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Dictionary of Geophysics, Astrophysic, and Astronomy.pdf

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gyroradius

tra would show a dip at frequencies higher than the Lyman-α emission line. The absence of such dip in the spectra of a quasar sample with

mean redshift z ! 2.6 leads to the upper limit nH (z = 2.64) ≤ 8.4 × 10−12hcm−3 for a ﬂat universe. This strict upper limit implies that

very little neutral hydrogen exists and what remains in the interstellar medium must be ionized. The physical mechanism that provided the energy necessary to reheat the interstellar medium is not known. Here h is the Hubble parameter H0/(100km/sec/Mpc).

GUT Grand Uniﬁed Theory. See grand uniﬁcation.

Guttenberg–Richter relation

The Guttenberg–Richter relation is

N = −b log m + a

where N is the number of earthquakes in a speciﬁed area and time interval with a magnitude

greater than m. This relation is applicable both regionally and globally. b-values are generally close to 0.9.

guyot In oceanographic geophysics, an isolated swell with a relatively large ﬂat top. Believed to be of volcanic origin.

gyrofrequency The natural (Larmor) frequency f of a charged particle in a magnetic ﬁeld. For nonrelativistic motion, f = qB/(2πm), where q is the charge, B is the magnitude of the magnetic ﬁeld, and m is the mass of the particle. In relativistic situations, the frequency depends on the particle’s velocity through the Lorentz factor γ which enters the deﬁnition of the momentum. Applied to electrons in the Earth’s magnetic ﬁeld. See Larmor frequency.

gyroradius See Larmor radius.

halocline

Hadley (cell) circulation A circulation in the meridional plane known to exist in the tropics due to the ascending warm air near the equator and descending cold air in high latitudes.

Hadley cell Convection cells within the atmosphere of a body. On planets where most of the atmospheric heating is produced by the sun (such as Earth, Venus, and Mars), air over the equatorial regions will be hotter than air over the poles. This hotter air is less dense than cooler air and thus rises, eventually losing heat as it moves toward the polar regions. Over the poles, the air becomes colder and more dense, thus sinking towards the surface. The cooler air moves back along the planet’s surface towards the equator, where it warms up and the cycle repeats. This basic cycle of warm air rising over the equator and cooler air sinking over the poles is called Hadley Circulation. Rapid rotation and variations in surface temperature (caused, for example, by oceans vs. continents) complicate this basic pattern. Hadley cells fairly accurately describe the atmospheric circulation only for Venus, although they form the basics for physical studies of other planetary atmospheres.

hadron Any particle that interacts with the strong nuclear force. Hadrons are divided into two groups: baryons (“heavy ones,” consisting of three quarks), which are fermions and obey the exclusion principle, and mesons which are bosons, and consist of a quark anti-quark pair. See fermion, boson, quark.

hail Large frozen pellets (greater than 5 mm in diameter) of water that occur in thunderstorms, in which updrafts keep the hail suspended at an altitude with freezing temperatures for long periods of time, growing the hailstone until it ﬁnally falls out of the cloud.

hailstone A single unit of hail.

Hale cycle The observation of sunspot numbers alone reveals an 11-year cycle. In contrast, the Hale cycle is a 22-year cycle which in addition to sunspot numbers also considers polarity patterns. While after 11 years the polarity of the sun is reversed, the original polarity pattern is restored only after 22 years. The Hale cycle therefore is also called the magnetic cycle of the sun.

Hale–Nicholson Polarity Law In a given solar cycle, examination of solar magnetograms reveals a distinctive alternation of positive and negative polarities in active regions. In the sun’s northern hemisphere the positive polarity is located in the “preceding” (westerly) part of the active region and the negative polarity is located in the “following” (easterly) part. The sense is reversed in the southern hemisphere. The hemispherical polarity patterns alternate with each successive activity cycle. This behavior of alternating active region magnetic polarities is known as the Hale–Nicholson Polarity Law.

