In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light propagates through the material. It is defined as n = c v, where c is the speed of light in vacuum and v is the phase velocity of light in the medium. For example, the refractive index of water is 1.333, meaning that light travels 1.333 times as fast in vacuum as in water. The refractive index determines how much the path of light is bent, or refracted, when entering a material; this is described by Snell's law of refraction, n1 sinθ1 = n2 sinθ2, where θ1 and θ2 are the angles of incidence and refraction of a ray crossing the interface between two media with refractive indices n1 and n2. The refractive indices determine the amount of light, reflected when reaching the interface, as well as the critical angle for total internal reflection and Brewster's angle; the refractive index can be seen as the factor by which the speed and the wavelength of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is v = c/n, the wavelength in that medium is λ = λ0/n, where λ0 is the wavelength of that light in vacuum.
This implies that vacuum has a refractive index of 1, that the frequency of the wave is not affected by the refractive index. As a result, the energy of the photon, therefore the perceived color of the refracted light to a human eye which depends on photon energy, is not affected by the refraction or the refractive index of the medium. While the refractive index affects wavelength, it depends on photon frequency and energy so the resulting difference in the bending angle causes white light to split into its constituent colors; this is called dispersion. It can be observed in prisms and rainbows, chromatic aberration in lenses. Light propagation in absorbing materials can be described using a complex-valued refractive index; the imaginary part handles the attenuation, while the real part accounts for refraction. The concept of refractive index applies within the full electromagnetic spectrum, from X-rays to radio waves, it can be applied to wave phenomena such as sound. In this case the speed of sound is used instead of that of light, a reference medium other than vacuum must be chosen.
The refractive index n of an optical medium is defined as the ratio of the speed of light in vacuum, c = 299792458 m/s, the phase velocity v of light in the medium, n = c v. The phase velocity is the speed at which the crests or the phase of the wave moves, which may be different from the group velocity, the speed at which the pulse of light or the envelope of the wave moves; the definition above is sometimes referred to as the absolute refractive index or the absolute index of refraction to distinguish it from definitions where the speed of light in other reference media than vacuum is used. Air at a standardized pressure and temperature has been common as a reference medium. Thomas Young was the person who first used, invented, the name "index of refraction", in 1807. At the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers; the ratio had the disadvantage of different appearances. Newton, who called it the "proportion of the sines of incidence and refraction", wrote it as a ratio of two numbers, like "529 to 396".
Hauksbee, who called it the "ratio of refraction", wrote it as a ratio with a fixed numerator, like "10000 to 7451.9". Hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1. Young did not use a symbol for the index of refraction, in 1807. In the next years, others started using different symbols: n, m, µ; the symbol n prevailed. For visible light most transparent media have refractive indices between 1 and 2. A few examples are given in the adjacent table; these values are measured at the yellow doublet D-line of sodium, with a wavelength of 589 nanometers, as is conventionally done. Gases at atmospheric pressure have refractive indices close to 1 because of their low density. All solids and liquids have refractive indices above 1.3, with aerogel as the clear exception. Aerogel is a low density solid that can be produced with refractive index in the range from 1.002 to 1.265. Moissanite lies at the other end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, but some high-refractive-index polymers can have values as high as 1.76.
