text:
Gravity of Earth
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Earth's gravity measured by NASA GRACE mission, showing deviations from the theoretical gravity of an idealized smooth Earth, the so-called earth ellipsoid. Red shows the areas where gravity is stronger than the smooth, standard value, and blue reveals areas where gravity is weaker. (Animated version.)[1]
The gravity of Earth, which is denoted by g, refers to the acceleration that is imparted to objects due to the distribution of mass within the Earth. In SI units this acceleration is measured in metres per second squared (in symbols, m/s2 or m·s−2) or equivalently in newtons per kilogram (N/kg or N·kg−1). Near the Earth's surface, gravitational acceleration is approximately 9.8 m/s2, which means that, ignoring the effects of air resistance, the speed of an object falling freely will increase by about 9.8 metres (32 ft) per second every second. This quantity is sometimes referred to informally as little g (in contrast, the gravitational constant G is referred to as big G).
The precise strength of Earth's gravity varies depending on location. The nominal "average" value at the Earth's surface, known as standard gravity is, by definition,[2] 9.80665 m/s2 (about 32.1740 ft/s2). This quantity is denoted variously as gn, ge (though this sometimes means the normal equatorial value on Earth, 9.78033 m/s2), g0, gee, or simply g (which is also used for the variable local value). The weight of an object on the Earth's surface is the downwards force on that object, given by Newton's second law of motion, or F = ma (force = mass × acceleration). Gravitational acceleration contributes to the total acceleration, but other factors, such as the rotation of the Earth, also contribute, and, therefore, affect the weight of the object.
Contents
[hide]
1 Variation in gravity and apparent gravity
1.1 Legal definition
1.2 Latitude
1.3 Altitude
1.4 Depth
1.5 Local topography and geology
1.6 Other factors
1.7 Comparative gravities in various cities around the world
1.8 Mathematical models
1.8.1 Latitude model
1.8.2 Free air correction
1.8.3 Slab correction
2 Estimating g from the law of universal gravitation
3 See also
4 References
5 External links
Variation in gravity and apparent gravity[edit]
A perfect sphere of uniform mass density, or whose density varies solely with distance from the centre (spherical symmetry), would produce a gravitational field of uniform magnitude at all points on its surface, always pointing directly towards the sphere's centre. The Earth is not spherically symmetric, but is slightly flatter at the poles while bulging at the Equator: an oblate spheroid. There are consequently slight deviations in both the magnitude and direction of gravity across its surface. The net force (or corresponding net acceleration) as measured by a scale and plumb bob is called "effective gravity" or "apparent gravity". Effective gravity includes other factors that affect the net force. These factors vary and include things such as centrifugal force[3] at the surface from the Earth's rotation and the gravitational pull of the Moon and Sun.
Effective gravity on the Earth's surface varies by around 0.7%, from 9.7639 m/s2 on the Nevado Huascarán mountain in Peru to 9.8337 m/s2 at the surface of the Arctic Ocean.[4] In large cities, it ranges from 9.7760[5] in Kuala Lumpur, Mexico City, and Singapore to 9.825 in Oslo and Helsinki.
Legal definition[edit]
In 1901 the third General Conference on Weights and Measures defined a standard gravitational acceleration for the surface of the earth: gn = 9,80665 m/s2. The larger variations described below were not yet known. Therefore this definition is not a value of any particular place or carefully worked out average, but an agreement for a value to use if a better actual local value is not known or not important.[6]
Latitude[edit]
The differences of Earth's gravity around the Antarctic continent.
The surface of the Earth is rotating, so it is not an inertial frame of reference. At latitudes nearer the Equator, the outward centrifugal force produced by Earth's rotation is larger than at polar latitudes. This counteracts the Earth's gravity to a small degree – up to a maximum of 0.3% at the Equator – and reduces the apparent downward acceleration of falling objects.