Halley’s comet A comet with a period of 74 to 79 years which was identiﬁed with several historical passages (including a visit coincident with the defeat of King Harold in 1066) of bright comets by Edmund Halley (1656–1742), validated when the comet reappeared after Halley’s death in 1758. The most recent perihelion passage of Halley’s comet occurred on February 9, 1986.

Halley’s identiﬁcation of the comet and suggestion of perturbations on its orbit provided an explanation of comets in the context of Newtonian mechanics and Newtonian gravity.

halo Arcs or spots of light in the sky, under suitable conditions even a bright circle around the sun or the moon. Halos are caused by the refraction of sunlight on ice crystals in the atmosphere; thus, halos can be observed best in cold climates where ice crystals also form in the lower (and denser) troposphere. The angular extend of a halo is always about 22◦; however, depending on crystal shape and orientation, additional arcs and even a wider ring can form.

halocline The region of large vertical gradient of density due to salinity in oceans.

Hα condensation

Hα condensation The downﬂow of Hα emitting material in the chromospheric portions of solar ﬂares. Typical downﬂow velocities are of the order 50 kms−1 and are observed as redshifts in Hα line proﬁles.

Hα radiation An absorption line of neutral hydrogen (Balmer α) which lies in the red part of the visible spectrum at 6563 Å. At this wavelength, Hα is an ideal line for observations of the solar chromosphere. In Hα, active regions appear as bright plages while ﬁlaments appear as dark ribbons.

Hamiltonian In simple cases, a function of the coordinates xα, the canonical momenta pσ conjugate to xσ , and the parameter (time) t.

H= H xα , pβ , t

=	dxα		− L xδ ,	dxγ	, t
	pα
		dt		dt

where

∂L pα = ∂ dxα ,

and L is the Lagrangian. This relation is inverted to express the right side of the equation in terms of xα , pβ , and t. Such a transformation is called a Legendre transformation. This can be done only if L is not homogeneous of degree 1 in dxdtα ; special treatments are needed in that case.

The standard action principle, written in terms of the Hamiltonian, provides the equations of motion.

I = pαx˙α − H xγ , pδ , t dt

is extremized, subject to xα being ﬁxed at the endpoints. This yields

−p˙α	−	∂H	= 0

		∂xα
x˙α	−	∂H	= 0 .

		∂pα

From these can be formed an immediate implication:

∂H

∂H dpσ

∂H dxµ

∂t

∂pσ dt

∂xµ dt

∂H

∂t

since the last two terms on the right side of the equation cancel, in view of the equations of motion. Hence H = constant if H is not an explicit function of t. Also, notice from the equations of motion that conserved quantities are easily found. If H is independent of xσ then pσ is a constant of the motion. See Lagrangian.

Hamiltonian and momentum constraints in general relativity The Einstein ﬁeld equations, as derived by varying the Einstein–Hilbert action SEH , are a set of 10 partial differential equations for the metric tensor g. Since the theory is invariant under general coordinate transformations, one expects the number of these equations as well as the number of components of g to be redundant with respect to the physical degrees of freedom. Once put in the ADM form, SEH shows no dependence on the time derivatives of the lapse (α) and shift (βi , i = 1, 2, 3) functions; their conjugated momenta πα and πβi vanish (primary constraints). This reﬂects the independence of true dynamics from rescaling the time variable t and relabeling space coordinates on the space-like hypersurfaces t of the 3+1 slicing of space-time.