For infrared light refractive indices can be higher. Germanium is transparent in the wavelength region from 2 to 14 µm and has a refractive index of about 4. A type of new materials, called topological insulator, was found holding higher refractive index of up to 6 in near to mid infrared frequency range. Moreover, topological insulator material are transparent; these excellent properties make them a type of significant materials for infrared optics. According to the theory of relativity, no information can travel faster than the speed of light in vacuum, but this does not mean that the refractive index cannot be lower than 1; the refractive index measures the phase velocity of light. The phase velocity is the speed at which the crests of the wave move and can be faster than the speed of light in vacuum, thereby give a refractive index below 1; this can occur close to resonance frequencies, for absorbing media, in plasmas, for X-rays. In the X-ray regime the refractive indices are
Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of height. This refraction is due to the velocity of light through air, decreasing with increased density. Atmospheric refraction near the ground produces mirages; such refraction can raise or lower, or stretch or shorten, the images of distant objects without involving mirages. Turbulent air can make distant objects appear to shimmer; the term applies to the refraction of sound. Atmospheric refraction is considered in measuring the position of both celestial and terrestrial objects. Astronomical or celestial refraction causes astronomical objects to appear higher above the horizon than they are. Terrestrial refraction causes terrestrial objects to appear higher than they are, although in the afternoon when the air near the ground is heated, the rays can curve upward making objects appear lower than they are. Refraction not only affects visible light rays, but all electromagnetic radiation, although in varying degrees.
For example, in the visible spectrum, blue is more affected than red. This may cause astronomical objects to appear dispersed into a spectrum in high-resolution images. Whenever possible, astronomers will schedule their observations around the times of culmination, when celestial objects are highest in the sky. Sailors will not shoot a star below 20° above the horizon. If observations of objects near the horizon cannot be avoided, it is possible to equip an optical telescope with control systems to compensate for the shift caused by the refraction. If the dispersion is a problem, atmospheric refraction correctors can be employed as well. Since the amount of atmospheric refraction is a function of the temperature gradient, temperature and humidity, the amount of effort needed for a successful compensation can be prohibitive. Surveyors, on the other hand, will schedule their observations in the afternoon, when the magnitude of refraction is minimum. Atmospheric refraction becomes more severe when temperature gradients are strong, refraction is not uniform when the atmosphere is heterogeneous, as when turbulence occurs in the air.
This causes suboptimal seeing conditions, such as the twinkling of stars and various deformations of the Sun's apparent shape soon before sunset or after sunrise. Astronomical refraction deals with the angular position of celestial bodies, their appearance as a point source, through differential refraction, the shape of extended bodies such as the Sun and Moon. Atmospheric refraction of the light from a star is zero in the zenith, less than 1′ at 45° apparent altitude, still only 5.3′ at 10° altitude. On the horizon refraction is greater than the apparent diameter of the Sun, so when the bottom of the sun's disc appears to touch the horizon, the sun's true altitude is negative. If the atmosphere vanished at this moment, one couldn't see the sun, as it would be below the horizon. By convention and sunset refer to times at which the Sun's upper limb appears on or disappears from the horizon and the standard value for the Sun's true altitude is −50′: −34′ for the refraction and −16′ for the Sun's semi-diameter.
The altitude of a celestial body is given for the center of the body's disc. In the case of the Moon, additional corrections are needed for the Moon's horizontal parallax and its apparent semi-diameter. Refraction near the horizon is variable, principally because of the variability of the temperature gradient near the Earth's surface and the geometric sensitivity of the nearly horizontal rays to this variability; as early as 1830, Friedrich Bessel had found that after applying all corrections for temperature and pressure at the observer precise measurements of refraction varied by ±0.19′ at two degrees above the horizon and by ±0.50′ at a half degree above the horizon. At and below the horizon, values of refraction higher than the nominal value of 35.4′ have been observed in a wide range of climates. Georg Constantin Bouris measured refraction of as much of 4° for stars on the horizon at the Athens Observatory and, during his ill-fated Endurance expedition, Sir Ernest Shackleton recorded refraction of 2°37′:“The sun which had made ‘positively his last appearance’ seven days earlier surprised us by lifting more than half its disk above the horizon on May 8.