The second major reason for the difference in gravity at different latitudes is that the Earth's equatorial bulge (itself also caused by centrifugal force from rotation) causes objects at the Equator to be farther from the planet's centre than objects at the poles. Because the force due to gravitational attraction between two bodies (the Earth and the object being weighed) varies inversely with the square of the distance between them, an object at the Equator experiences a weaker gravitational pull than an object at the poles.
In combination, the equatorial bulge and the effects of the surface centrifugal force due to rotation mean that sea-level effective gravity increases from about 9.780 m/s2 at the Equator to about 9.832 m/s2 at the poles, so an object will weigh about 0.5% more at the poles than at the Equator.[3][7]
The same two factors influence the direction of the effective gravity (as determined by a plumb line or as the perpendicular to the surface of water in a container). Anywhere on Earth away from the Equator or poles, effective gravity points not exactly toward the centre of the Earth, but rather perpendicular to the surface of the geoid, which, due to the flattened shape of the Earth, is somewhat toward the opposite pole. About half of the deflection is due to centrifugal force, and half because the extra mass around the Equator causes a change in the direction of the true gravitational force relative to what it would be on a spherical Earth.
Altitude[edit]
The graph shows the variation in gravity relative to the height of an object
Gravity decreases with altitude as one rises above the Earth's surface because greater altitude means greater distance from the Earth's centre. All other things being equal, an increase in altitude from sea level to 9,000 metres (30,000 ft) causes a weight decrease of about 0.29%. (An additional factor affecting apparent weight is the decrease in air density at altitude, which lessens an object's buoyancy.[8] This would increase a person's apparent weight at an altitude of 9,000 metres by about 0.08%)
It is a common misconception that astronauts in orbit are weightless because they have flown high enough to escape the Earth's gravity. In fact, at an altitude of 400 kilometres (250 mi), equivalent to a typical orbit of the Space Shuttle, gravity is still nearly 90% as strong as at the Earth's surface. Weightlessness actually occurs because orbiting objects are in free-fall.[9]
The effect of ground elevation depends on the density of the ground (see Slab correction section). A person flying at 30 000 ft above sea level over mountains will feel more gravity than someone at the same elevation but over the sea. However, a person standing on the earth's surface feels less gravity when the elevation is higher.
The following formula approximates the Earth's gravity variation with altitude:
g h = g 0 ( r e r e + h ) 2 {\displaystyle g_{h}=g_{0}\left({\frac {r_{\mathrm {e} }}{r_{\mathrm {e} }+h}}\right)^{2}}
Where
gh is the gravitational acceleration at height h above sea level.
re is the Earth's mean radius.
g0 is the standard gravitational acceleration.
The formula treats the Earth as a perfect sphere with a radially symmetric distribution of mass; a more accurate mathematical treatment is discussed below.
Depth[edit]
See also: Shell theorem
An approximate value for gravity at a distance r from the centre of the Earth can be obtained by assuming that the Earth's density is spherically symmetric. The gravity depends only on the mass inside the sphere of radius r. All the contributions from outside cancel out as a consequence of the inverse-square law of gravitation. Another consequence is that the gravity is the same as if all the mass were concentrated at the centre. Thus, the gravitational acceleration at this radius is[10]
g ( r ) = − G M ( r ) r 2 . {\displaystyle g(r)=-{\frac {GM(r)}{r^{2}}}.}
where G is the gravitational constant and M(r) is the total mass enclosed within radius r. If the Earth had a constant density ρ, the mass would be M(r) = (4/3)πρr3 and the dependence of gravity on depth would be
g ( r ) = 4 π 3 G ρ r . {\displaystyle g(r)={\frac {4\pi }{3}}G\rho r.}
