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56-592: The horizon is the line at which the sky and the Earth's surface appear to meet. Horizon or The Horizon may also refer to: Horizon The true horizon is a theoretical line, which can only be observed to any degree of accuracy when it lies along a relatively smooth surface such as that of Earth's oceans . At many locations, this line is obscured by terrain , and on Earth it can also be obscured by life forms such as trees and/or human constructs such as buildings. The resulting intersection of such obstructions with

112-489: A parallel of latitude is given by p = N cos ⁡ ( φ ) {\displaystyle p=N\cos(\varphi )} . The Earth's meridional radius of curvature at the equator equals the meridian's semi-latus rectum : The Earth's prime-vertical radius of curvature at the equator equals the equatorial radius, N e = a . The Earth's polar radius of curvature (either meridional or prime-vertical) is: The principal curvatures are

168-497: A boat ( h B =1.7   m) can just see the tops of trees on a nearby shore ( h L =10   m), the trees are probably about D BL =16 km away. Referring to the figure at the right, and using the approximation above , the top of the lighthouse will be visible to a lookout in a crow's nest at the top of a mast of the boat if where D BL is in kilometres and h B and h L are in metres. As another example, suppose an observer, whose eyes are two metres above

224-447: A distant object is visible above the horizon. Suppose an observer's eye is 10 metres above sea level, and he is watching a ship that is 20 km away. His horizon is: kilometres from him, which comes to about 11.3 kilometres away. The ship is a further 8.7 km away. The height of a point on the ship that is just visible to the observer is given by: which comes to almost exactly six metres. The observer can therefore see that part of

280-403: A more precise value for its polar radius is needed. The geocentric radius is the distance from the Earth's center to a point on the spheroid surface at geodetic latitude φ , given by the formula: where a and b are, respectively, the equatorial radius and the polar radius. The extrema geocentric radii on the ellipsoid coincide with the equatorial and polar radii. They are vertices of

336-448: A perfect sphere. Local topography increases the variance, resulting in a surface of profound complexity. Our descriptions of Earth's surface must be simpler than reality in order to be tractable. Hence, we create models to approximate characteristics of Earth's surface, generally relying on the simplest model that suits the need. Each of the models in common use involve some notion of the geometric radius . Strictly speaking, spheres are

392-448: A planet causes it to approximate an oblate ellipsoid /spheroid with a bulge at the equator and flattening at the North and South Poles , so that the equatorial radius a is larger than the polar radius b by approximately aq . The oblateness constant q is given by where ω is the angular frequency , G is the gravitational constant , and M is the mass of the planet. For

448-416: A right triangle, with the sum of the radius and the height as the hypotenuse. With referring to the second figure at the right leads to the following: The exact formula above can be expanded as: where R is the radius of the Earth ( R and h must be in the same units). For example, if a satellite is at a height of 2000 km, the distance to the horizon is 5,430 kilometres (3,370 mi); neglecting

504-466: A sphere in many ways. This section describes the common ways. The various radii derived here use the notation and dimensions noted above for the Earth as derived from the WGS-84 ellipsoid; namely, A sphere being a gross approximation of the spheroid, which itself is an approximation of the geoid, units are given here in kilometers rather than the millimeter resolution appropriate for geodesy. In geophysics,

560-543: A sphere. The word horizon derives from the Greek ὁρίζων κύκλος ( horízōn kýklos ) 'separating circle', where ὁρίζων is from the verb ὁρίζω ( horízō ) 'to divide, to separate', which in turn derives from ὅρος ( hóros ) 'boundary, landmark'. Historically, the distance to the visible horizon has long been vital to survival and successful navigation, especially at sea, because it determined an observer's maximum range of vision and thus of communication , with all

616-423: Is more directly comparable to the geographical distance on a map. It can be formulated in terms of γ in radians , then Solving for s gives The distance s can also be expressed in terms of the line-of-sight distance d ; from the second figure at the right, substituting for γ and rearranging gives The distances d and s are nearly the same when the height of the object is negligible compared to

