DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

A space that is isotropic with respect to all of its
points is also homogeneous, but the converse is
not true. See isotropy, Killing vector.
homologous flares Solar flares that occur
repetitively in the same active region, with essentially
the same position and with a common
pattern of development.
homologous temperature (T/Tm) The temperature
(T ), normalized by the “melting temperature”,
Tm.
homothety A one-parameter family of maps
of space-times with the property
Lξgik =σgik .
Here ξ is the generator and gik the metric. L
denotes the Lie derivative along the vector ξ.
Homothety isaspecialone-parameterconformal
group when σ is a constant.
homotopy group Mathematical structure
that allows classification of topological defects
formed during phase transitions. Denoting M
as the vacuum manifold, one can define the zeroth
homotopy group π0 as a counter of its disjoint
parts if it is disconnected. For a connected
manifold, it reduces to the identity element.
Inequivalent paths on a plane with a hole. The closed
curves in full line, not encircling the hole, are all equiv-
alent, and so are those in dashed line that wind once
around the hole. However, because of the hole, it is
not possible to smoothly transform full line curves into
dashed lines. The first homotopy group of the plane
with a hole is Z, the group of integers, because the
curves can wind an integer a number of times around
the hole, in one way or another (hence there is a pos-
itive or negative sign associated to each curve).
© 2001 by CRC Press LLC
horizontal branch star
The first, or fundamental, homotopy group
π1 counts the inequivalent closed paths on M:
for a plane with a hole for instance, all paths
encircling the hole are said to be equivalent dependingonthenumberoftimestheypassaround
it (see figure). As they can also wind clockwise
or counter-clockwise inequivalently one has, for
a plane with one hole, π1 ∼ Z, the group of integers.
Similarly, one can define the nth homotopy
groupπn of M by means of the inequivalentndimensional
closed hypersurfaces that exist on
M.
The topological defect classification stems
from these homotopy groups: when π0(M) is
non-trivial, then domain walls must form, while
strings, monopoles, and textures will exist, respectively,
if π1(M), π2(M), and π3(M) are
non-trivial. See cosmic phase transition, cosmic
topological defect, spontaneous symmetry
breaking.
hopper dredge A dredge that includes a hopper
or bin for storage of sediment that will later
be pumped out. Used to remove sand from channel
entrances and harbors.
horizon On the Earth, or other planet, the distance
at which a line from the viewpoint above
the surface tangentially touches the surface of
the planet. Hence, the maximum distance that
can be seen from that viewpoint. The distance to
the horizon depends on the height of the viewpoint;
for 2 m above sea level the distance on
Earth is approximately 5 km. In general relativity
one defines an event horizon, which bounds
partsofaspacetimewhichcanneverbeobserved
from outside the horizon; and particle horizon,
which delimits the boundary visible up to a particular
time at a particular location in space. See
apparenthorizon, blackholehorizon, eventhorizon,
particle horizon.
horizontal branch star A star of low mass
( < ∼ 1.5 solar masses) and low heavy-element
content (0.2%) in the evolutionary phase where
the two main sources of energy are helium burning
in the core and CNO-cycle hydrogen burning
in a thin shell. The star is about 30 times as
bright as it was on the main sequence, and the
phase lasts about 3% as long. The name comes
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