g at depth d is given by g'=g(1-d/R) where g is acceleration due to gravity on surface of the earth, d is depth and R is radius of Earth. If the density decreased linearly with increasing radius from a density ρ0 at the centre to ρ1 at the surface, then ρ(r) = ρ0 − (ρ0 − ρ1) r / re, and the dependence would be
g ( r ) = 4 π 3 G ρ 0 r − π G ( ρ 0 − ρ 1 ) r 2 r e . {\displaystyle g(r)={\frac {4\pi }{3}}G\rho _{0}r-\pi G\left(\rho _{0}-\rho _{1}\right){\frac {r^{2}}{r_{\mathrm {e} }}}.}
The actual depth dependence of density and gravity, inferred from seismic travel times (see Adams–Williamson equation), is shown in the graphs below.
Earth's radial density distribution according to the Preliminary Reference Earth Model (PREM).[11]
Earth's gravity according to the Preliminary Reference Earth Model (PREM).[11] Two models for a spherically symmetric Earth are included for comparison. The dark green straight line is for a constant density equal to the Earth's average density. The light green curved line is for a density that decreases linearly from centre to surface. The density at the centre is the same as in the PREM, but the surface density is chosen so that the mass of the sphere equals the mass of the real Earth.
Local topography and geology[edit]
See also: Physical geodesy
Local differences in topography (such as the presence of mountains), geology (such as the density of rocks in the vicinity), and deeper tectonic structure cause local and regional differences in the Earth's gravitational field, known as gravitational anomalies.[12] Some of these anomalies can be very extensive, resulting in bulges in sea level, and throwing pendulum clocks out of synchronisation.
The study of these anomalies forms the basis of gravitational geophysics. The fluctuations are measured with highly sensitive gravimeters, the effect of topography and other known factors is subtracted, and from the resulting data conclusions are drawn. Such techniques are now used by prospectors to find oil and mineral deposits. Denser rocks (often containing mineral ores) cause higher than normal local gravitational fields on the Earth's surface. Less dense sedimentary rocks cause the opposite.
Other factors[edit]
In air, objects experience a supporting buoyancy force which reduces the apparent strength of gravity (as measured by an object's weight). The magnitude of the effect depends on air density (and hence air pressure); see Apparent weight for details.
The gravitational effects of the Moon and the Sun (also the cause of the tides) have a very small effect on the apparent strength of Earth's gravity, depending on their relative positions; typical variations are 2 µm/s2 (0.2 mGal) over the course of a day.
Comparative gravities in various cities around the world[edit]
Tools exist for calculating the strength of gravity at various cities around the world.[13] The effect of latitude can be clearly seen with gravity in high-latitude cities: Anchorage (9.826 m/s2), Helsinki (9.825 m/s2), being about 0.5% greater than that in cities near the equator: Kuala Lumpur (9.776 m/s2), Manila (9.780 m/s2). The effect of altitude can be seen in Mexico City (9.776 m/s2; altitude 2,240 metres (7,350 ft)), and by comparing Denver (9.798 m/s2; 1,616 metres (5,302 ft)) with Washington, D.C. (9.801 m/s2; 30 metres (98 ft)), both of which are near 39° N. Measured values can be obtained from Physical and Mathematical Tables by T.M. Yarwood and F. Castle, Macmillan, revised edition 1970.[14]
Mathematical models[edit]
Latitude model[edit]
Main article: Theoretical gravity
If the terrain is at sea level, we can estimate g { ϕ } {\displaystyle g\{\phi \}} , the acceleration at latitude ϕ {\displaystyle \phi } :
g { ϕ } = 9.780327 m ⋅ s − 2 ( 1 + 0.0053024 sin 2  ϕ − 0.0000058 sin 2  2 ϕ ) , = 9.780327 m ⋅ s − 2 ( 1 + 0.0052792 sin 2  ϕ + 0.0000232 sin 4  ϕ ) , = 9.780327 m ⋅ s − 2 ( 1.0053024 − 0.0053256 cos 2  ϕ + 0.0000232 cos 4  ϕ ) , = 9.780327 m ⋅ s − 2 ( 1.0026454 − 0.0026512 cos  2 ϕ + 0.0000058 cos 2  2 ϕ ) {\displaystyle {\begin{aligned}g\{\phi \}&=9.780327\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}\,\,\left(1+0.0053024\,\sin ^{2}\phi -0.0000058\,\sin ^{2}2\phi \right),\\&=9.780327\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}\,\,\left(1+0.0052792\,\sin ^{2}\phi +0.0000232\,\sin ^{4}\phi \right),\\&=9.780327\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}\,\,\left(1.0053024-0.0053256\,\cos ^{2}\phi +0.0000232\,\cos ^{4}\phi \right),\\&=9.780327\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}\,\,\left(1.0026454-0.0026512\,\cos 2\phi +0.0000058\,\cos ^{2}2\phi \right)\end{aligned}}} .