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672-555: Is possible to combine the principal radii of curvature above in a non-directional manner. The Earth's Gaussian radius of curvature at latitude φ is: Where K is the Gaussian curvature , K = κ 1 κ 2 = det B det A {\displaystyle K=\kappa _{1}\,\kappa _{2}={\frac {\det \,B}{\det \,A}}} . The Earth's mean radius of curvature at latitude φ is: The Earth can be modeled as

728-571: Is sometimes used as a unit of measurement in astronomy and geophysics , a conversion factor used when expressing planetary properties as multiples or fractions of a constant terrestrial radius; if the choice between equatorial or polar radii is not explicit, the equatorial radius is to be assumed, as recommended by the International Astronomical Union (IAU). Earth's rotation , internal density variations, and external tidal forces cause its shape to deviate systematically from

784-403: Is the angular dip of the horizon. It is related to the horizon zenith angle z {\displaystyle z} by: For a non-negative height h {\displaystyle h} , the angle z {\displaystyle z} is always ≥ 90°. To compute the greatest distance D BL at which an observer B can see the top of an object L above the horizon, simply add

840-581: Is the radius of a sphere having the same volume as the ellipsoid ( R 3 ). All three values are about 6,371 kilometres (3,959 mi). Other ways to define and measure the Earth's radius involve either the spheroid's radius of curvature or the actual topography . A few definitions yield values outside the range between the polar radius and equatorial radius because they account for localized effects. A nominal Earth radius (denoted R E N {\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }} )

896-428: Is usually considered to be 6,371 kilometres (3,959 mi) with a 0.3% variability (±10 km) for the following reasons. The International Union of Geodesy and Geophysics (IUGG) provides three reference values: the mean radius ( R 1 ) of three radii measured at two equator points and a pole; the authalic radius , which is the radius of a sphere with the same surface area ( R 2 ); and the volumetric radius , which

952-681: The Global Positioning System gained importance, true global models were developed which, while not as accurate for regional work, best approximate the Earth as a whole. The following radii are derived from the World Geodetic System 1984 ( WGS-84 ) reference ellipsoid . It is an idealized surface, and the Earth measurements used to calculate it have an uncertainty of ±2 m in both the equatorial and polar dimensions. Additional discrepancies caused by topographical variation at specific locations can be significant. When identifying

1008-616: The second fundamental form for a surface (Equation (123) in ): e, f, and g are elements of the shape tensor: n = N | N | {\displaystyle n={\frac {N}{|N|}}} is the unit normal to the surface at r {\displaystyle r} , and because ∂ r ∂ φ {\displaystyle {\frac {\partial r}{\partial \varphi }}} and ∂ r ∂ λ {\displaystyle {\frac {\partial r}{\partial \lambda }}} are tangents to

1064-417: The Earth ⁠ 1 / q ⁠ ≈ 289 , which is close to the measured inverse flattening ⁠ 1 / f ⁠ ≈ 298.257 . Additionally, the bulge at the equator shows slow variations. The bulge had been decreasing, but since 1998 the bulge has increased, possibly due to redistribution of ocean mass via currents. The variation in density and crustal thickness causes gravity to vary across

1120-402: The Earth at that point" . It is also common to refer to any mean radius of a spherical model as "the radius of the earth" . When considering the Earth's real surface, on the other hand, it is uncommon to refer to a "radius", since there is generally no practical need. Rather, elevation above or below sea level is useful. Regardless of the model, any of these geocentric radii falls between

1176-484: The Earth's surface for hundreds of kilometres. Opposite conditions occur, for example, in deserts, where the surface is very hot, so hot, low-density air is below cooler air. This causes light to be refracted upward, causing mirage effects that make the concept of the horizon somewhat meaningless. Calculated values for the effects of refraction under unusual conditions are therefore only approximate. Nevertheless, attempts have been made to calculate them more accurately than