A glow on the northern horizon resolved itself into the sun at 11 am that day. A quarter of an hour the unreasonable visitor disappeared again, only to rise again at 11:40 am, set at 1 pm, rise at 1:10 pm and set lingeringly at 1:20 pm; these curious phenomena were due to refraction. The temperature was 15° below 0° Fahr. and we calculated that the refraction was 2° above normal.” Day-to-day variations in the weather will affect the exact times of sunrise and sunset as well as moon-rise and moon-set, for that reason it is not meaningful to give rise and set times to greater precision than the nearest minute. More precise calculations can be useful for determining day-to-day changes in rise and set times that would occur with the standard value for refraction if it is understood
In radio communication, skywave or skip refers to the propagation of radio waves reflected or refracted back toward Earth from the ionosphere, an electrically charged layer of the upper atmosphere. Since it is not limited by the curvature of the Earth, skywave propagation can be used to communicate beyond the horizon, at intercontinental distances, it is used in the shortwave frequency bands. As a result of skywave propagation, a signal from a distant AM broadcasting station, a shortwave station, or – during sporadic E propagation conditions a distant VHF FM or TV station – can sometimes be received as as local stations. Most long-distance shortwave radio communication – between 3 and 30 MHz – is a result of skywave propagation. Since the early 1920s amateur radio operators, limited to lower transmitter power than broadcast stations, have taken advantage of skywave for long distance communication. Skywave propagation is distinct from: groundwave propagation, where radio waves travel near Earth's surface without being reflected or refracted by the atmosphere – the dominant propagation mode at lower frequencies, line-of-sight propagation, in which radio waves travel in a straight line, the dominant mode at higher frequencies.
The ionosphere is a region of the upper atmosphere, from about 80 km to 1000 km in altitude, where neutral air is ionized by solar photons and cosmic rays. When high frequency signals enter the ionosphere obliquely, they are back-scattered from the ionized layer as scatter waves. If the midlayer ionization is strong enough compared to the signal frequency, a scatter wave can exit the bottom of the layer earthwards as if reflected from a mirror. Earth's surface diffusely reflects the incoming wave back towards the ionosphere. Like a rock "skipping" across water, the signal may "bounce" or "skip" between the earth and ionosphere two or more times. Since, at shallow incidence, losses remain quite small, signals of only a few watts can sometimes be received many thousands of miles away as a result; this is. If the ionization is not great enough, the scatter wave is deflected downwards, subsequently upwards such that it exits the top of the layer displaced. Sky wave propagation occurs in the waveguide formed by the ground and ionosphere, each serving as reflectors.
With a single "hop," path distances up to 3500 km may be reached. Transatlantic connections are obtained with two or three hops; the layer of ionospheric plasma with equal ionization is not fixed, but undulates like the surface of the ocean. Varying reflection efficiency from this changing surface can cause the reflected signal strength to change, causing "fading" in shortwave broadcasts. VHF signals with frequencies above about 30 MHz penetrate the ionosphere and are not returned to the Earth's surface. E-skip is a notable exception, where VHF signals including FM broadcast and VHF TV signals are reflected to the Earth during late spring and early summer. E-skip affects UHF frequencies, except for rare occurrences below 500 MHz. Frequencies below 10 MHz, including broadcasts in the mediumwave and shortwave bands, propagate most efficiently by skywave at night. Frequencies above 10 MHz propagate most efficiently during the day. Frequencies lower than 3 kHz have a wave length longer than the distance between the Earth and the ionosphere.
The maximum usable frequency for skywave propagation is influenced by sunspot number. Skywave propagation is degraded – sometimes – during geomagnetic storms. Skywave propagation on the sunlit side of the Earth can be disrupted during sudden ionospheric disturbances; because the lower-altitude layers of the ionosphere disappear at night, the refractive layer of the ionosphere is much higher above the surface of the Earth at night. This leads to an increase in the "skip" or "hop" distance of the skywave at night. Amateur radio operators are credited with the discovery of skywave propagation on the shortwave bands. Early long-distance services used surface wave propagation at low frequencies, which are attenuated along the path. Longer distances and higher frequencies using this method meant more signal attenuation. This, the difficulties of generating and detecting higher frequencies, made discovery of shortwave propagation difficult for commercial services. Radio amateurs conducted the first successful transatlantic tests in December 1921, operating in the 200 meter mediumwave band —the shortest wavelength available to amateurs.