This is the International Gravity Formula 1967, the 1967 Geodetic Reference System Formula, Helmert's equation or Clairaut's formula.[15]
An alternate formula for g as a function of latitude is the WGS (World Geodetic System) 84 Ellipsoidal Gravity Formula:[16]
g { ϕ } = G e [ 1 + k sin 2  ϕ 1 − e 2 sin 2  ϕ ] , {\displaystyle g\{\phi \}=\mathbb {G} _{e}\left[{\frac {1+k\sin ^{2}\phi }{\sqrt {1-e^{2}\sin ^{2}\phi }}}\right],\,\!}
where,
a , b {\displaystyle a,\,b} are the equatorial and polar semi-axes, respectively;
e 2 = 1 − ( b / a ) 2 {\displaystyle e^{2}=1-(b/a)^{2}} is the spheroid's eccentricity, squared;
G e , G p {\displaystyle \mathbb {G} _{e},\,\mathbb {G} _{p}\,} is the defined gravity at the equator and poles, respectively;
k = b G p − a G e a G e {\displaystyle k={\frac {b\,\mathbb {G} _{p}-a\,\mathbb {G} _{e}}{a\,\mathbb {G} _{e}}}} (formula constant);
then, where G p = 9.8321849378 m ⋅ s − 2 {\displaystyle \mathbb {G} _{p}=9.8321849378\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}} ,[16]
g { ϕ } = 9.7803253359 m ⋅ s − 2 [ 1 + 0.00193185265241 sin 2  ϕ 1 − 0.00669437999013 sin 2  ϕ ] {\displaystyle g\{\phi \}=9.7803253359\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}\left[{\frac {1+0.00193185265241\,\sin ^{2}\phi }{\sqrt {1-0.00669437999013\,\sin ^{2}\phi }}}\right]} .
The difference between the WGS-84 formula and Helmert's equation is less than 0.68 μm·s−2.
Free air correction[edit]
Main article: Free-air anomaly
The first correction to be applied to the model is the free air correction (FAC) that accounts for heights above sea level. Near the surface of the Earth (sea level), gravity decreases with height such that linear extrapolation would give zero gravity at a height of one half of the earth's radius - (9.8 m·s−2 per 3,200 km.)[17]
Using the mass and radius of the Earth:
r E a r t h = 6.371 ⋅ 10 6 m {\displaystyle r_{\mathrm {Earth} }=6.371\cdot 10^{6}\,\mathrm {m} }
m E a r t h = 5.9722 ⋅ 10 24 k g {\displaystyle m_{\mathrm {Earth} }=5.9722\cdot 10^{24}\,\mathrm {kg} }
The FAC correction factor (Δg) can be derived from the definition of the acceleration due to gravity in terms of G, the Gravitational Constant (see Estimating g from the law of universal gravitation, below):
g 0 = G m E a r t h / r E a r t h 2 = 9.8196 m s 2 {\displaystyle g_{0}=G\,m_{\mathrm {Earth} }/r_{\mathrm {Earth} }^{2}=9.8196\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}}
where:
G = 6.67384 ⋅ 10 − 11 m 3 k g ⋅ s 2 . {\displaystyle G=6.67384\cdot 10^{-11}\,{\frac {\mathrm {m} ^{3}}{\mathrm {kg} \cdot \mathrm {s} ^{2}}}.}
At a height h above the nominal surface of the earth gh is given by:
g h = G m E a r t h / ( r E a r t h + h ) 2 {\displaystyle g_{h}=G\,m_{\mathrm {Earth} }/\left(r_{\mathrm {Earth} }+h\right)^{2}}
So the FAC for a height h above the nominal earth radius can be expressed:
Δ g h = [ G m E a r t h / ( r E a r t h + h ) 2 ] − [ G m E a r t h / r E a r t h 2 ] {\displaystyle \Delta g_{h}=\left[G\,m_{\mathrm {Earth} }/\left(r_{\mathrm {Earth} }+h\right)^{2}\right]-\left[G\,m_{\mathrm {Earth} }/r_{\mathrm {Earth} }^{2}\right]}
This expression can be readily used for programming or inclusion in a spreadsheet. Collecting terms, simplifying and neglecting small terms (h<<rEarth), however yields the good approximation:
Δ g h ≈ − G m E a r t h r E a r t h 2 ⋅ 2 h r E a r t h {\displaystyle \Delta g_{h}\approx -\,{\dfrac {G\,m_{\mathrm {Earth} }}{r_{\mathrm {Earth} }^{2}}}\cdot {\dfrac {2\,h}{r_{\mathrm {Earth} }}}}
Using the numerical values above and for a height h in metres:
Δ g h ≈ − 3.086 ⋅ 10 − 6 h {\displaystyle \Delta g_{h}\approx -3.086\cdot 10^{-6}\,h}
Grouping the latitude and FAC altitude factors the expression most commonly found in the literature is:
g { ϕ , h } = g { ϕ } − 3.086 ⋅ 10 − 6 h {\displaystyle g\{\phi ,h\}=g\{\phi \}-3.086\cdot 10^{-6}h}
