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Dictionary of Geophysics, Astrophysic, and Astronomy.pdf

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mascon

cally identical to the composition of the Martian atmosphere. The 13th meteorite is much older than the other 12 (dating from 4.5 × 109 yrs ago) and may contain evidence of fossilized Martian life. These meteorites are believed to have been ejected from the planet’s surface during the formation of impact craters on Mars. The subsequent discovery of lunar meteorites indicates that this is a common process, at least for the smaller solar system bodies.

mascon “Mass concentration”; concentration of density beneath the surface of the moon, detectable because of its effect on satellite orbits. Mascons associated with lunar craters typically lie under the center of the craters; there is usually a mass deﬁcit in a ring associated with the crater wall.

maser Acronym for microwave ampliﬁcation by stimulated emission of radiation: ampliﬁcation of radiation coming from excited states of molecules. While a maser can be a laboratory instrument, especially interesting are the natural astrophysical masers. Ampliﬁed lines have been observed at frequencies in the range of 1 to 100 Ghz, in association with dense molecular clouds associated to star forming regions, or in circumstellar envelopes of cold late-type stars, such as giant and supergiant M stars, carbon and S stars, where diatomic or more complex molecules are not dissociated by the radiation from the star. Masers are produced by stimulated emission: a photon of frequency matching the frequency of a particular transition between two states induces the emission of a second photon, whose frequency and phase are identical to the ﬁrst. Net ampliﬁcation of the line radiation is achieved by population inversion, a condition realized when the higher energy level is more populated than the lower by a “pumping” source of energy, like a background radiation ﬁeld. Astrophysical masers have been observed at several frequencies corresponding to rotational transition of diand tri-atomic molecules. The most luminous masers have been observed in external galaxies at the frequency of 22.235 Ghz corresponding to a rotational transition of the water vapor molecule; they are known as water mega-masers.

mass coefﬁcient See inertia coefﬁcient.

mass-defect The binding energy Eb of a system expressed in terms of Einstein’s famous formula Eb = Mbc2. (Mb is taken positive if the system is bound.) In Newtonian gravity, and in Einstein’s general relativity,

Mgrav = M0 − Mb

where M0 is the sum of the masses of the system’s constituents if it were inﬁnitely dispersed, and Mgrav is the active gravitational mass that determines, e.g., planetary orbits. In other relativistic theories of gravity this simple relation does not hold, and different kinds of binding energy make different contributions to the active gravitational mass.

mass extinctions Extinctions of a large percentage of ﬂora and/or fauna have occurred throughout Earth’s history. The deﬁnition of a mass extinction is that more than 25% of the existing species disappear from the Earth within a short time period (generally < 105 yrs). Since the boundaries between terrestrial geologic epochs are deﬁned by changes in the fossil record, many of these boundaries are coincident with mass extinctions. Mass extinctions result from major environmental changes that occur rapidly enough that many species do not have time to adapt. Several mass extinctions are now associated with large impact events on the Earth, the most famous being the extinction of the dinosaurs and other species at the end of the Cretaceous Period (65×106 yrs ago) by the creation of the approximately 200-km-diameter Chicxulub impact crater in Mexico. Debris tossed into the atmosphere by these large impacts decreases the amount of sunlight reaching the surface, causing a collapse of the food chain and accompanying starvation, which leads to the mass extinctions. Mass extinctions are also suspected to occur when plate motions cause changes in ocean currents, which leads to climate change, and perhaps by periods of enhanced volcanic activity.

massive ghost A theoretical massive particle with unphysical properties which spoils the physical content of a quantum particle theory.

Maunder Minimum

For instance, in a model higher derivative quantum gravity, the massive spin-2 ghosts have negative kinetic energy and they interact with physical particles such as gravitons. As a result they lead to the instability of the classical solutions and, on a quantum level, to the loss of conservation probably in interaction (loss of unitarity of the physical S matrix). See higher derivative theories.

mass-luminosity relation The relation between luminosity and mass for main sequence stars. As the mass of a star increases, the pressure in its core, and hence its temperature, also increases. Hence, as the mass of a star increases, it burns hydrogen more rapidly in its core and is consequently brighter. The massluminosity relation for main sequence stars is: L Mx where x is roughly 3. This relation only holds when comparing stars lying on the main-sequence (hydrogen burning stars). A low mass giant star can be brighter than a higher mass main sequence star.