In 1922 hundreds of North American amateurs were heard in Europe at 200 meters and at least 30 North American amateurs heard amateur signals from Europe. The first two-way communications between North American and Hawaiian amateurs began in 1922 at 200 meters. Although operation on wavelengths shorter than 200 meters was technically illegal, amateurs began to experiment with those wavelengths using newly available vacuum tubes shortly after World War I. Extreme interference at the upper edge of the 150-200 meter band—the official wavelengths allocated to amateurs by the Second National Radio Conference in 1923—forced amateurs to shift to shorter and shorter wavelengths.
In physics, attenuation or, in some contexts, extinction is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, water and air attenuate both light and sound at variable attenuation rates. Hearing protectors help reduce acoustic flux from flowing into the ears; this phenomenon is measured in decibels. In electrical engineering and telecommunications, attenuation affects the propagation of waves and signals in electrical circuits, in optical fibers, in air. Electrical attenuators and optical attenuators are manufactured components in this field. In many cases, attenuation is an exponential function of the path length through the medium. In chemical spectroscopy, this is known as the Beer–Lambert law. In engineering, attenuation is measured in units of decibels per unit length of medium and is represented by the attenuation coefficient of the medium in question. Attenuation occurs in earthquakes. One area of research in which attenuation plays a prominent role, is in ultrasound physics.
Attenuation in ultrasound is the reduction in amplitude of the ultrasound beam as a function of distance through the imaging medium. Accounting for attenuation effects in ultrasound is important because a reduced signal amplitude can affect the quality of the image produced. By knowing the attenuation that an ultrasound beam experiences traveling through a medium, one can adjust the input signal amplitude to compensate for any loss of energy at the desired imaging depth. Ultrasound attenuation measurement in heterogeneous systems, like emulsions or colloids, yields information on particle size distribution. There is an ISO standard on this technique. Ultrasound attenuation can be used for extensional rheology measurement. There are acoustic rheometers that employ Stokes' law for measuring extensional viscosity and volume viscosity. Wave equations which take acoustic attenuation into account can be written on a fractional derivative form, see the article on acoustic attenuation or e.g. the survey paper.
Attenuation coefficients are used to quantify different media according to how the transmitted ultrasound amplitude decreases as a function of frequency. The attenuation coefficient can be used to determine total attenuation in dB in the medium using the following formula: Attenuation = α ⋅ ℓ ⋅ f Attenuation is linearly dependent on the medium length and attenuation coefficient, –approximately– on the frequency of the incident ultrasound beam for biological tissue. Attenuation coefficients vary for different media. In biomedical ultrasound imaging however, biological materials and water are the most used media; the attenuation coefficients of common biological materials at a frequency of 1 MHz are listed below: There are two general ways of acoustic energy losses: absorption and scattering, for instance light scattering. Ultrasound propagation through homogeneous media is associated only with absorption and can be characterized with absorption coefficient only. Propagation through heterogeneous media requires taking into account scattering.
Fractional derivative wave equations can be applied for modeling of lossy acoustical wave propagation, see acoustic attenuation and Ref. Main article: Electromagnetic absorption by waterShortwave radiation emitted from the sun have wavelengths in the visible spectrum of light that range from 360 nm to 750 nm; when the sun's radiation reaches the sea-surface, the shortwave radiation is attenuated by the water, the intensity of light decreases exponentially with water depth. The intensity of light at depth can be calculated using the Beer-Lambert Law. In clear open waters, visible light is absorbed at the longest wavelengths first. Thus, red and yellow wavelengths are absorbed at higher water depths, blue and violet wavelengths reach the deepest in the water column; because the blue and violet wavelengths are absorbed last compared to the other wavelengths, open ocean waters appear deep-blue to the eye. In near-shore waters, sea water contains more phytoplankton than the clear central ocean waters.
