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Lg wave

ber is expected to approach 1, and the turbulent diffusivities should become independent of the considered substance C, if turbulent transport is due to intermingling neighboring water parcels. For double-diffusion, in water the so-called “apparent” diffusivities of salt and heat are identical only in the asymptotic case of stability ratio Rρ → 1.

Lg wave When we observe a shallow earthquake at a point with epicentral distance of several hundred to several thousand kilometers, there are some cases that predominant and substantially short-period (2 10 s) seismic waves compared to usual surface waves appear. The wave was discovered by F. Press in 1952. The apparent velocity of the Lg wave is about 3.5 km/s, and the wave is considered to be generated by multi-reﬂection of body waves within a continental crustal structure, namely, the granitic layer. Characteristics of Lg wave are similar to Love wave, and the name Lg was taken from the ﬁrst letter of Love wave, followed by that of granite. It is known that Lg wave does not propagate in oceanic crustal structures. This is thought to be because the oceanic crust is thin and the trapped waves leak into oceanic water.

libration The behavior of an angular variable in a dynamical system which exhibits a small oscillation around an equilibrium conﬁguration. The moon is known to be in synchronous rotation: It completes one orbit around the Earth in 29.5 days while it rotates once on its own axis so that in general only 50% of the lunar surface should be visible from the Earth. However, it exhibits three kinds of libration, the libration in longitude about the selenographic origin over ±7◦45 due to its orbital eccentricity, the libration in latitude about the selenographic origin over ±6◦44 due to its rotational axis being tilted with respect to the normal to its orbital plane, and the diurnal libration about the selenographic origin over ±1◦ simply due to the rotation of the Earth causing parallax to the earthbound observer. In the orbital motion of an artiﬁcial satellite around the Earth, it is known that its argument of perigee librates around 90◦ when its orbital inclination stays in the neighborhood of 63.4◦ as the speciﬁc shape of the Earth causes gravitational perturbations. An-

other example is a Trojan asteroid the average ecliptic longitude of which librates around that of Jupiter ±60◦ (Lagrange points L4 and L5) due to Jovian gravitational perturbations.

Lie derivative Derivative operator associated with a one-parameter group of diffeomorphisms. Let ξa be the generator of the group. The Lie derivative of a ﬁeld 8(x) of geometric objects at point x of a differentiable manifold is deﬁned by the limit

8(x) − 8 (x)

Lξ 8 = lim

t→0 t

where 8 (x) is the image of the ﬁeld at x + ξt under the diffeomorphism map. Expressed in components, the Lie derivative of a tensor is

L T ab...p = ξu ∂ T ab...p

ξ qr...t ∂xu qr...t

− T ub...p ∂ ξa qr...t ∂xu

− · · · − T ab...uqr...t ∂u ξp

∂x

+ T ab...p ∂ ξu ur...t ∂xq

+ · · · + T ab...pqr...u ∂ t ξu .

∂x

In theoretical physics the Lie derivative is important in establishing classiﬁcations of spacetimes in general relativity (isometry groups, or groups of motions, and the Bianchi types in spatially homogeneous cosmologies); in formulation of the Noether theorem in ﬁeld theory; in the theory of reference frames; in some methods of generation of exact solutions of Einstein’s equations; in some approaches to the second quantization postulate in quantum ﬁeld theory, and in physical applications of the theory of Lie groups. See tensor, vector.

lift coefﬁcient A dimensionless coefﬁcient that appears in the equation for lift force induced by a ﬂow around an object. The lift force is typically vertical, but in general need only be perpendicular to the ﬂow direction.

lifting condensation level The level at which a parcel of moist air lifted adiabatically becomes saturated.

light pollution

light bridge Observed in white light, a bright tongue or streak penetrating or crossing a sunspot umbra. The appearance of a light bridge is frequently a sign of impending region division or dissolution.

light cone The 3-dimensional hypersurface in spacetime which is generated by light rays (i.e., light-like geodesics) originating at a single event p (future light cone) and meeting at this event (past light cone); the event p is referred to as the vertex of the light cone. In the ﬂat spacetime of special relativity such hypersurfaces with vertices at different events of the ﬂat spacetime are isometric (i.e., are identical copies of each other). In general spacetimes, the hypersurfaces may be curved and may end at folds, or self-intersections. In curved spacetimes, light cones with different vertices are in general not isometric.

The notion of a light cone is important in observational astronomy: the image of the sky seen by an observer at the moment t = t0 is a projection of the observer’s past light cone at t = t0 onto the celestial sphere. All the events in our actual spacetime that could have been observed by a given observer up to the moment t = t0 on his clock lie on or inside the past light cone of the event t = t0 of the observer’s history, a restatement of the fact that an event q can be observed by an observer O only by the time when the light-ray emitted at q reaches O.

light-curve A plot of light received from a star vs. time. A variable star is one that has a light-curve with peaks and troughs. Often these features are periodic (in the case of eclipsing binaries) or nearly (or quasi-) periodic in the case of variations in an isolated star, such as sunspots (starspots) or stellar oscillations.

