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The study of systems consisting of arbitrary sets of elements of unspecified type, together with certain operations satisfying prescribed lists of axioms. Abstract algebra has been developed since the mid-1920s and has become a basic idiom of…

An increase in the expansion rate of the universe, attributed to dark energy. In 1998, astronomers presented evidence that the universe's expansion is not slowing down, as would be expected due to the gravity from the abundances of matter strewn…

The study of the universe and the objects in it through scientific investigation. The goal of astronomical research is to understand celestial objects and the nature and evolution of the cosmos (Fig. 1). Because much of contemporary astronomy uses…

The theory that the universe began in a hot, dense state and has since expanded and cooled. The big bang is the generally accepted cosmological theory for the origin, properties, and evolution of the universe (Fig. 1). Cosmology Universe 1 Timeline…

A discipline that combines biology, mathematics, and computer science to acquire, store, analyze, and disseminate complex biological data. The term bioinformatics was coined by Paulien Hogeweg and Ben Hesper in the early 1970s to describe the "the…

A branch of mathematics that was first developed systematically, because of its applications to logic, by the English mathematician George Boole, around 1850. Closely related are its applications to sets and probability.Boolean algebra also…

In its simplest form, a vector is a directed line segment. Physical quantities, such as velocity, acceleration, force, and displacement, are vector quantities, or simply vectors, because they can be represented by directed line segments. The algebra…

A branch of mathematics that deals with intrinsic properties of curves and surfaces in three-dimensional Euclidean space. The intrinsic properties are those which are independent of the geometrical object's orientation or location in space. The…

In arithmetic and algebra, the process of finding one of two factors of a number (or polynomial) when their product and one of the factors are given. The symbol ÷ now used mostly in elementary English and American arithmetics to denote division…

A System that locates suitably equipped users by means of measurements made by electronic sensors. Typical measurements include distance or bearing from a vehicle to a known point, or direct determination of the present position of the vehicle in a…

Planets beyond our solar system. For millennia, humankind has known about most of the major planets in our solar system because these worlds are visible to the naked eye. Exoplanets, however, were not indisputably known until 1995, when astronomers…

A method of quantitative multivariate analysis with the goal of representing the interrelationships among a set of continuously measured variables (usually represented by their intercorrelations) by a number of underlying, linearly independent…

The largest visible-light observatory ever placed into space. The Hubble Space Telescope's orbit, some 612 km (380 mi) above Earth's surface (Fig. 1), keeps it beyond almost all of Earth's atmosphere, at a location where its view of the heavens is…

A space telescope mission dedicated to detecting exoplanets. The tenth mission launched under the National Aeronautics and Space Administration's (NASA's) Discovery Program, Kepler was the agency’s first mission dedicated to the search for planets…

A topological group with only countably many connected components whose identity component is open and is an analytic group. An analytic group or connected Lie group is a topological group with the additional structure of a smooth manifold such that…

That branch of mathematics which deals with solutions of systems of linear equations and the related geometric notions of vector spaces and linear transformations. It is fundamental in the theory of the calculus of functions of several variables and…

The planet fourth outward from the Sun and the planet most like Earth in terms of geology, climate, and suitability for life. Mars is nicknamed the "Red Planet" for its reddish hue (Fig. 1). Besides Earth, Mars is the most-studied world in the solar…

A relatively large, rounded, celestial body that developed around a star and cannot sustain nuclear fusion reactors in its interior. Although humankind has known about the existence of planets in our solar system for centuries, recent discoveries…

The largest dwarf planet in the solar system and a member of the Kuiper Belt of icy objects mostly beyond the orbit of Neptune. Pluto (Fig. 1) was discovered on February 18, 1930, by U.S. astronomer Clyde W. Tombaugh at the Lowell Observatory in…

A natural object, not human-made, in orbit around at least one more massive object, called a primary, when this primary is not a star. An object orbiting a star, a pair of stars, a galaxy, or a black hole will not be referred to as a natural…