Chlorophyll-a pigments in the phytoplankton absorb light, the plants themselves scatter light, making coastal waters less clear than open waters. Chlorophyll-a absorbs light most in the shortest wavelengths of the visible spectrum. In near-shore waters where there are high concentrations of phytoplankton, the green wavelength reaches the deepest in the water column and the color of water to an observer appears green-blue or green; the energy with which an earthquake affects a location depends on the running distance. The attenuation in the signal of ground motion intensity plays an important role in the assessment of possible strong groundshaking. A seismic wave loses energy; this phenomenon is tied into the dispersion of the seismic energy with the distance. There are two types of dissipated energy: geometric dispersion caused by distribution of the seismic energy to greater volumes dispersion as heat called intrinsic attenuation or anelastic attenuat
In physics, a surface wave is a 90 degree wave that propagates along the interface between differing media. A common example is gravity waves along the surface such as ocean waves. Gravity waves can occur within liquids, at the interface between two fluids with different densities. Elastic surface waves can travel along the surface such as Rayleigh or Love waves. Electromagnetic waves can propagate as "surface waves" in that they can be guided along a refractive index gradient or along an interface between two media having different dielectric constants. In radio transmission, a ground wave is a guided wave that propagates close to the surface of the Earth. In seismology, several types of surface waves are encountered. Surface waves, in this mechanical sense, are known as either Love waves or Rayleigh waves. A seismic wave is a wave that travels through the Earth as the result of an earthquake or explosion. Love waves have transverse motion, whereas Rayleigh waves have both longitudinal and transverse motion.
Seismic waves are measured by a seismograph or seismometer. Surface waves span a wide frequency range, the period of waves that are most damaging is 10 seconds or longer. Surface waves can travel around the globe many times from the largest earthquakes. Surface waves are caused when P S waves come to the surface. Examples are the waves at the surface of air. Another example is internal waves, which can be transmitted along the interface of two water masses of different densities. In theory of hearing physiology, the traveling wave of Von Bekesy, resulted from an acoustic surface wave of the basilar membrane into the cochlear duct, his theory purported to explain every feature of the auditory sensation owing to these passive mechanical phenomena. Jozef Zwislocki, David Kemp, showed that, unrealistic and that active feedback is necessary. Ground wave refers to the propagation of radio waves parallel to and adjacent to the surface of the Earth, following the curvature of the Earth; this radiative ground wave is known as the Norton surface wave.
Other types of surface wave are the non-radiative Zenneck surface wave or Zenneck–Sommerfeld surface wave, the trapped surface wave and the gliding wave. See Dyakonov surface waves propagating at the interface of transparent materials with different symmetry. Lower frequency radio waves, below 3 MHz, travel efficiently as ground waves. In ITU nomenclature, this includes: medium frequency, low frequency low frequency, ultra low frequency, super low frequency low frequency waves. Ground propagation works because lower-frequency waves are more diffracted around obstacles due to their long wavelengths, allowing them to follow the Earth's curvature; the Earth has one refractive index and the atmosphere has another, thus constituting an interface that supports the guided wave's transmission. Ground waves propagate in vertical polarization, with their magnetic field horizontal and electric field vertical. With VLF waves, the ionosphere and earth's surface act as a waveguide. Conductivity of the surface affects the propagation of ground waves, with more conductive surfaces such as sea water providing better propagation.