where g { ϕ , h } {\displaystyle g\{\phi ,h\}} = acceleration in m·s−2 at latitude ϕ {\displaystyle \ \phi } and altitude h in metres. Alternatively (with the same units for h) the expression can be grouped as follows:
g { ϕ , h } = g { ϕ } − 3.155 ⋅ 10 − 7 h m s 2 {\displaystyle g\{\phi ,h\}=g\{\phi \}-3.155\cdot 10^{-7}h\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}}
Slab correction[edit]
Main article: Bouguer anomaly
Note: The section uses the galileo (symbol: "Gal"), which is a cgs unit for acceleration of 1 centimetre/second2.
For flat terrain above sea level a second term is added for the gravity due to the extra mass; for this purpose the extra mass can be approximated by an infinite horizontal slab, and we get 2πG times the mass per unit area, i.e. 4.2×10−10 m3·s−2·kg−1 (0.042 μGal·kg−1·m2) (the Bouguer correction). For a mean rock density of 2.67 g·cm−3 this gives 1.1×10−6 s−2 (0.11 mGal·m−1). Combined with the free-air correction this means a reduction of gravity at the surface of ca. 2 µm·s−2 (0.20 mGal) for every metre of elevation of the terrain. (The two effects would cancel at a surface rock density of 4/3 times the average density of the whole earth. The density of the whole earth is 5.515 g·cm−3, so standing on a slab of something like iron whose density is over 7.35 g·cm−3 would increase one's weight.)
For the gravity below the surface we have to apply the free-air correction as well as a double Bouguer correction. With the infinite slab model this is because moving the point of observation below the slab changes the gravity due to it to its opposite. Alternatively, we can consider a spherically symmetrical Earth and subtract from the mass of the Earth that of the shell outside the point of observation, because that does not cause gravity inside. This gives the same result.
Estimating g from the law of universal gravitation[edit]
From the law of universal gravitation, the force on a body acted upon by Earth's gravity is given by
F = G m 1 m 2 r 2 = ( G m 1 r 2 ) m 2 {\displaystyle F=G\,{\frac {m_{1}m_{2}}{r^{2}}}=\left(G\,{\frac {m_{1}}{r^{2}}}\right)m_{2}}
where r is the distance between the centre of the Earth and the body (see below), and here we take m1 to be the mass of the Earth and m2 to be the mass of the body.
Additionally, Newton's second law, F = ma, where m is mass and a is acceleration, here tells us that
F = m 2 g {\displaystyle F=m_{2}\,g\,}
Comparing the two formulas it is seen that:
g = G m 1 r 2 {\displaystyle g=G\,{\frac {m_{1}}{r^{2}}}}
So, to find the acceleration due to gravity at sea level, substitute the values of the gravitational constant, G, the Earth's mass (in kilograms), m1, and the Earth's radius (in metres), r, to obtain the value of g:
g = G m 1 r 2 = 6.67384 ⋅ 10 − 11 m 3 ⋅ k g − 1 ⋅ s − 2 5.9722 ⋅ 10 24 k g ( 6.371 ⋅ 10 6 m ) 2 = 9.8196 m ⋅ s − 2 {\displaystyle g=G\,{\frac {m_{1}}{r^{2}}}=6.67384\cdot 10^{-11}\,\mathrm {m} ^{3}\cdot \mathrm {kg} ^{-1}\cdot \mathrm {s} ^{-2}\,\,\,{\frac {5.9722\cdot 10^{24}\,\mathrm {kg} }{(6.371\cdot 10^{6}\,\mathrm {m} )^{2}}}=9.8196\,\,{\mbox{m}}\cdot {\mbox{s}}^{-2}}
Note that this formula only works because of the mathematical fact that the gravity of a uniform spherical body, as measured on or above its surface, is the same as if all its mass were concentrated at a point at its centre. This is what allows us to use the Earth's radius for r.
The value obtained agrees approximately with the measured value of g. The difference may be attributed to several factors, mentioned above under "Variations":
The Earth is not homogeneous
The Earth is not a perfect sphere, and an average value must be used for its radius
This calculated value of g only includes true gravity. It does not include the reduction of constraint force that we perceive as a reduction of gravity due to the rotation of Earth, and some of gravity being counteracted by centrifugal force.
There are significant uncertainties in the values of r and m1 as used in this calculation, and the value of G is also rather difficult to measure precisely.
If G, g and r are known then a reverse calculation will give an estimate of the mass of the Earth. This method was used by Henry Cavendish.
See also[edit]
Earth sciences portal
Gravitation portal
Earth's magnetic field
