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Vasyliunas theorem

with gαβ the components of the inverse metric

tensor (α = 0, 1, 2, 3, corresponding to t, x, y, z), ηαβ = diagonal(−1, 1, 1, 1) in the usual

rectangular coordinates (thus √|g| = 1 in these coordinates).

In this case the variations are of the form

φ(t, x, y, z) → φ(t, x, y, z) + δφ(t, x, y, z), and an additional requirement is that the integration be considered over a finite 4-dimensional domain, and the variation vanish on the spatial boundary of the domain (the field values are held fixed there), as well as at the time end values. The equation of motion is then

α αφ ≡ φ =

 

 

 

 

−

∂2

∂2

∂2

∂2

φ = 0 .

 

+

 

+

 

+

 

∂t2

∂x2

∂y2

∂z2

See action, Lagrangian.

 

 

 

 

Vasyliunas theorem

See Grad–Vasyliunas

theorem.

 

 

 

 

 

 

vector An abstraction of the concept of displacement from one position to the other. In Euclidean 3-space, for instance, one can introduce coordinates x, y, z, and the displacement vector has components (δx, δy, δz). Formally, the vector is abstracted from the concept of tangent vector to a curve. A curve, parameterized by a parameter t, has a tangent, which can be thought of as a derivative along the curve: d/dt acting on a function f can be thought of in elementary terms as

ti ∂/∂xi

def

(f ) = df/dt

where the left side is written in a locally defined coordinate system {xi }. The {∂/∂xi } can be thought of as a set of basis vectors in this coordinate system. The geometrical quantity is the vector itself, independent of which coordinate system is used to express its components, however. Vectors in general can be viewed as directional derivative operators. For our tangent vector example, one writes

t = t

i

∂

= t

i ∂

 

∂xi

 

∂xi

where {xi } is a new coordinate system. Using the chain rule:

∂

 

∂xj

 

∂

 

=

 

 

∂xi

∂xi

∂xj

produces the rule to obtain the components of t in one coordinate basis, given the components in another basis:

ti ∂xj = tj . ∂xi

A vector is sometimes called a contravariant vector, to distinguish it from a one-form, which is sometimes called a covariant vector. See curve, one-form, tensor.

vector cross product For two 3-space vectors A , B, of lengths |A|, |B|, making an angle θ to one another; the vector of length |A||B| sin θ, pointing in the direction perpendicular to both A and B, with direction given by the right-hand rule.

vector field A smooth collection of vectors in some region of space such that the integral curves they define have no intersections in that region.

vector magnetograph Magnetograph which uses Zeeman-sensitive lines in the solar atmosphere (mostly in the photosphere) to measure all three components of the magnetic field on the surface of the sun.

velocity The (limit of the) vector difference in position at two instants, divided by the time between the instants. Here the limit is taken as the time interval goes to zero, so the definition defines velocity v as the vector time derivative of the position vector x. Units: cm/sec, m/sec.

velocity curve A plot of the velocity of stars observed in a spiral galaxy, measured along a major axis of the light distribution. Near the center the velocity is zero. On one side of the center the velocity is away from the observer, on the other side it is toward. The observed speed typically initially grows linearly with distance from the center. There follows a long plateau of constant velocity, typically 200 to 300 km/sec.

© 2001 by CRC Press LLC

Venus

A drop with increasing distance, as expected for Newtonian motion around a central condensed mass, is almost never seen.

velocity dispersion A parameter describing the range of velocities around a mean velocity value in a system of stars or galaxies. For example, the radial velocity dispersion can be measured for the stars in an elliptical galaxy and for the galaxies in a cluster. In the first case, the radial velocity dispersion is derived from the width of spectral lines, whose broadening is due to the motion of a large number of unresolved stars in the galaxy; in the second case the radial velocity dispersion is derived from the redshift of each individual galaxy. From the measurement of the velocity dispersion, the mass of the aggregation can be estimated.

velocity distribution In kinetic theory, the density f of particles in velocity space. The velocity distribution (or velocity distribution function) is a function of the velocity v and spatial coordinates x and the time t and is commonly normalized so that f (v, x, t)d3vd3x is the number of particles contained in the volume element d3vd3x. In a plasma each charge species (electrons and the various ions) is characterized by its own velocity distribution. The velocity distribution must satisfy an appropriate kinetic equation, such as the Boltzmann equation or Vlasov equation.

velocity strengthening Under certain conditions, the coefficient of static friction of the interface between two rocks increases with increasing speed of relative slip. This phenomenon of the sliding surface becoming stronger with slip rate is called velocity strengthening. If the coefficient of friction decreases with increasing slip rate, the phenomenon is called velocity weakening. Velocity weakening is an unstable process that results in fast slips such as in earthquake faulting.

velocity weakening See velocity strengthen- ing.