Collectively, the various biological sciences concerned with the study of organisms when exposed to the environment of space, or of a spacecraft beyond the immediate influence of Earth and its atmosphere. Space biology dates back to November 1957,…

A spherical, self-luminous, astronomical body that generates energy through thermonuclear fusion. Over 6000 stars can be seen in the night sky with the unaided eye, shining as fixed points due to their tremendous distances from Earth (Fig. 1). These…

Groups of stars held together by mutual gravitational attraction. There are two basic morphological types: open clusters and globular clusters. Typical densities of field stars near the Sun are 0.1 star per cubic parsec. (One parsec equals 1.9 ×…

A device that automatically measures the angular separation of stellar observations with respect to a reference platform. It is also referred to as an astrotracker.By utilizing the star tracker in conjunction with a precise time reference…

A galaxy that is observed to be undergoing an unusually high rate of formation of stars. It is often defined as a galaxy that, if it continues to form stars at the currently observed rate, will exhaust its entire supply of star-forming material, the…

A magnetic field, far stronger than the Earth's magnetic field, which is possessed by many stars. Magnetic fields are important throughout the life cycle of a star. Initially, magnetic fields regulate how quickly interstellar clouds collapse into…

One of the categories into which stars may be classified, based on their place in the evolution of the galaxy that they occupy. The stellar component of the Milky Way Galaxy consists of three populations: the thin disk, the thick disk, and the halo.…

The spinning of stars, due to their angular momentum. Stars do not necessarily rotate as solid bodies, and their angular momentum may be distributed nonuniformly, depending on radius or latitude. However, it is nearly impossible to resolve the…

The assertion that under certain light restrictions the surface integral of (∇ × ℑ) · ν over a surface patch S is equal to the line integral of ℑ · τ taken around C, the boundary curve of S, provided the sense of transcription of C is right-handed…

An evolving star of luminosity class IV. Such a star is brighter than the main-sequence dwarfs and fainter than the true giants in its spectral class, lying between the two on the Hertzsprung-Russell diagram. The classic subgiants fall in a small…

One of the four fundamental operations of arithmetic and algebra. The first printed use of the symbol − to denote subtraction is in Johann Widman's Behennde und hüpsche Rechnung, Leipzig, 1489. Subtraction is often regarded as an operation inverse…

The star around which the Earth revolves and the source of our planet's light and heat. Containing 99.8% of the solar system's mass, the Sun holds all the system's planets and objects under its gravitational sway. The Sun measures 1.4 million km…

An instrument for telling time by the Sun. It is composed of a style that casts a shadow and a dial plate, which is the surface upon which hour lines are marked and upon which the shadow falls. The style lies parallel to Earth's axis. The…

A member of a class of evolved stars that occupy the top of the Hertzsprung-Russell diagram to the right of the main sequence. The absolute visual magnitudes (MV ) of supergiants range approximately between −4 and −10, and they are the largest and…

A system of convective cells, called supergranules, with diameters of about 30,000 km (20,000 mi) and lifetimes of 1 or 2 days, which cover the visible disk of the Sun, the photosphere. Smaller convection cells called granules, which are about 1500…

Proper motion of an astronomical object apparently exceeding the velocity of light, c. This phenomenon is relatively common in the nuclei of quasars, many of which exhibit systematic changes in their images over periods of months to years. In some…

A two-dimensional geometric figure (a collection of points) in three-dimensional space. The simplest example is a plane—a flat surface. Some other common surfaces are spheres, cylinders, and cones, the names of which are also used to describe the…

A double star system in the late stage of stellar evolution. A symbiotic star is a binary system and not a single star. The symbiotic phase represents a brief span in the life of the binary. Symbiotic stars are rare objects, and their distances are…

The alignment of three celestial objects within a solar system. Syzygy is most often used to refer to the alignment of the Sun, Earth, and Moon at the time of new or full moon. Although syzygy is strongly associated with these two lunar phases in…