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1232-588: The Earth's surface is about where h is height above sea level and R is the Earth radius . The expression can be simplified as: where the constant equals k = 3.57 km/m = 1.22 mi/ft . In this equation, Earth's surface is assumed to be perfectly spherical, with R equal to about 6,371 kilometres (3,959 mi). Assuming no atmospheric refraction and a spherical Earth with radius R=6,371 kilometres (3,959 mi): On terrestrial planets and other solid celestial bodies with negligible atmospheric effects,

1288-553: The Earth. Gravitational attraction from the Moon or Sun can cause the Earth's surface at a given point to vary by tenths of a meter over a nearly 12-hour period (see Earth tide ). Given local and transient influences on surface height, the values defined below are based on a "general purpose" model, refined as globally precisely as possible within 5 m (16 ft) of reference ellipsoid height, and to within 100 m (330 ft) of mean sea level (neglecting geoid height). Additionally,

1344-433: The air above it, a cold, dense layer of air forms close to the surface, causing light to be refracted downward as it travels, and therefore, to some extent, to go around the curvature of the Earth. The reverse happens if the ground is hotter than the air above it, as often happens in deserts, producing mirages . As an approximate compensation for refraction, surveyors measuring distances longer than 100 meters subtract 14% from

1400-408: The aircraft. Pilots can also retain their spatial orientation by referring to the horizon. In many contexts, especially perspective drawing , the curvature of the Earth is disregarded and the horizon is considered the theoretical line to which points on any horizontal plane converge (when projected onto the picture plane) as their distance from the observer increases. For observers near sea level,

1456-465: The appearance of the horizon. Usually, the density of the air just above the surface of the Earth is greater than its density at greater altitudes. This makes its refractive index greater near the surface than at higher altitudes, which causes light that is travelling roughly horizontally to be refracted downward. This makes the actual distance to the horizon greater than the distance calculated with geometrical formulas. With standard atmospheric conditions,

1512-444: The calculated curvature error and ensure lines of sight are at least 1.5 metres from the ground, to reduce random errors created by refraction. If the Earth were an airless world like the Moon, the above calculations would be accurate. However, Earth has an atmosphere of air , whose density and refractive index vary considerably depending on the temperature and pressure. This makes the air refract light to varying extents, affecting

1568-477: The center of the true horizon is below the observer and below sea level . Its radius or horizontal distance from the observer varies slightly from day to day due to atmospheric refraction , which is greatly affected by weather conditions. Also, the higher the observer's eyes are from sea level, the farther away the horizon is from the observer. For instance, in standard atmospheric conditions , for an observer with eye level above sea level by 1.8 metres (6 ft),

1624-422: The difference between this geometrical horizon (which assumes a perfectly flat, infinite ground plane) and the true horizon (which assumes a spherical Earth surface) is imperceptible to the unaided eye. However, for someone on a 1,000 m (3,300 ft) hill looking out across the sea, the true horizon will be about a degree below a horizontal line. In astronomy, the horizon is the horizontal plane through

1680-420: The difference is about 8%. This changes the factor of 3.57, in the metric formulas used above, to about 3.86. For instance, if an observer is standing on seashore, with eyes 1.70 m above sea level, according to the simple geometrical formulas given above the horizon should be 4.7 km away. Actually, atmospheric refraction allows the observer to see 300 metres farther, moving the true horizon 5 km away from

1736-502: The distance to the horizon can easily be calculated. The tangent-secant theorem states that Make the following substitutions: with d, D, and h all measured in the same units. The formula now becomes or where R is the radius of the Earth. The same equation can also be derived using the Pythagorean theorem . At the horizon, the line of sight is a tangent to the Earth and is also perpendicular to Earth's radius. This sets up

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1792-464: The distance to the horizon for a "standard observer" varies as the square root of the planet's radius. Thus, the horizon on Mercury is 62% as far away from the observer as it is on Earth, on Mars the figure is 73%, on the Moon the figure is 52%, on Mimas the figure is 18%, and so on. If the Earth is assumed to be a featureless sphere (rather than an oblate spheroid ) with no atmospheric refraction, then

