DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

variable star A star that changes its flux over
time, which can be hours or years. Variable stars
are classified as either extrinsic or intrinsic, depending
on the cause for the variability being
due to the environment or interior processes of
the star, respectively. Three major groups of variable
stars are: eclipsing, cataclysmic, and pulsating.
An eclipsing variable is really two stars
with different temperature types that, due to fortuitous
line of sight with Earth, orbit in front of
and in back of each other, thus changing the light
the system puts out for observers on Earth. Pulsating
variables alternately expand and contract
theiratmosphereandthepulsationsarefueledby
a driver under the photosphere that operates on
the opacity changes of partially ionized H and
He. The cataclysmic variables, which include
novae, dwarf novae, recurrent novae, and flare
stars, can change their brightness by 10 magnitudes
or more. The novae cataclysmic variables
are binary stars whose increase in brightness
arises because of the hydrogen-rich material
of the companion star igniting (fusion) on
the surface of the hot white dwarf. Other minor
types of variables include the R CrB stars
that are believed to puff out great clouds of dark
carbon soot periodically, causing their light to
dim.
variable stars [cataclysmic variables] See
cataclysmic variables.
variable stars [geometric variables] Binary
systems whose stars actually present constant
luminosity, but are periodically eclipsing each
other when seen from Earth, thus exhibiting apparent
variability.
variable stars [peculiar variables] Stars
that present an inhomogenous surface luminosity
and, therefore, are observed as variables due
to their rotation.
variable stars [pulsating variables] Stars
that present variable luminosity due to instabilities
in their internal structure.
variational principle A way of connecting
an integral definition of some physical orbit,
with the differential equations describing evolution
along that orbit. Varying a path or vary-
© 2001 by CRC Press LLC
xi ,tiC
variational principle
ing the values of field parameters around an extremizing
configuration in an action (which is
defined via an integral) leads to the condition
that the first-order variation vanishes (since we
vary around an extremum). The vanishing of
this first-order variation is typically a differential
equation, the equation defining the field configuration
which produces the extremum. In point
mechanics, one may compute the action:
xf ,tf
I = L x, dx
dt , d2
x
··· dt
dt2 (where x = x i ,i = 1 ···N, the dimension of
the space). By demanding that the curve C extremize
the quantity I, compared to other curves
that pass through the given endpoints at the given
times, one is led to differential conditions on the
integrand
∂L d ∂L d2
− +
∂xi dt ∂ ˙x i dt2 ∂2L ∂ ¨x
−···=0,i = 1 ···N.
Here the partial derivatives relate to the explicit
appearance of ˙x i = dxi
dt , ¨xi = d2x i
dt2 , ···.Newton’s
laws for the motion of point in conservative
field follow from the simple Lagrangian
L = T − V,
where T = Tij (xl ) ˙x i ˙x j is quadratic in the ˙x i ,
and V is a function of the xi only. Then, one
obtains
− ∂V
d
− Til
∂xi dt
dxl
= 0 .
dt
For the simplest case of Til = 1 2 mδil, one finds:
m ¨x i =− ∂V
∂xi .
The quantity I = Ldt is called the action.
For classical and quantum field theories, one
defines a Lagrangian density. The Lagrangian
is then defined as an integral of this density. For
instance, the action for a free (real) scalar field
satisfying a massless wave equation is
I = L |g|d 4 x,
where
L = ∇αφ∇βφg αβ
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