A young low-mass star characterized by variability, the presence of hydrogen emission lines, and often other signatures of circumstellar gas and dust, including bright x-ray, ultraviolet, infrared, and radio emissions. T Tauri stars are named after…

A member of the rho-theta family of air navigation systems which define an aircraft's position by its distance and bearing to a single beacon. Such systems inherently answer the navigator's questions of “in what direction” and “how far,” without…

A term describing a relationship of two figures (usually of the same dimension) in the neighborhood of a common point. The figures are tangent at a point P if they touch at P but do not intersect in a sufficiently small neighborhood of P. To be more…

The Bull, a large zodiacal northern constellation prominent in the evening winter sky (see illustration). Taurus is usually said to be the form assumed by Zeus in Greek mythology in order to carry off the princess Europa. Zodiac Modern boundaries…

The systematic study of tensors which led to an extension and generalization of vectors, begun in 1900 by two Italian mathematicians, G. Ricci and T. Levi-Civita, following G. F. B. Riemann's proposal concerning a generalization of Euclidean…

A proposition arrived at by the methods of logical deduction from a set of basic postulates or axioms accepted as primitive and therefore not subject to deductive proof. So long as a theorem is part of a purely formal system, it is not meaningful to…

The branch of mathematics concerned with finding facts concerning the roots of algebraic equations and finding methods for obtaining them. The most important type of algebraic equation is the polynomial equation in one unknown which is an expression…

The largest moon of Saturn and the only satellite in the solar system with a substantial atmosphere, as well as the only object, besides Earth, known to host stable bodies of liquid on its surface. Titan, 5150 km (3200 mi) in diameter, is the second…

The study and application of topological transformation groups. Topological dynamics originated in the late-nineteenth-century investigations of the qualitative behavior of the solutions to the differential equations of classical mechanics by Henri…

The branch of mathematics that studies the qualitative properties of spaces, as opposed to the more delicate and refined geometric or analytic properties. While there are earlier results that belong to the field, the beginning of the subject as a…

A surface obtained by rotating a circle about a line that lies in the plane of the circle but that has no points in common with the circle (see illustration). The equations x = u cos v, y = u sin v, z = [r2 − (u − b)2]1/2, b > r > 0, represent the…

A plane curve for which the length of any tangent between the curve and a fixed line is constant c. If the x axis is the fixed line, its differential equation is Eq. (1). 1 With ψ, the inclination of the tangent, as pa­ram­eter, this yields Eqs.…

The passage of one astronomical object across another. The term "transit" implies that the foreground object is smaller than the background object (Fig. 1). A related term, "eclipse," implies that the foreground object hides a background object that…

The study of triangles and the trigonometric functions. One common use for trigonometry is to measure heights and distances that are awkward or impossible to measure by ordinary means. Surveyors use it to find heights of mountains and distances…

Asteroids located near the equilateral Lagrangian stability points of a Sun-planet system (Fig. 1). As shown by J. L. Lagrange in 1772, these are two of the five stable points in the circular, restricted, three-body system, the other three points…

The Great Bear, a circumpolar constellation for observers at mid-northern latitudes or higher latitudes (see illustration). Part of this constellation is the prominent asterism (star pattern) known as the Big Dipper, whose bowl of four stars is…

The Little Bear, a circumpolar constellation from northern latitudes (see illustration). Part of this constellation is the prominent asterism (star pattern) known as the Little Dipper, whose stars are fainter than the corresponding stars of the Big…

A term that became part of mathematical language during the development of analytic geometry (no equivalent word is found in the writings of Archimedes and other Greek mathematicians), and which is frequently not defined in modern works, apparently…

A star with a detectable change in brightness that is often accompanied by other physical changes. Every star varies in brightness sometime during its life, usually in the early stages (while it is forming) or in the late stages (close to its…

A mathematical element used to describe quantities having magnitude and direction. It can be represented by a directed line segment. Many physical quantities, for example, velocity, acceleration, and force, are vectors. Vectors are widely used in…