mass transfer Mass transfer occurs in binary star systems when one star overﬁlls its Roche-lobe and accretes onto its companion. Binary mass transfer is separated into three cases depending upon the evolutionary phase of the mass-losing star (Case A, Case B, Case C). Case A denotes the mass transfer that occurs during hydrogen burning. Case B mass transfer occurs after hydrogen burning, but before helium core ignition. Case C mass transfer denotes any mass transfer which occurs after helium core ignition. If the orbital angular momentum of the system is conserved during the mass transfer phase, the mass transfer is denoted “conservative”. The orbital separation of the binary is uniquely determined:

a		M1	−2	M2	−2
a0	= M01			M02	,
					,

where the subscript 0 denotes the initial conditions. Note that if the mass-losing star is less massive than the accreting star, the orbit widens, and if it is more massive, the orbit decreases. In nature, however, orbital angular momentum is lost from the system either as mass escapes the

binary or as mass forms an accretion disk and spins up the accreting star.

mass wasting Also called mass movement, mass wasting is the downhill movement of soil or fractured rock under the inﬂuence of gravity. Common features associated with mass wasting include landslides (including rockslides, mudﬂows, Earthﬂows, and debris avalanches), rock falls (also called talus slopes), soil creep (the very slow (usually imperceptible) downhill ﬂow of soil under the inﬂuence of gravity), and soliﬂuction (movement of frozen soil).

matter density perturbations Inhomogeneities in the matter distribution in the universe. The density ﬁeld δ(x) = (ρ(x) − ρo)/ρo is usually characterized by its Fourier transform

. It is often assumed that the Fourier modes

δ(k)

are Gaussian random variables. In this case, the power spectrum P (k) =< |δ(k)|2 >, sufﬁces to characterize all the statistical properties of the density ﬁeld. Two theories compete to explain the origin of density perturbations: (1) they were generated during inﬂation, favored by the data on cosmic microwave background temperature anisotropies, or (2) they are the result of distortions produced by topological defects (cosmic strings or domain walls) on a homogeneous background. Inﬂation generically predicts that today P (k) kn with n 1 on scales close to the horizon size.

The exact shape of the power spectrum is one of the most challenging problems of observational cosmology. Given an initial density ﬁeld, density perturbations can be evolved in time and the power spectrum at present can be computed. The ﬁnal processed spectrum depends on the geometry of the universe, the baryon fraction, and the amount and nature of the dark matter. In the ﬁgure on page 308 we plot a compilation of the power spectrum obtained from the distribution of galaxies and clusters in different catalogs. The largest scale sampled by the data is 600h−1 Mpc. The amplitude is different since clusters and galaxies are biased tracers of the matter distribution. See biasing parameter.

Maunder Minimum The period from 1645 to 1715 during which the number of sunspots on the solar disk was severely depressed and,

maximal extension of a space-time

	106
			ACOE
			ACOR
			APMT
	105		APMgal.3D
	105		APMgal.2D
			CfA2
]			LCRS
3			LCRS
Mpc			IRASP
Mpc			IRASTE
3	104		IRASTE
[h	104
P(k)
	103
	102	101	100
		k [h Mpc1 ]

Observed power spectrum of galaxies and clusters of galaxies: ACO-E and ACO-R are spectra for AbellACO clusters as derived with two different techniques; APM-T is the spectrum of APM clusters; APM-gal.3D and APM-gal.2D are spectra of APM galaxies found from 3-D and 2-D data; CfA2 is the spectrum of the SSRS+CfA2 130 Mpc/h sample; LCRS is the spectrum of the Las Campanas Redshift Survey; IRAS-P and IRAS-TE are spectra of IRAS galaxies found by two different groups. Notice the difference in amplitude when the power spectrum corresponds to galaxies and clusters. The difference is due to bias.

despite frequent and systematic attempts to observe them, sunspots were detected only occasionally. During this period, when sunspots were detected, they were observed at relatively low latitudes. The solar rotation was slower than today, in addition, and the differential rotation was more pronounced. This period of unusual low solar activity is also documented indirectly by an unusually small number of aurora observations and unusual high counts in cosmogenic nuclides such as 14C and 10Be. The 10Be concentrations in ice cores suggest that a solar cycle still was existent but its duration was reduced to 9 years. The combination of these observations with models of solar activity suggests that during the Maunder Minimum the solar constant might have been reduced by 0.2 to 0.5%. The Maunder Minimum coincides with an unusual cold time period, the Little Ice Age.

maximal extension of a space-time A spacetime is called extendible if it has an extension. A maximal extension of a space-time is an extension that is not extendible. See extensions of space-times.

maximally rotating black hole A Kerr black hole for which the angular momentum per unit mass (which in geometrized units has dimensions of [mass]) reaches a maximum value equal to the mass of the black hole, in these units. Thus, the maximum angular momentum expected from the solution of Einstein’s ﬁeld equation for a rotating black hole is equal to GM2/c, and corresponds to a black hole rotating at the speed of light at a radius equal to GM/c2, i.e., at the event horizon for this maximally rotating case. (In the nonrotating case the event horizon is at r = 2GM/c2.) As shown by Thorne, a rotating black hole interacting with its environment can come close to, but cannot reach, this limit.

maximally symmetric space A metric space which admits the maximum number N of linearly independent Killing vectors. If D is the dimension of space, then N = D (D + 1)/2 and the space is isotropic with respect to each point; therefore, it is also homogeneous and has constant curvature.