Increasing the conductivity in a surface results in less dissipation. The refractive indices are subject to temporal changes. Since the ground is not a perfect electrical conductor, ground waves are attenuated as they follow the earth’s surface; the wavefronts are vertical, but the ground, acting as a lossy dielectric, causes the wave to tilt forward as it travels. This directs some of the energy into the earth where it is dissipated, so that the signal decreases exponentially. Most long-distance LF. Mediumwave radio transmissions, including AM broadcast band, travel both as groundwaves and, for longer distances at night, as skywaves. Ground losses become lower at lower frequencies increasing the coverage of AM stations using the lower end of the band; the VLF and LF frequencies are used for military communications with ships and submarines. The lower the frequency the better the waves penetrate sea water. ELF waves have been used to communicate with submerged submarines. Ground waves have been used in over-the-horizon radar, which operates at frequencies between 2–20 MHz over the sea, which has a sufficiently high conductivity to convey them to and from a reasonable distance.
In the development of radio, ground waves were used extensively. Early commercial and professional radio services relied on long wave, low frequencies and ground-wave propagation. To prevent interference with these services and experimental transmitters were restricted to the high frequencies, felt to be useless since their ground-wave range was limited. Upon discovery of the other propagation modes possible at medium wave and short wave frequencies, the advantages of HF for commercial and military purposes became apparent. Amateur experimentation was confined only to authorized frequencies in the range. Mediumwave and shortwave reflect off the ionosphere at night, known as skywave. During daylight hours, the lower D layer of the ionosphere absorbs lower frequency energy; this prevents skywave propaga
Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle. In classical physics, the diffraction phenomenon is described as the interference of waves according to the Huygens–Fresnel principle that treats each point in the wave-front as a collection of individual spherical wavelets; these characteristic behaviors are exhibited when a wave encounters an obstacle or a slit, comparable in size to its wavelength. Similar effects occur when a light wave travels through a medium with a varying refractive index, or when a sound wave travels through a medium with varying acoustic impedance. Diffraction has an impact on the acoustic space. Diffraction occurs with all waves, including sound waves, water waves, electromagnetic waves such as visible light, X-rays and radio waves. Since physical objects have wave-like properties, diffraction occurs with matter and can be studied according to the principles of quantum mechanics.
Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660. While diffraction occurs whenever propagating waves encounter such changes, its effects are most pronounced for waves whose wavelength is comparable to the dimensions of the diffracting object or slit. If the obstructing object provides multiple spaced openings, a complex pattern of varying intensity can result; this is due to the addition, or interference, of different parts of a wave that travel to the observer by different paths, where different path lengths result in different phases. The formalism of diffraction can describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer can all be analyzed using diffraction equations; the effects of diffraction are seen in everyday life. The most striking examples of diffraction are those.
This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired. Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges; the speckle pattern, observed when laser light falls on an optically rough surface is a diffraction phenomenon. When deli meat appears to be iridescent, diffraction off the meat fibers. All these effects are a consequence of the fact. Diffraction can occur with any kind of wave. Ocean waves diffract around other obstacles. Sound waves can diffract around objects, why one can still hear someone calling when hiding behind a tree. Diffraction can be a concern in some technical applications; the effects of diffraction of light were first observed and characterized by Francesco Maria Grimaldi, who coined the term diffraction, from the Latin diffringere,'to break into pieces', referring to light breaking up into different directions.
The results of Grimaldi's observations were published posthumously in 1665. Isaac Newton attributed them to inflexion of light rays. James Gregory observed the diffraction patterns caused by a bird feather, the first diffraction grating to be discovered. Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two spaced slits. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, made public in 1815 and 1818, thereby gave great support to the wave theory of light, advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory. In traditional classical physics diffraction arises because of the way; the propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a point source for a secondary spherical wave. The wave displacement at any subsequent point is the sum of these secondary waves.