The fifth brightest of all stars and the third brightest in the northern sky. It will be the north polar star in about 12,000 years. In moving through the Milky Way Galaxy, the Sun is generally heading toward the position now occupied by Vega. At a…

The Virgin, perhaps Astraea, in Greek mythology the goddess of justice, whose scales (the constellation Libra) are adjacent. Also, she could be Ceres, goddess of agriculture, or her daughter Proserpina who, because she ate from a forbidden…

The nearest large cluster of galaxies and one of the best studied. It dominates the distribution of bright galaxies in the sky, forming a conspicuous clump in the distribution of easily visible elliptical and spiral galaxies. The clump lies mostly…

A theorem in classical mechanics which relates the kinetic energy of a system to the virial of Clausius, as defined below. The theorem can be generalized to quantum mechanics and has widespread application. It connects the average kinetic and…

The apparent change in direction of a source of light caused by an observer's component of motion perpendicular to the impinging rays.To visualize the effect, first imagine a stationary telescope (illus.a) aimed at a luminous source such as a star,…

A discipline that combines the means to know the location and heading of an aircraft with respect to some map or coordinate system, with guidance which means steering the aircraft along some desired path toward a destination. The construction of the…

The branch of mathematics dealing with the solution of equations. These equations involve unknowns, or variables, along with fixed numbers from a specified system (Fig. 1). The origins of algebra were based on the need to develop equations that…

The study of zero sets of polynomial equations. Examples of zero sets of polynomial equations studied in algebraic geometry are the parabola y − x2 = 0, thought of as sitting in the (x, y)-plane, and the locus of all points (t, t2, t3, t4), which is…

The closest spiral galaxy to the Milky Way Galaxy. The name of the Andromeda Galaxy (Fig. 1) derives from its appearance in the constellation of Andromeda. The galaxy is also known as M31 and NGC 224, based on its entries in two widely used…

A gravitationally bound, condensed body whose mass is too small to sustain nuclear fusion reactions in its core. Brown dwarfs are often likened to "failed stars." The approximate mass range for brown dwarfs is between 13 and 80 times the mass of…

The branch of mathematics dealing with two fundamental operations, differentiation and integration, which are carried out on functions. The subject of calculus (Fig. 1), as traditionally developed in college textbooks, is partly an elementary…

An algorithm that gives a name to each day. These day names are generally distinct, but not always. For example, the day of the week is a calendar, in a trivial sense, with infinitely many days having the same day name. The design of calendars has…

A heart-shaped curve generated by a point of a circle that rolls (without slipping) on a fixed circle of the same diameter. In point-wise construction of the curve, let O be a fixed point of a circle C of diameter a, and Q a variable point of C. Lay…

A theory of mathematical structure in which smooth continuous inputs lead to discontinuous responses. Water suddenly boils, ice melts, a building crashes to the ground, or the earth unexpectedly buckles and quakes. The French mathematician René Thom…

A force whose line of action is always directed toward a fixed point. The central force may attract or repel. The point toward or from which the force acts is called the center of force. If the central force attracts a material particle, the path of…

Light emitted by a high-speed charged particle when the particle passes through a transparent, nonconducting, solid material at a speed greater than the speed of light in the material. The blue glow observed in the water of a nuclear reactor, close…

A quantitative measure of a star's color. Even a casual look at the night sky reveals the stars to be colored. From spectral analysis and the laws of radiation, it has been known since early in the twentieth century that such color is closely…

The branch of mathematics which studies arrangements of elements (usually a finite number) into sets under certain prescribed constraints. Problems combinatorialists attempt to solve include the enumeration problem (how many such arrangements are…

The theory of complex systems. Complexity theory arises from the need to understand the richness in structure and behavior often seen in large systems. The property that distinguishes complex systems from systems that are large but simple is the…

The solid of revolution obtained by revolving a right triangle about one of its shorter sides is called a cone, or more precisely a right circular cone (see illustration). More generally, the term cone is used in solid geometry to describe a solid…