For D = 3, one has N = 6 and the spacetime is described by one of the Robertson– Walker metrics. See homogeneity, isotropy, Killing vector, Robertson–Walker cosmological models.

Maximum Observable Frequency (MOF)

Deﬁned as the highest frequency at which a reﬂection from the ionosphere occurs during oblique incidence sounding between two points. The MOF is a measurement made on a real propagation path using an oblique sounder capable of exciting all possible paths for the circuit. It may also refer to an identiﬁed mode, in which case the mode would be shown with the MOF. See ionospheric radio propagation path, oblique ionogram.

maximum probable wave (or probable maximum wave) A description of the largest wave that is expected to exist in a particular sea condition. Used for design purposes.

Maxwell–Boltzmann distribution An expression for the fraction of molecules f (v) in a gas that have velocity v within a small speciﬁed

mean ﬁeld dynamo

interval, divided by that interval:

f (v) = 4π	2	πRT	3	−2RT
			v2e
	M		2	Mv2

where M is the molar mass, R is the molar gas constant, and T is the temperature.

The Maxwell–Boltzmann distribution is a classical distribution; it does not take account of quantum indistinguishability, which are described by Bose–Einstein distribution for integer spin quantum particles, and by the Fermi– Dirac distribution for half-integer spin quantum particles.

Maxwell material Also called a Maxwell ﬂuid or Maxwell solid. A viscoelastic material that behaves as a linearly elastic solid in response to high frequency loading but as an incompressible, linearly viscous ﬂuid in response to low frequency loading. The strain rate for the Maxwell material is a combination of the rate of elastic strain according to Hooke’s law and the rate of viscous strain according to Newtonian ﬂow law. The relation between volumetric strain θ = εii and pressure p = σii/3 is elastic: θ = p/K, where K is the bulk modulus. The constitutive relation for the deviatoric strain ε ij = εij − θδij /3 and deviatoric stress

σ ij = σij − pδij

∂ε

∂σ

∂t

2µ

where G is shear modulus, and µ is viscosity. The quantity τM = µ/G is called the Maxwell relaxation time, a time that roughly deﬁnes the transition from predominantly elastic to predominantly viscous behavior after a suddenly imposed constant loading. In modeling the rheology of the Earth or other planets, the above constitutive relation is often modiﬁed by replacing the Newtonian ﬂow law with a nonlinear ﬂow law (such as the Power law). In geophysics, such nonlinearly viscoelastic materials are usually still referred to as the Maxwell materials.

mean anomaly, M Measured in degrees, the ratio M/360 is equal to the ratio of the time elapsed since last periapse to the orbital period. Thus, at periapse, M = 0 , and at apoapse, M =

180. For a circular orbit the mean anomaly and the true anomaly are the same.

mean celestial equator The great circle on the celestial sphere perpendicular to the mean celestial pole. Its intersections with the ecliptic deﬁne the mean vernal equinox, and the mean autumnal equinox.

mean celestial pole The mean direction of Earth’s north rotation pole when the short timescale (days to decades) variations, called nutation, are averaged out. This hypothetical pole executes a circle of radius approximately 23.44◦ about the North Ecliptic Pole, in about 25,800 y. (There is a corresponding mean south celestial pole, opposite to the north one.) The precise derivation of the motion of this pole is given by J.H. Lieske, T. Lederle, W. Fricke and B. Morando in Astron. and Astrophys., 58, 1 (1977).

mean cosine of scattering angle The integral over all directions of the volume scattering function multiplied by the cosine of the scattering angle, divided by the integral over all directions of the volume scattering function; also called the single-scattering asymmetry factor.

mean diameter, mean grain size The mean particle size in a soil sample. Not as frequently used as median grain size, d50.

mean ﬁeld dynamo A mathematical simpliﬁcation of the physics governing the behavior of magnetic ﬁeld in a dynamo such as that in the Earth’s core so that the evolution of the magnetic ﬁeld can be simulated. Essentially, the system is considered to be axisymmetric; that is, variations along lines of latitude are neglected. However, it turns out that a purely axisymmetric dynamo cannot exist (Cowling’s theorem), as a vital part of the axisymmetric portion of the magnetic ﬁeld can only be generated from longitudinal variations in the magnetic ﬁeld. In the mean ﬁeld dynamo, such effects (which are commonly assumed to be associated with ﬂuid turbulence) are parameterized. This simpliﬁcation of the equations allows dynamo solutions to be found that satisfy both the equation governing the evolution of the magnetic ﬁeld and

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