We denote by πij the momenta conjugate to the 3-metric components γij . Once primary constraints are satisﬁed, the canonical Hamiltonian then reads

HG + HM =
t	d3x α (HG + HM ) + βi HGi + HMi ,

where the gravitational Hamiltonian density is

HG =

8πGγ −1/2	γikγjl + γilγjk
				1			1/2(3)
−γij γkl π	ij	π	kl	−		γ		R ,
					16πG

and the gravitational momentum densities are

HGi =

− 2π\|ijj = −	1	πjk .
	16πGγ il 2γjl,k − γjk,l

The corresponding quantities for matter have been denoted by the subscript M in place of G and their explicit expressions depend on the particular choice of matter ﬁelds that one wishes to consider.

harmonic analysis

Conservation of the primary constraints, namely arbitrariness of the momentum associated with α and βi, leads to the vanishing of the Poisson brackets between the canonical Hamiltonian and, respectively, the lapse and shift functions. These are the secondary constraints

HG + HM = 0

HGi + HMi = 0 ,

the ﬁrst of which is known as the Hamiltonian constraint, the second ones as the momentum constraints. One ﬁnds that, when the space-time coordinates are (t, x) deﬁned by the 3 + 1 splitting, the Hamiltonian constraint and the momentum constraints are equivalent to the Einstein ﬁeld equations, respectively, for the G00 and the G0i Einstein tensor. See ADM form of the Einstein–Hilbert action, ADM mass, Einstein equations, initial data, tensor.

Hamilton–Jacobi Theory In classical mechanics, a method of solution of Hamiltonian systems that makes use of the fact that the Hamiltonian is the generator of inﬁnitesimal canonical transformations in time. Consider a canonical transformation to a set of variables {Ql , Pn} where the new Hamiltonian K is identically zero. Then Ql , Pn are constants (because the right side of Hamilton’s equations is zero). K = 0 implies the equation

0 = ∂F + H . ∂t

Thus H , which is known to generate inﬁnitesimal canonical transformations in time, is integrated up in time to produce the generating function F . F therefore represents a canonical transformation from the current phase space coordinates qk , pl evolving in time, to (functions of) the constants of the motion Ql , Pk given by

pl	=		∂S

			∂ql
Pk	=	−		∂S

				∂Qk

where S (called the action) is the solution to the ﬁrst equation. In order to solve the ﬁrst

equation, we write: F = F(ql , Qk , t); H =

H(ql , pn , t) ≡ H(ql , ∂q∂Fn , t).

Thus, the ﬁrst equation is a (generally nonlinear) partial differential equation for S. There are n+1 derivatives of F appearing in the ﬁrst equation, so n + 1 integration constants, but because only the derivatives of S are ever used, there are n signiﬁcant constants that we take to be the Qk, functions of the constants of the notion, i.e., of the initial data. Once the ﬁrst equation is solved for S(ql , Qk , t), then the third equation gives a connection between the constant Pk, the constant Qk, and ql and t, which gives the explicit time evolution of ql. Since the right side of the second equation is a function of ql(t), and of constants Qk , Pk, the second equation gives the explicit time evolution of pl. A variant of this method writes f = f (pk , Qk , t); then the

transformation equations are:
			ql	=	−	∂S

						∂pl
			Pk	=	−	∂S		,

						∂Qk
and	the		Hamilton–Jacobi				equation is writ-
ten	with								l
ten	with		the substitution				H(q , pm , t) ≡
	∂F		the substitution				H(q , pm , t) ≡
H(−∂ql		, pm , t) as

∂F	+ H	−	∂F		= 0 .
				, pm , t
∂t			∂ql

The analysis follows analogously to the ﬁrst one.

hard freeze A freeze in which surface vegetation is destroyed, and water in puddles and the Earth itself are frozen solid.

hard radiation Ionizing radiation.

Hard X-ray Telescope (HXT) Telescope on the Yohkoh spacecraft designed to provide imaging of solar ﬂares at hard X-ray energies. Consists of four distinct energy channels: L (13.9 – 22.7 keV), M1 (22.7 – 32.7 keV), M2 (32.7 – 52.7 keV), H (52.7 – 92.8 keV).

harmonic analysis A technique for identiﬁcation of the relative strength of harmonic components in a tidal signal. Involves assuming the tide signal to be the sum of a series of sinusoids and determining the amplitude of each component.

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