Gravity (Gravitation)
Gravity anomaly, Bouguer anomaly
Gravitation of the Moon
Gravitational acceleration
Gravity Field and Steady-State Ocean Circulation Explorer
Gravity Recovery and Climate Experiment
Newton's law of universal gravitation
References[edit]
Jump up ^ NASA/JPL/University of Texas Center for Space Research. "PIA12146: GRACE Global Gravity Animation". Photojournal. NASA Jet Propulsion Laboratory. Retrieved 30 December 2013.
Jump up ^ The international system of units (SI) (PDF) (2008 ed.). United States Department of Commerce, NIST Special Publication 330. p. 51.
^ Jump up to: a b Boynton, Richard (2001). "Precise Measurement of Mass" (PDF). Sawe Paper No. 3147. Arlington, Texas: S.A.W.E., Inc. Retrieved 2007-01-21.
Jump up ^ Hirt, Christian; Claessens, Sten; Fecher, Thomas; Kuhn, Michael; Pail, Roland; Rexer, Moritz (August 28, 2013). "New ultrahigh-resolution picture of Earth's gravity field". Geophysical Research Letters. 40 (16): 4279–4283. Bibcode:2013GeoRL..40.4279H. doi:10.1002/grl.50838.
Jump up ^ "Wolfram|Alpha Gravity in Kuala Lumpur", Wolfram Alpha, accessed May 2017
Jump up ^ Resolution of the 3rd CGPM (1901), page 70 (in cm/s2). BIPM – Resolution of the 3rd CGPM
Jump up ^ "Curious About Astronomy?", Cornell University, retrieved June 2007
Jump up ^ "I feel 'lighter' when up a mountain but am I?", National Physical Laboratory FAQ
Jump up ^ "The G's in the Machine", NASA, see "Editor's note #2"
Jump up ^ Tipler, Paul A. (1999). Physics for scientists and engineers. (4th ed.). New York: W.H. Freeman/Worth Publishers. pp. 336–337. ISBN 9781572594913.
^ Jump up to: a b A. M. Dziewonski, D. L. Anderson (1981). "Preliminary reference Earth model" (PDF). Physics of the Earth and Planetary Interiors. 25 (4): 297–356. Bibcode:1981PEPI...25..297D. ISSN 0031-9201. doi:10.1016/0031-9201(81)90046-7.
Jump up ^ Watts, A. B.; Daly, S. F. (May 1981). "Long wavelength gravity and topography anomalies" (PDF). Annual Review of Earth and Planetary Sciences. 9: 415–418. Bibcode:1981AREPS...9..415W. doi:10.1146/annurev.ea.09.050181.002215.
Jump up ^ Gravitational Fields Widget as of Oct 25th, 2012 – WolframAlpha
Jump up ^ T.M. Yarwood and F. Castle, Physical and Mathematical Tables, revised edition, Macmillan and Co LTD, London and Basingstoke, Printed in Great Britain by The University Press, Glasgow, 1970, pp 22 & 23.
Jump up ^ International Gravity formula
^ Jump up to: a b Department of Defense World Geodetic System 1984 ― Its Definition and Relationships with Local Geodetic Systems,NIMA TR8350.2, 3rd ed., Tbl. 3.4, Eq. 4-1
Jump up ^ The rate of decrease is calculated by differentiating g(r) with respect to r and evaluating at r=rEarth.
External links[edit]
Altitude gravity calculator
GRACE – Gravity Recovery and Climate Experiment
GGMplus high resolution data (2013)
Geoid 2011 model Potsdam Gravity Potato
[hide]
v
t
e
Geophysics
Overview
Outline
Geophysicists
Subfields
Geophysical fluid dynamics
Geodesy
Geodynamics
Geomagnetism
Mathematical geophysics
Mineral physics
Near-surface geophysics
Paleomagnetism
Seismology
Tectonophysics
Physical phenomena
Chandler wobble
Coriolis effect
Earth's magnetic field
Geodynamo
Geothermal gradient
Gravity of Earth
Mantle convection
Precession of the equinoxes
Seismic wave
Tides
Organizations
American Geophysical Union
American Meteorological Society
Canadian Geophysical Union
Environmental and Engineering Geophysical Society
European Association of Geoscientists and Engineers
European Geosciences Union
International Association of Geomagnetism and Aeronomy
International Union of Geodesy and Geophysics
Royal Astronomical Society
Society of Exploration Geophysicists
Seismological Society of America
Retrieved from "https://en.wikipedia.org/w/index.php?title=Gravity_of_Earth&oldid=800304615"
Categories:
Gravimetry
Gravitation
Earth
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
Languages
العربية
বাংলা
Bân-lâm-gú
Български
Čeština
Deutsch
Esperanto
فارسی
Հայերեն
हिन्दी
Hrvatski
Italiano
Magyar
മലയാളം
日本語
Occitan
ਪੰਜਾਬੀ
Português
Русский
Slovenčina
Српски / srpski
తెలుగు
Türkçe
Українська
Tiếng Việt
粵語
中文
Edit links
This page was last edited on 12 September 2017, at 18:06.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view

