliospheric magnetic field The magnetic field that fills the heliosphere. Because coronal and heliospheric plasmas are excellent electrical conductors, the magnetic field is “frozen into” the expanding gas. Solar wind gas, once it has accelerated away from its coronal-hole origin, is hypersonic and hyper-Alfvénic, so its kinetic energy exceeds its magnetic energy and the field is passively carried along by the wind. The field lines are anchored in a rotating solar source, but carried along by a wind flowing in the outward radial direction, and may be idealized as lying on cones of constant heliographic latitude within which they are twisted to form a global spiral pattern.

The angle ψ between the field and the radial direction (called the Parker spiral angle) is given by tanψ = r-cosλ/V , where r is heliocentric distance, - is the angular velocity of rotation of the sun, λ is heliographic latitude, and V is the solar wind flow speed. In the ecliptic plane at 1 AU, ψ is of order 45◦, and as r→∞ the field is transverse to the flow direction. In situ observations of the heliospheric field are in accord with this idealized global picture, although modest quantitative deviations have been reported.

Although the magnetic field at the sun’s surface is very complicated, there is an underlying dipole pattern except perhaps for brief periods near the time of maximum sunspot activity, when the solar magnetic dynamo reverses its polarity. When the underlying dipole component is present, the polarity of the heliospheric magnetic field at high northern or southern heliographic latitudes coincides with the polarity of the magnetic field in the corresponding highlatitude regions of the solar surface. For example, the Ulysses spacecraft’s mid-1990s observations of high-latitude fields show outward polarity at northern latitudes and inward polarity at southern latitudes; these polarities will be reversed in the next sunspot cycle.

Thus, the heliospheric magnetic field may be characterized in the first approximation as consisting of two hemispheres consisting of oppositely directed spiraling field lines. These two hemispheres are separated by a thin current sheet, called the heliospheric current sheet.

© 2001 by CRC Press LLC

helium

Now the heliospheric field is not perfectly aligned with the solar poles and equator (equivalently, the underlying solar dipole is tilted with respect to the rotation axis), so that the heliospheric current sheet is tilted with respect to the heliographic equator. Moreover, the solar surface field is always far from being a perfect dipole, and the heliospheric current sheet itself is warped.

The sun rotates, and the heliospheric magnetic field configuration must rotate as well. Hence, a fixed observer near the equatorial plane will be immersed first in field of one polarity (inward, say), then in field of outward polarity, then inward again, etc. The net result is that the magnetic record shows two or more sectors of opposite polarity, and this pattern is usually approximately repeated in the next solar rotation. The phenomenon is ubiquitous at low heliographic latitudes, and in particular at each planet the magnetic field in the solar wind will exhibit alternating polarity. This organization of the low-latitude heliospheric field into sectors of opposite polarity is known as the interplanetary magnetic sector structure. The heliospheric current continues to exist at the largest heliocentric distances where measurements have been made, and presumably extends out to the heliospheric current shock.

The heliospheric magnetic field, like all quantities observed in the solar wind, also exhibits a rich variety of transitory variations. The large-scale morphology described above is a background on which waves, turbulence and other transient structures are superposed.

heliospheric stream structure Longitudinal organization of the solar wind into faster and slower streams (also called interplanetary stream structure). Apart from the occasional violent outbursts associated with coronal mass ejections, the fastest solar wind comes from coronal holes. Coronal holes most often are found at relatively high solar latitudes, so that on average the high-latitude wind is faster than the wind at equatorial latitudes (velocity 800 km/s as against 400 km/s). However, the polar coronal holes often have equatorward extensions for a substantial range of longitudes, so that there is a substantial longitudinal variation in the equatorial solar wind speed.

For much of the solar activity cycle this longitudinal structure is long-lived enough that the wind forms stream patterns that approximately corotate with the sun. It should be stressed that this corotation is wave-like in the sense that although the pattern may corotate out to large heliocentric distances, the plasma itself, which is subject only to relatively weak magnetic torques, does not corotate. In fact, beyond a few solar radii the flow must be essentially radial, so the rotation of the sun sets up a situation in which fast wind will overtake slow wind from below.

The interaction between fast and slow wind occurs over several astronomical units in heliocentric distance. By 1 a.u. there is substantial compression near the stream interfaces. By 5 a.u. the interaction has proceeded to the point that regions of compressed plasma bounded by shocks, the so-called corotating interaction regions or CIRs, are common; the amplitude of the stream-structure velocity variation at 5 a.u. is substantially smaller than at 1 a.u. This erosion of the velocity structure continues as the wind flows farther out, and beyond about 10 a.u. the CIRs are no longer apparent.

heliospheric termination shock The shock wave associated with the transition from supersonic to subsonic flow in the solar wind; see solar wind, heliosphere. The location of the termination shock is estimated to be at 100 astronomical units from the sun. Outbound heliospheric spacecraft have passed 60 astronomical units heliocentric distance, but have not yet encountered the termination shock. There is optimism that Voyagers 1 and 2 will encounter the shock within the next decade or two.

helium (From Greek. helios, the sun.) An odorles colorless gas; the second lightest (after hydrogen) gas, atomic number: 2. Consisting of two isotopes: 3He, and 4He. 4He is the second most abundant element in the universe (presumably formed early on in the Big Bang), but is very rare on Earth (partial pressure ≈ 10−5 atm) at the surface of the Earth. The helium content of the atmosphere is about 1 part in 200,000.

Helium was first detected in 1868 in the solar spectrum and in 1895 in uranium-containing minerals. In 1907, alpha particles were demon-

© 2001 by CRC Press LLC

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