ventifacts Rocks that display a flat surface due to eolian erosion. When the environment has one or two prevailing wind directions, rocks

will be sandblasted with wind-blown material until the face of the rock becomes smooth. Ventifacts often have two smooth faces (sometimes more, if the rock has been moved from its original orientation) separated by sharp edges.

ventilated thermocline A layered model of the upper ocean, in which interfaces between some layers intersect the surface. This model better describes the three-dimensional structure of the thermocline than unventilated quasi-geostrophic theories of the upper ocean, which describe the vertical structure of the thermocline, but ignore the considerable horizontal structure present in the ocean. A ventilated thermocline model results in more realistic flow domains. A constantly refreshed, ventilated region contains fluid layers all in motion. On the periphery of the ventilated region are shadow zones which bound regions of constant potential vorticity or stagnant regions. The final possible ventilated zone is a region of Ekman upwelling. See also thermocline, Ekman pumping.

Venturi tube A device consisting of a tube of smaller diameter in the middle than at the ends. When a fluid flows through such a tube, the pressure in the central portion is reduced according to Bernouli’s principle. Measurement of the pressure drop gives a measure of the fluid flux through the tube. A Venturi tube may also be used to entrain other fluids into the main stream: atmospheric pressure forces the secondary fluid into an opening at the low pressure point in the Venturi tube. Applications in practice include internal combustion carburetors (gasoline vapor into air) and natural gas cook stoves (air into natural gas).

Venus The second planet from the sun.

Named after the Roman god of love, Venus. It has a mass M = 4.869 × 1027 g and a ra-

dius of R = 6052 km, giving it a mean density of 5.24 g cm−3 and a surface gravity of 0.91 that of Earth. Its rotational period of 243 days is longer than its orbital period around the sun, 224.7 days. The rotation axis has an obliquity of 177.3◦, so its rotation is retrograde. The slow rotation means that the planet’s oblateness is very nearly 0. Venus’ orbit around the sun is characterized by a mean distance of 0.723 AU, an

© 2001 by CRC Press LLC

Venusian Tessera

eccentricity of e = 0.0068, and an orbital inclination of i = 3.395◦. Its synodic period is 584 days. The average albedo of 0.75 refers to the cloud tops. These clouds, predominantly sulfuric acid drops, are embedded in an atmosphere primarily composed of CO2. A strong greenhouse effect maintains an average surface temperature of close to 750 K. Venus’ interior structure is probably similar to that of the Earth, with a core composed of a mixture of Fe, Ni, and S, surrounded by a silicate mantle. Radar maps of the surface show it to be uncratered and relatively young (< 109 years at the oldest) and covered with lava flows that are still active in some spots, but with no evidence of plate tectonics. Venus has been the subject of many missions including Mariner 2, Pioneer Venus, Venera 7, Venera 9, and Magellan. Because its moment of inertia is not known, the relative sizes of the core and mantle can only be estimated from the mean density of the planet. Venus has no satellites.

Venusian Tessera A distinct type of highland region on the surface of Venus belonging to the Plateau Highlands class (as opposed to Volcanic Rises). Plateau Highlands are topographically rugged, characterized by steep margins, and lack volcanic landforms (lava flows are rare; small volcanic constructs do not occur regularly and there are no shield volcanoes). Tessera form complex ridged terrain within the highlands. They may have been created by shortening the lithosphere so that the surface fractures and buckles, and the crust thickens, similar to the mountain-building process on Earth. In the absence of plate tectonics on Venus, however, a tessera-like landscape is formed.

vernal equinox The epoch at the end of the Northern hemisphere winter on which the sun is located at the intersection of the celestial equator and the ecliptic; on this day the night and day are of equal length throughout the Earth. The date of the vernal equinox is at the end of the Southern hemisphere summer. The vernal equinox also refers to the direction to the sun at this instant: 0hRA, 0◦ declination. See autumnal equinox.

Very Long Baseline Interferometry (VLBI)

A technique for measurement of small scale (milliarc second) features in radio sources by

carrying out interferometry between widely separated radio telescope sites, up to thousands of kilometers apart.

Vesta The 3rd largest asteroid known; the fourth asteroid to be discovered, in 1807. It is located in the Asteroid Belt between Mars and Jupiter. Vesta has a mean diameter of 576 km and mass of 2.76×1020 kg and rotates every 5.34 hours. As it rotates, Vesta’s brightness changes which suggests that the surface is not uniform either in terrain or composition. Images of Vesta from the Hubble Space Telescope show dark and light patches indicating a very diverse terrain, consisting of exposed mantle, lava flows, and a large impact basin. Spectra suggest that the compound pyroxene is present, which is typically found in lava flows. The composition of Vesta is sufficiently different from that of other known asteroids, so that Vesta has its own class. Orbit: semimajor axis 2.3615 AU, eccentricity 0.09, inclination to the ecliptic 7◦.13405, period 3.629 years.

virga Rain (or perhaps snow) falling from a cloud but evaporating before it reaches the ground.