A special operation in mathematics in which a set of points in one coordinate system is mapped or transformed into a corresponding set in another coordinate system, preserving the angle of intersection between pairs of curves.A mapping or…

Schemes for locating points in a given space by means of numerical quantities specified with respect to some frame of reference. These quantities are the coordinates of a point. To each set of coordinates there corresponds just one point in any…

The Southern Cross, a small southern constellation. Because of the absence of a star at the center of the cross, this constellation really looks like a kite (see illustration). To its side, the Coalsack is a giant dark silhouette of dust against the…

A unit of time equal to the period of rotation of Earth. Different sorts of day are distinguished, according to how the period of rotation is reckoned with respect to one or another direction in space. A day is normally defined as 86,400 SI (Système…

A form of navigation that determines position of a craft by advancing a previous position to a new one on the basis of assumed distance and direction moved. The name probably stems from the early practice of determining speed by throwing overboard a…

An applied branch of decision theory. Decision analysis offers individuals and organizations a methodology for making decisions; it also offers techniques for modeling decision problems mathematically and finding optimal decisions numerically.…

A broad range of concepts which have been developed to both describe and prescribe the process of decision making, where a choice is made from a finite set of possible alternatives. Normative decision theory describes how decisions should be made in…

A mathematical operation performed on a function to determine the effect of a change in the value of the independent variable. Functions of one variable are considered in this article. For differentiation of functions of several variablesPartial…

Equipment used for determining the direction from which a received signal is being radiated. Direction finders serve a variety of applications. Direction finders can be used for navigation purposes to determine the heading of a vehicle relative to a…

For a polynomial f(x) = an xn + an−1 xn−1 + + a1 x + a0, the discriminant is given by the expression where x1, x2, …, xn are the roots of the equation f(x) = 0. There are n(n − 1)/2 terms (xi − xj)2 in the product corresponding to all possible…

An internationally standardized navigation system which allows an aircraft to measure its distance from a selected ground-based beacon. Such beacons are used throughout the world by all airliners, most of the military aircraft of the West, and a…

A star whose state of evolution resembles that of the Sun. The term “dwarf star” derives from the early work of Ejnar Hertzsprung and Henry Norris Russell, who distinguished two kinds of stars: large ones, called giants (or supergiants), and smaller…

A mathematical abstraction corresponding to an expression of the type βγ + δ∊ + · · · , in which the elements (dyad symbols) consist of two vector symbols in juxtaposition without the intervention of either the dot (·) or cross (×). Essentially a…

One of the solutions of an eigenvalue equation. A parameter-dependent equation that possesses nonvanishing solutions only for particular values (eigenvalues) of the parameter is an eigenvalue equation, the associated solutions being the…

One of the values of the parameter in an eigenvalue equation for which the equation has a solution. A linear eigenvalue equation is a parameter-dependent equation of the form (1) 1that possesses nonvanishing solutions, |v(n), only for particular…

A member of the class of curves that are intersections of a plane with a cone of revolution. The ellipse is obtained when the plane cuts all the elements of one nappe, and does not go through the apex. In the illustration, denote the distance…

The elliptic functions were developed originally by K. F. Gauss, N. H. Abel, K. G. J. Jacobi, and others in order to solve the problem of finding the arc length of an ellipse, but they proved to have remarkable applications in physics, probability…

A source of data pertaining to features of various objects, usually astronomical, that depend on time. The ephemeris can be a printed or computer-readable table such as on a compact disk, or it can be a set of numerical data accessible by computer…

A curve traced by a point on a circle that rolls on the convex side of a fixed circle. The term is also occasionally applied to the curve generated by a point on the prolongation of the radius of a circle as the circle rolls on a straight line. If…

The annual, cyclic variation between mean solar time shown on uniformly running clocks and apparent solar time displayed on sundials.In the course of the Sun's daily east-to-west transit of the sky, the Sun crosses the meridian, an imaginary line…

A branch of probability and statistics concerned with deriving information about properties of random variables, stochastic processes, and systems based on observed samples. Some of the important applications of estimation theory are found in…