1848-406: The distance to the horizon is If d is in nautical miles , and h in feet, the constant factor is about 1.06, which is close enough to 1 that it is often ignored, giving: These formulas may be used when h is much smaller than the radius of the Earth (6371 km or 3959 mi), including all views from any mountaintops, airplanes, or high-altitude balloons. With the constants as given, both

1904-440: The distances to the horizon from each of the two points: For example, for an observer B with a height of h B =1.70 m standing on the ground, the horizon is D B =4.65 km away. For a tower with a height of h L =100 m, the horizon distance is D L =35.7 km. Thus an observer on a beach can see the top of the tower as long as it is not more than D BL =40.35 km away. Conversely, if an observer on

1960-496: The ellipse and also coincide with minimum and maximum radius of curvature. There are two principal radii of curvature : along the meridional and prime-vertical normal sections . In particular, the Earth's meridional radius of curvature (in the north–south direction) at φ is: where e {\displaystyle e} is the eccentricity of the earth. This is the radius that Eratosthenes measured in his arc measurement . If one point had appeared due east of

2016-459: The equator is slightly shorter in the north–south direction than in the east–west direction. In summary, local variations in terrain prevent defining a single "precise" radius. One can only adopt an idealized model. Since the estimate by Eratosthenes , many models have been created. Historically, these models were based on regional topography, giving the best reference ellipsoid for the area under survey. As satellite remote sensing and especially

2072-418: The eyes of the observer. It is the fundamental plane of the horizontal coordinate system , the locus of points that have an altitude of zero degrees. While similar in ways to the geometrical horizon, in this context a horizon may be considered to be a plane in space, rather than a line on a picture plane. Ignoring the effect of atmospheric refraction , distance to the true horizon from an observer close to

2128-463: The horizon is at a distance of about 4.8 kilometres (3 mi). When observed from very high standpoints, such as a space station , the horizon is much farther away and it encompasses a much larger area of Earth's surface. In this case, the horizon would no longer be a perfect circle, not even a plane curve such as an ellipse, especially when the observer is above the equator, as the Earth's surface can be better modeled as an oblate ellipsoid than as

2184-443: The level ground, uses binoculars to look at a distant building which he knows to consist of thirty storeys , each 3.5 metres high. He counts the stories he can see and finds there are only ten. So twenty stories or 70 metres of the building are hidden from him by the curvature of the Earth. From this, he can calculate his distance from the building: which comes to about 35 kilometres. It is similarly possible to calculate how much of

2240-403: The metric and imperial formulas are precise to within 1% (see the next section for how to obtain greater precision). If h is significant with respect to R , as with most satellites, then the approximation is no longer valid, and the exact formula is required. Another relationship involves the great-circle distance s along the arc over the curved surface of the Earth to the horizon; this

2296-439: The observer. This correction can be, and often is, applied as a fairly good approximation when atmospheric conditions are close to standard . When conditions are unusual, this approximation fails. Refraction is strongly affected by temperature gradients, which can vary considerably from day to day, especially over water. In extreme cases, usually in springtime, when warm air overlies cold water, refraction can allow light to follow

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2352-420: The obvious consequences for safety and the transmission of information that this range implied. This importance lessened with the development of the radio and the telegraph , but even today, when flying an aircraft under visual flight rules , a technique called attitude flying is used to control the aircraft, where the pilot uses the visual relationship between the aircraft's nose and the horizon to control

2408-412: The only solids to have radii, but broader uses of the term radius are common in many fields, including those dealing with models of Earth. The following is a partial list of models of Earth's surface, ordered from exact to more approximate: In the case of the geoid and ellipsoids, the fixed distance from any point on the model to the specified center is called "a radius of the Earth" or "the radius of

2464-400: The other, one finds the approximate curvature in the east–west direction. This Earth's prime-vertical radius of curvature , also called the Earth's transverse radius of curvature , is defined perpendicular ( orthogonal ) to M at geodetic latitude φ and is: N can also be interpreted geometrically as the normal distance from the ellipsoid surface to the polar axis. The radius of