Virgo A gravitational wave observatory under construction near the Italian town Cascina. The device is a laser interferometer, with two arms at right angles. The length of each arm of the laser interferometer device is 3 km. Operation is expected to start in 2001. See LIGO.

Virgocentric flow The motion of the galaxies in the local group toward the Virgo Cluster. The gravitational force exerted by the mass of the Virgo attracts all surrounding galaxies, including the galaxies of the local group. The velocity of approach toward Virgo lies in the range vr ≈ 100 to 400 km s−1. A correction for Virgocentric flow should be applied to the radial velocity measured for nearby galaxies, especially if the recessional velocity is used as an indicator of distance according to Hubble’s Law.

Virgo constellation (1.) A zodiac constellation most visible in spring, covering the area of sky approximately ranging from 12 to 15 h in right ascension and from ≈ −15◦ to ≈ +15◦

© 2001 by CRC Press LLC

viscoelastic material

in declination. Virgo (Latin for virgin) can be identified by looking for a “Y”- shaped configuration formed by four stars between Libra and Leo. The brightest star of Virgo (α Virginis, Spica, apparent magnitude = 1.0) is located at the lower tip of the “Y”. The sky region of the Virgo constellation includes the Virgo cluster of galaxies. See also Virgo Supercluster.

(2.) Sixth sign of the zodiac; constellation on the ecliptic around RA 13h; the second largest constellation in the sky. Brightest star , Spica (α−Virginis) at RA 13h25m11s .5, dec −11◦9m41s , spectral class B, VB 0.98. The sun passes through this constellation from late September through October. The Virgo Supercluster of galaxies overlaps the northern border of Virgo into the neighboring constellation of Coma Berenices.

Virgo Supercluster (1.) A supercluster roughly centered on the Virgo Cluster of galaxies, which includes the galaxy and which is accordingly known as the local supercluster. The Virgo Supercluster, first introduced by G. de Vaucouleurs, has a clumpy structure which includes several groups and clusters of galaxies, and whose somewhat flattened distribution shows a preferential plane defining a “supergalactic equator” and a “supergalactic” system of coordinates. The galaxy is located in the outskirts of the Local Supercluster, at approximately 15 to 20 Mpc from the Virgo cluster. See also supercluster.

(2.) The Virgo Supercluster contains about 2000 member galaxies. The giant elliptical (E1) galaxy M87 (NGC4486), MV 8.6 at RA 12h30m.8, dec +12◦24 , 18Mpc distant, is at the physical center of the Virgo cluster and of the Virgo (or Coma–Virgo) Supercluster. The supercluster’s enormous mass modifies the local Hubble flow, thus causing an effective matter flow towards itself (the so-called Virgo-centric flow). Our local group has apparently acquired a Virgocentric velocity (a modification of our Hubble flow) of up to 400 km/sec toward the Virgo cluster.

The actual recession velocity of the Virgo Supercluster is of order 1100 km/sec. However, the Virgo supercluster is very tightly bound, with high peculiar velocities, to over 1500 km/sec with respect to the cluster’s center of mass.

Hence, some of the supercluster’s members show blue shifts (moving toward us); the largest blue shift is exhibited by IC3258, which is approaching us at 517 km/sec; while others show large red shifts, up to 2535 km/sec (NGC4388).

virtual geomagnetic pole If the Earth’s magnetic field was a pure geocentric dipole, then knowledge of the direction of the magnetic field at any position on the surface (i.e., declination D, the angle that the horizontal component of the field makes with a line pointing north, and inclination I, the angle the field makes with the horizontal) can be used to calculate the position of the magnetic north pole:

θm = cot−1

1

tan I

 

2

θp = cos−1 (cos θ cos θm + sin θ sin θm cos D)

α

=

sin−1

sin θm sin D

 

sin θp

 

 

φp − φ = α

cos θm > cos θ cos θp

π + φ − φp = α cos θm < cos θ cos θp

where (φ, θ) are the coordinates (longitude, colatitude) of observation, and (φp, θp) are the coordinates of the magnetic north pole. It is possible using various techniques to estimate from measurements the original inclination and declination of the magnetic field from a rock at the time of its formation. By assuming that the field at the time was a geocentric dipole, one may infer the orientation of the magnetic field at that time. By further assuming that the dipole was oriented parallel to the Earth’s rotation axis, one may infer the latitude of the rock at the time of its formation (θm) as well as the horizontal orientation. There are various techniques for correcting for the effects of folding, and it can be argued that it is likely that the Earth’s field, when averaged over time, is sufficiently close to an axial geocentric dipole that hence using many samples, an accurate estimate of paleolatitude and orientation can be made.

viscoelastic material A material that deforms as an elastic solid at certain time scales but as a viscous fluid at other time scales. The application of a constant stress causes an immediate deformation that disappears if the stress

© 2001 by CRC Press LLC

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