For planetary bodies, the variation in brightness due to an object’s rotation. A light-curve can be due to shape effects (seeing different cross-sectional areas of an object as it rotates), albedo changes on the surface (seeing different bright and dark patches as the object rotates), or some combination of the two.

light deﬂection The curving of light rays by the gravitational ﬁeld. In the absence of all gravitational ﬁelds, i.e., in the ﬂat spacetime of

special relativity, light rays are straight. However, in curved spacetime containing a massive body, a light ray passing by the massive body curves toward the body. If a body of mass M is spherical, and the closest distance between the ray and the center of the body is r, then the angle of deﬂection @φ is given by the approximate Einstein formula @φ = 4GM/(c2r), where G is the gravitational constant and c is the velocity of light. The formula applies only when r 2GM/c2 (i.e., when the angle @φ is small). For a ray grazing the surface of the sun, @φ = 1.75 sec of arc. This effect was measured by Eddington in 1919, and more recently conﬁrmed with greater precision for radio-waves from radio-sources; deﬂection of light by the sun has been veriﬁed to parts in 1000 by observations of the direction to extragalactic radio sources, as the sun passes near their position in the sky. An observer placed far enough behind the deﬂecting mass can see light rays that passed on opposite sides of the deﬂector. For such an observer, the deﬂector would be a gravitational lens. In the case of distant sources, deﬂection by intervening galaxies or clusters of galaxies causes lensing, leading to the appearance of multiple images, and of rings or arcs of distorted images. Light rays approaching the very strong gravitational ﬁeld of a black hole can describe very complicated orbits.

light-harvesting complex A pigment-pro- tein complex containing chlorophyll that supplies additional energy in photosynthesis.

lightlike current (cosmic string)	See elec-
tric regime (cosmic string).

lightning Electric discharge in the atmosphere that produces a lightning ﬂash. Two fundamental types can be distinguished: ground ﬂashes where the discharge occurs between the ground and the cloud with the ﬂash either propagating upward or downward, and cloud ﬂashes or intracloud ﬂashes, where the discharge occurs between clouds and the ﬂash does not strike the ground.

light pollution Deleterious effects of outdoor lighting on astronomical observations (which require dark sky and which are severely affected

light year (l.y.)

by speciﬁc spectral lines arising from outdoor lighting using ionized gas lamps).

light year (l.y.) A unit of distance used in astronomy, deﬁned as the distance light travels

in one year in a vacuum. Its approximate value is 9.46073 × 1015m.

LIGO (Laser Interferometric Gravitational Observatory) A laboratory for detecting gravitational waves by using laser interferometry. At each of two sites, one at Livingston, Louisiana, the other at Hanford, Washington there is an L-shaped Fabri–Perot light resonator (a version of a Michelson interferometer). The length of the arms is 4 km. The Washington site also contains a shorter (A = 2 km) interferometer inside the same vacuum pipe. Laser interferometer gravitational wave detection occurs by observing an alteration of the relative length of the two arms of the interferometer caused by the passage of the wave. Operation is expected to start in 2001. See GEO, Virgo.

limb The edge of the visible disk of an astronomical object. For instance, one refers to the limb of the sun, or to the limb of the moon.

limb brightening The apparent brightness increase observed in long wavelength (1 mm) photospheric observations towards the limb of the sun. The transition from limb darkening to limb brightening provides a diagnostic for determining the location of the temperature minimum in the photosphere.

limb darkening The reﬂectivity of a body with an atmosphere is determined in part by the absorption and reﬂectivity properites of the atmosphere. For a spherical body, the amount of atmosphere encountered by a light ray will increase towards the limb of the body. The reﬂectivity of the body will therefore vary from the center of the disk to its limb. For most atmospheres this results in a lowering of the reﬂectivity, hence the term limb darkening. In the sun, the fall off in photospheric brightness towards the solar limb due to the effects of the decreasing temperature of the higher photospheric layers. Predominantly observed in white light

and at wavelengths, λ < 1 mm, i.e., visible light and infrared.