2520-399: The polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km (3,950 to 3,963 mi). Hence, the Earth deviates from a perfect sphere by only a third of a percent, which supports the spherical model in most contexts and justifies the term "radius of the Earth". While specific values differ, the concepts in this article generalize to any major planet . Rotation of

2576-452: The position of an observable location, the use of more precise values for WGS-84 radii may not yield a corresponding improvement in accuracy . The value for the equatorial radius is defined to the nearest 0.1 m in WGS-84. The value for the polar radius in this section has been rounded to the nearest 0.1 m, which is expected to be adequate for most uses. Refer to the WGS-84 ellipsoid if

2632-406: The radius (that is, h  ≪  R ). When the observer is elevated, the horizon zenith angle can be greater than 90°. The maximum visible zenith angle occurs when the ray is tangent to Earth's surface; from triangle OCG in the figure at right, where h {\displaystyle h} is the observer's height above the surface and γ {\displaystyle \gamma }

2688-401: The radius can be estimated from the curvature of the Earth at a point. Like a torus , the curvature at a point will be greatest (tightest) in one direction (north–south on Earth) and smallest (flattest) perpendicularly (east–west). The corresponding radius of curvature depends on the location and direction of measurement from that point. A consequence is that a distance to the true horizon at

2744-581: The roots of Equation (125) in: where in the first fundamental form for a surface (Equation (112) in ): E, F, and G are elements of the metric tensor : r = [ r 1 , r 2 , r 3 ] T = [ x , y , z ] T {\displaystyle r=[r^{1},r^{2},r^{3}]^{T}=[x,y,z]^{T}} , w 1 = φ {\displaystyle w^{1}=\varphi } , w 2 = λ , {\displaystyle w^{2}=\lambda ,} in

2800-401: The second fundamental form gives the distance from r + d r {\displaystyle r+dr} to the plane tangent at r {\displaystyle r} . The Earth's azimuthal radius of curvature , along an Earth normal section at an azimuth (measured clockwise from north) α and at latitude φ , is derived from Euler's curvature formula as follows: It

2856-490: The second term in parentheses would give a distance of 5,048 kilometres (3,137 mi), a 7% error. If the observer is close to the surface of the Earth, then it is valid to disregard h in the term (2 R + h ) , and the formula becomes- Using kilometres for d and R , and metres for h , and taking the radius of the Earth as 6371 km, the distance to the horizon is Using imperial units , with d and R in statute miles (as commonly used on land), and h in feet,

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2912-407: The ship that is more than six metres above the level of the water. The part of the ship that is below this height is hidden from him by the curvature of the Earth. In this situation, the ship is said to be hull-down . Due to atmospheric refraction the distance to the visible horizon is further than the distance based on a simple geometric calculation. If the ground (or water) surface is colder than

2968-497: The simple approximation described above. Earth radius Earth radius (denoted as R 🜨 or R E ) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid (an oblate ellipsoid ), the radius ranges from a maximum ( equatorial radius , denoted a ) of nearly 6,378 km (3,963 mi) to a minimum ( polar radius , denoted b ) of nearly 6,357 km (3,950 mi). A globally-average value

3024-404: The sky is called the visible horizon . On Earth, when looking at a sea from a shore, the part of the sea closest to the horizon is called the offing . The true horizon surrounds the observer and it is typically assumed to be a circle, drawn on the surface of a perfectly spherical model of the relevant celestial body, i.e., a small circle of the local osculating sphere . With respect to Earth,

3080-505: The surface and in time, so that the mean sea level differs from the ellipsoid. This difference is the geoid height , positive above or outside the ellipsoid, negative below or inside. The geoid height variation is under 110 m (360 ft) on Earth. The geoid height can change abruptly due to earthquakes (such as the Sumatra-Andaman earthquake ) or reduction in ice masses (such as Greenland ). Not all deformations originate within

3136-402: The surface, is normal to the surface at r {\displaystyle r} . With F = f = 0 {\displaystyle F=f=0} for an oblate spheroid, the curvatures are and the principal radii of curvature are The first and second radii of curvature correspond, respectively, to the Earth's meridional and prime-vertical radii of curvature. Geometrically,

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