When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns have a series of maxima and minima. In the modern quantum mechanical understanding of light propagation through a slit every photon has what is known as a wavefunction which describes its path from the emitter through the slit to the screen; the wavefunction is determined by the physical surroundings such as slit geometry, screen distance and initial conditions when the photon is created. In important experiments the existence of the photon's wavef
Figure of the Earth
Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model; the sphere is an approximation of the figure of the Earth, satisfactory for many purposes. Several models with greater accuracy have been developed so that coordinate systems can serve the precise needs of navigation, cadastre, land use, various other concerns. Earth's topographic surface is apparent with its variety of land forms and water areas; this topographic surface is the concern of topographers and geophysicists. While it is the surface on which Earth measurements are made, mathematically modeling it while taking the irregularities into account would be complicated; the Pythagorean concept of a spherical Earth offers a simple surface, easy to deal with mathematically. Many astronomical and navigational computations use a sphere to model the Earth as a close approximation. However, a more accurate figure is needed for measuring distances and areas on the scale beyond the purely local.
Better approximations can be had by modeling the entire surface as an oblate spheroid, using spherical harmonics to approximate the geoid, or modeling a region with a best-fit reference ellipsoids. For surveys of small areas, a planar model of Earth's surface suffices because the local topography overwhelms the curvature. Plane-table surveys are made for small areas without considering the size and shape of the entire Earth. A survey of a city, for example, might be conducted this way. By the late 1600s, serious effort was devoted to modeling the Earth as an ellipsoid, beginning with Jean Picard's measurement of a degree of arc along the Paris meridian. Improved maps and better measurement of distances and areas of national territories motivated these early attempts. Surveying instrumentation and techniques improved over the ensuing centuries. Models for the figure of the earth improved in step. In the mid- to late 20th century, research across the geosciences contributed to drastic improvements in the accuracy of the figure of the Earth.
The primary utility of this improved accuracy was to provide geographical and gravitational data for the inertial guidance systems of ballistic missiles. This funding drove the expansion of geoscientific disciplines, fostering the creation and growth of various geoscience departments at many universities; these developments benefited many civilian pursuits as well, such as weather and communication satellite control and GPS location-finding, which would be impossible without accurate models for the figure of the Earth. The models for the figure of the Earth vary in the way they are used, in their complexity, in the accuracy with which they represent the size and shape of the Earth; the simplest model for the shape of the entire Earth is a sphere. The Earth's radius is the distance from Earth's center to about 6,371 kilometers. While "radius" is a characteristic of perfect spheres, the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts and justifying the term "the radius of the Earth".
The concept of a spherical Earth dates back to around the 6th century BC, but remained a matter of philosophical speculation until the 3rd century BC. The first scientific estimation of the radius of the Earth was given by Eratosthenes about 240 BC, with estimates of the accuracy of Eratosthenes’s measurement ranging from 2% to 15%; the Earth is only spherical, so no single value serves as its natural radius. Distances from points on the surface to the center range from 6,353 km to 6,384 km. Several different ways of modeling the Earth as a sphere each yield a mean radius of 6,371 kilometers. Regardless of the model, any radius falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km; the difference 21 kilometers correspond to the polar radius being 0.3% shorter than the equator radius. Since the Earth is flattened at the poles and bulges at the Equator, geodesy represents the figure of the Earth as an oblate spheroid; the oblate spheroid, or oblate ellipsoid, is an ellipsoid of revolution obtained by rotating an ellipse about its shorter axis.
It is the regular geometric shape. A spheroid describing the figure of the Earth or other celestial body is called a reference ellipsoid; the reference ellipsoid for Earth is called an Earth ellipsoid. An ellipsoid of revolution is uniquely defined by two quantities. Several conventions for expressing the two quantities are used in geodesy, but they are all equivalent to and convertible with each other: Equatorial radius a, polar radius b. Eccentricity and flattening are different ways of expressing; when flattening appears as one of the defining quantities in geodesy it is expressed by its reciprocal. For example, in the WGS 84 spheroid used by today's GPS systems, the reciprocal of the flattening 1 / f is set to be 298.257223563. The difference between a sphere and a reference ellipsoid for Earth is small, only about one part in 300. Flattening was computed from grade measurements. Nowadays, geodetic networks and satellite geodesy are used. In practice, many reference ellip