DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

symmetry
posite walls of the cube, then only rotations by
multiples of 90 ◦ are symmetries. If the axis
passes through the centers of diagonally opposite
edges of the cube, then only rotations by
multiples of 180 ◦ are symmetries.) Continuous
symmetries of an n-dimensional space can be
labeled by up to 1 2n(n + 1) parameters. For example,
among two-dimensional surfaces some
have no symmetry (think about the surface of a
loaf of bread), some have one-parametric symmetry
(think about the surface of a perfectly
smooth and round doughnut, which is symmetric
with respect to rotations around an axis that
passes through its center: the parameter is the
angle of rotation), some have a two-parametric
symmetry (think about the surface of an infinite
cylinder, the parameters are the angle of rotation
about the axis of the cylinder and the distance
of displacement along the axis), and two
have three-parametric symmetries. These latter
are the plane and the surface of the sphere.
For the plane, the symmetries are displacements
along two perpendicular directions and rotations
around an axis perpendicular to the plane. For
the sphere, the symmetries are rotations around
three mutually perpendicular axes. A characteristic
property of each collection of symmetries
is the difference between the following two operations:
We take two of the basic symmetries
(call them a and b), apply them to a point of
the space in one order (first a, then b), and then
apply them in the reversed order (first b, then
a). The difference may be zero (like for the
two translations on a plane) or nonzero (for any
two rotations on a sphere or for a translation
and a rotation on a plane). The fact that on a
plane there exists a pair of symmetry transformations
whose final result does not depend on
the order in which they are executed, while no
such pair exists on a sphere, shows that these
two collections of symmetries are inequivalent.
Real objects usually have no symmetry in the
strict sense but may be approximately symmetric.
(Example: the surface of the Earth is nearly
a sphere. The departures from spherical shape
caused by rotational flattening and by the mountains
and other irregularities of the surface can
be neglected for many purposes.) Symmetries
are often assumed in physics in order to solve
problems that would be too difficult to consider
by exact methods without such an assumption.
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Small departures from symmetries can then be
taken into account by approximate calculations.
A typical example is the calculation of orbits of
the planets around the sun. Both in Newtonian
mechanics and in general relativity a first approximation
to the realistic solution was found
under the assumption of spherical symmetry,
which implies that the mass distribution inside
the sun is perfectly spherical, and that the orbit of
a given planet is not influenced by other planets.
Then, small perturbations away from spherical
symmetry caused by the other planets can be
considered and corrections to the orbits calculated.
Calculations without any symmetries assumed
have been attempted only by numerical
methods. A pronounced time dependent asymmetry
in a gravitating system typically results in
the emission of gravitational waves.
synchronous orbit For a small body orbiting
a more massive one, the orbital radius for which
the orbital period is equal to the rotation period
of the more massive body. A satellite in a zero
inclination synchronous orbit above the equator
will always appear to be above a particular point
on the more massive body.
synchronous rotation Rotation of a planet
or other secondary body so that the same side
always faces the primary; for instance, the moon
rotates synchronously around the Earth. In the
case of the moon, this is ensured by tidal locking;
for commercial communications satellites, it is
accomplished by active means, such as reaction
wheels or thrusters.
synchrotron radiation The radiation produced
by a charged particle as it gyrates around
magnetic field lines. The radiation is emitted
at the gyrofrequency, ω = qB/γ mc, where q
is the particle charge, B is the magnetic field
strength, m is the particle mass, γ is the Lorentz
factor, and c is the speed of light. For an electron
we have ω = 1.76 × 10 7 Bγ. Synchrotron radiation
from energetic electrons is an important
production mechanism for microwave emission
in the solar corona.
synchrotron self-Compton mechanism A
mechanism suggested for the production of high
energy photons in radio loud active galactic nu-