DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

Located near Socorro, NM at 34 ◦ 04 ′ 43".497N,
107 ◦ 37 ′ 03".819W. The array has a maximum
resolution of 0".04 and can be used over the frequency
range 300 to 50,000 MHz (90 to 0.7 cm).
Vlasov equation The kinetic equation governing
the velocity distributions of electrons and
ions in an environment where Coulomb collisions
may be neglected. For non-relativistic particles
of charge q and mass m in the presence
of an electric field E(x,t)and a magnetic field
B(x,t), the Vlasov equation is (cgs units)
∂ ∂ q
+ v · + E +
∂t ∂x m
v
c ×B
· ∂
f= 0 ,
∂v
where f(v, x,t) is the velocity distribution.
Here v and x are, respectively, the velocity and
spatial coordinates, and t is the time.
Moment equations for macroscopic quantities
are obtained by multiplying the Vlasov
equation by 1, v, etc. and integrating over velocity
space. For example, the zero-order moment
equation is
∂n
∂t
+∇·(nV) = 0 ,
where the zeroth moment of f
n= fd 3 v
is the particle number density, and the first moment
(divided by n)
V = 1
vfd
n
3 v
is the mean vector velocity. Another useful moment
is the stress (or pressure) tensor
P=m (v − V)(v − V)fd 3 v;
note that the stress tensor need not be isotropic
in a collisionless plasma.
The electric charge and current densities in
the plasma are
fαd 3 v
and
α
α
qα
© 2001 by CRC Press LLC
qα
vfαd 3 v,
Voigt profile
respectively, where the summation is taken over
all charge species α.
Vlasov–Maxwell equations The fundamental
equations governing the behavior of a
plasma in which Coulomb collisions are negligible.
A solution of the Vlasov–Maxwell system
is comprised of velocity distributions for
all charge-species and the associated electric
charge and current densities, together with electric
and magnetic fields, all consistent with the
Maxwell equations governing electrodynamics.
See Vlasov equation.
voids Large regions of the universe in
which the matter-density is distinctly lower than
the average. Voids are surrounded by relatively
thin layers into which nearly all galaxies
are crowded. Typical diameters of voids
are between 20h −1 Mpc and 60h −1 Mpc (h is
the dimensionless Hubble parameter, i.e., H0
/100/km/sec/Mpc), and the mass-density within
a void is believed to be about 10% of the cosmic
average matter-density. Hence, the large-scale
distribution of matter in the universe is somewhat
similar to a foam, with the voids being
analogs of the air-bubbles and the layers with
galaxies being analogs of the soap films.
Voids were observationally discovered in
1978–1979 and hailed at that time as a surprising
discovery. However, papers were already
published in 1934 that predicted that
the Friedmann–Lemaître cosmological models
were unstable against the formation of local
minima in matter-density. Had the faith in the
cosmological principle not prevailed later, voids
would have been, in fact, expected.
Voigt body Also called Kelvin body. A kind
of material with both properties of elastic body
and viscous fluid. A Voigt body is represented
by the sum of two terms for which stresses are
proportional to elastic strain and viscous strain
rate. Most elastic bodies containing many pores
filled with viscous fluid are examples of Voigt
bodies.
Voigt profile An expression that describes
the shape of an absorption line considering the
effects of natural and thermal Doppler broadening
mechanisms. The total absorption coeffi-
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