DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

equator. While geodetic latitude is used for
most mapping, geocentric latitude is useful for
describing the orbits of spacecraft and other
bodies near the Earth. The geocentric latitude
φ ′ for any point P is defined as the angle between
the line OP from Earth’s center O to the
point, and Earth’s equatorial plane, counted positive
northward and negative southward. See
the mathematical relationships under latitude.
Geocentric latitude φ ′ is the complement of
the usual spherical polar coordinate in spherical
geometry. Thus, if L is the longitude, and
r the distance from Earth’s center, then righthanded,
earth-centered rectangular coordinates
(X,Y,Z), with Z along the north, and X intersecting
the Greenwich meridian at the equator
are given by
X = r cos φ ′ cos(L)
Y = r cos φ ′ sin(L)
Z = r sin φ ′
geocorona The outermost layer of the exosphere,
consisting mostly of hydrogen, which
can be observed (e.g., from the moon) in the
ultraviolet glow of the Lyman α line. The hydrogen
of the geocorona plays an essential role
in the removal of ring current particles by charge
exchange following a magnetic storm and in
ENA phenomena.
geodesic The curve along which the distance
measured from a pointp to a pointq in a space
of n ≥ 2 dimensions is shortest or longest in
the collection of nearby curves. Whether the
geodesic segment is the shortest or the longest
arc from p to q depends on the metric of the
space, and in some spaces (notably in the spacetime
of the relativity theory) on the relation between
p and q. If the shortest path exists, then
the longest one does not exist (i.e., formally its
length is infinite), and vice versa (in the latter
case, quite formally, the “shortest path” would
have the “length” of minus infinity). Examples
of geodesics on 2-dimensional surfaces are
a straight line on a plane, a great circle on a
sphere, a screw-line on a cylinder (in this last
case, the screw-line may degenerate to a straight
line when p and q lie on the same generator
of the cylinder, or to a circle when they lie in
© 2001 by CRC Press LLC
geodesic
the same plane perpendicular to the generators).
In the spacetime of relativity theory, the points
(called events) p and q are said to be in a timelike
relation if it is possible to send a spacecraft
from p to q or from q to p that would move all
the way with a velocity smaller thanc (the velocity
of light). Example: A light signal sent from
Earth can reach Jupiter after a time between 30odd
minutes and nearly 50 minutes, depending
on the positions of Earth and Jupiter in their orbits.
Hence, in order to redirect a camera on a
spacecraft orbiting Jupiter in 10 minutes from
now, a signal faster than light would be needed.
The two events: “now” on Earth, and “now +
10 minutes” close to Jupiter are not in a timelike
relation. For eventsp andq that are in a timelike
relation, the geodesic segment joining p and q
is a possible path of a free journey between p
and q. (“Free” means under the influence of
gravitational forces only. This is in fact how
each spacecraft makes its journey: A rocket accelerates
to a sufficiently large initial velocity
at Earth, and then it continues on a geodesic in
our space time to the vicinity of its destination,
where it is slowed down by the rocket.) The
length of the geodesic arc is, in this case, the
lapse of time that a clock carried by the observer
would show for the whole journey (see relativity
theory, time dilatation, proper time, twin paradox),
and the geodesic arc has a greater length
than any nearby trajectory. The events p and
q are in a light-like (also called null) relation if
a free light-signal can be sent from p to q or
from q to p. (Here “free” means the same as
before, i.e., mirrors that would redirect the ray
are not allowed.) In this case the length of the
geodesic arc is equal to zero, and this arc is the
path which a light ray would follow when going
between p and q. The zero length means
that if it was possible to send an observer with a
clock along the ray, i.e., with the speed of light,
then the observer’s clock would show zero timelapse.
If p and q are neither in a timelike nor
in a light-like relation, then they are said to be
in a spacelike relation. Then, an observer exists
who would see, on the clock that he/she carries
along with him/her, the events p and q to occur
simultaneously, and the length of the geodesic
arc between p and q would be smaller than the
length of arc of any other curve between p and
q. Given a manifold with metric g, the equa-
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