Lindblad resonance A resonance in the orbital angular speed developed in non-axisym- metric gravitational potentials, such as the potential in a weakly barred galaxy, or of a planet slightly deviating from spherical symmetry, named after B. Lindblad (1965). In the case of a barred galaxy, the bar supposedly rotates like a rigid body with a “pattern speed” 6P and thus gives rise to a non-axisymmetric gravitational potential. In non-axisymmetric potentials, orbits are not generally closed; if the deviation from axisymmetry is small, their orbital motion can be thought of as due to the rotation associated to circular motion plus small radial oscillations. Lindblad resonances occur for stars orbiting at angular speed 6 = 6P ± mκ , where κ is the resonant angular frequency for radial oscillation, usually in the range 1 to 2 6P , and m is an integer. The plus sign and the minus sign deﬁne inner and outer Lindblad resonances, respectively. Radii at which such resonances occur are called Lindblad radii. Rings or ring-like features are expected to form at and close to the Lindblad radii.

linearized gravitation A treatment of weak gravitational ﬁelds. The space-time metric is written in the form gab = ηab + hab where ηab is the Minkowski (i.e., the ﬂat space, zero gravity) metric and the quantities hab, together with their derivatives, are treated as inﬁnitesimal quantities. This form of the metric is preserved by the inﬁnitesimal coordinate transformation xa = xa +ξa where the functions ξa are inﬁnitesimal. Linearized gravitation is widely used in various perturbation problems and is the basis of the post-Newtonian approximation scheme. It also predicts the gravitational radiation from motions accurately described in the Newtonian regime.

linear momentum Momentum p = γ mv, where

γ = 1/ 1 − v2/c2 ,

where c = speed of light.

For nonrelativistic motion, γ 1, and momentum p = mv.

Liouville’s theorem (the equation of continuity)

linear wave theory Also referred to as Airy wave theory, or Stokes ﬁrst-order theory. A theory for description of water waves on a free surface, such as waves on the ocean. Referred to as linear because of the fact that higher order terms in the boundary conditions are neglected during derivation of the solution.

line of apsides In celestial mechanics, the line connecting the two extrema of an elliptical (or nearly elliptical) orbit.

line of nodes The line connecting the center of mass of the solar system with the point on a planet’s orbit where the planet crosses the ecliptic in a northward direction. Can be similarly deﬁned for Earth centered orbits, and for artiﬁcial satellites.

LINER Acronym for Low Ionization Nuclear Emitting Region. LINERs are narrow emission line galaxies that show optical and UV spectra with notable differences from classical Seyfert-1 and Seyfert-2 galaxies, namely lower nuclear luminosity and stronger forbidden lines from low-ionization species such as neutral oxygen, singly ionized sulfur, and nitrogen. The ionization mechanism of the line-emitting gas is unclear. An appealing explanation sees LINERs as the lowest luminosity active galactic nuclei, photoionized by a non-stellar continuum weaker than that of more powerful active nuclei. Alternatively, gas may be heated mechanically by shocks, or may be photoionized in dense clouds embedding hot stars of early spectral types. LINERs are frequently observed in the nuclei of both spiral and elliptical galaxies, and might be detectable in nearly half of all spiral galaxies. See Seyfert galaxies.

line squall A squall or series of squalls that occur along a line and advance on a wide front, caused by the replacement of a warmer by colder body of air. Line squalls are often associated with the passage of a cold front; it deﬁnes the line of the cloud and wind structure. Warmer air is overrun by cold air to produce the line squall. A line squall may extend for hundreds of miles, with a sudden rapid change of wind direction, generally from the southeast or south to west, northwest and north, a rapid rise in barometric

pressure and relative humidity, a rapid fall in temperature, and violent changes in weather — heavy cloud, thunderstorms, heavy rain, snow, or hail.

line wing That portion of a line proﬁle (either absorption or emission) on either side of the central line core. The shape of a line proﬁle is related to the Maxwellian velocity distribution of gas particles (and therefore to the temperature of the gas), the density of the gas, and the natural line width (See curve of growth). The larger the spread in velocities, the broader the line wings; and the denser the gas, the deeper the central core of the line proﬁle. Even if the gas has a temperature of absolute zero (0 K), the line proﬁle would still have a non-zero width due to the quantum mechanical uncertainty in the energy (Heisenberg’s uncertainty principle); this is called the natural line width and has a Lorentzian shape. In realistic circumstances, i.e., gases with T > 0 K, a Maxwellian distribution in velocity broadens the line proﬁle, and is the dominant contributor to the line wings. Other phenomena also cause a broadening of the line proﬁle: pressure broadening, macroand micro-turbulence in the gas, and bulk rotation of the gas, as in a rotating star. Rotational broadening can be dramatic, depending on the rotation velocity of the star. In early type stars, (O and B) rotation velocities can be as high as 400 km/s. At this speed, the power in the line core is so smeared out into the wings that photospheric lines are completely washed out.

	1.0
	0.9	line core
		line core
	0.8
	line wing	line wing
	0.7
Intensity	0.6
	0.5
	0.4
	0.4
	0.3
	0.2
	0.1
	0.0
		Wavelength

Liouville’s theorem (the equation of continuity) The substrate of points in phase space