values: mw-page-base noprint

values: mw-head-base noprint

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

value: img

attrs: alt src width height class srcset data-file-width data-file-height

attrs: href class

attrs: href class title

value: sup

values:
/wiki/Metres_per_second_squared
mw-redirect
Metres per second squared

values:
/wiki/Air_resistance
mw-redirect
Air resistance

values: Edit section: Variation in gravity and apparent gravity

values:
/wiki/Spherical_symmetry
mw-redirect
Spherical symmetry

values:
/wiki/Oblate_spheroid
mw-redirect
Oblate spheroid

values:
/wiki/Nevado_Huascar%C3%A1n
mw-redirect
Nevado Huascarán

values: General Conference on Weights and Measures

values: Centrifugal force (rotating reference frame)

values: vertical-align: -2.505ex; width:19.165ex; height:6.676ex;

values: vertical-align: -2.171ex; width:17.616ex; height:6.009ex;

values: vertical-align: -2.005ex; width:15.242ex; height:5.509ex;

values: vertical-align: -2.171ex; width:35.294ex; height:6.009ex;

values: Edit section: Local topography and geology

values:
/wiki/Mineral_deposits
mw-redirect
Mineral deposits

values: Edit section: Comparative gravities in various cities around the world

values: vertical-align: -0.838ex; width:4.868ex; height:2.843ex;

values: vertical-align: -0.671ex; width:1.396ex; height:2.509ex;

attrs: columnalign rowspacing columnspacing displaystyle

values: right left right left right left right left right left right left

values: 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em

values: vertical-align: -6.505ex; width:79.223ex; height:14.176ex;

values:
/wiki/International_Gravity_Formula
mw-redirect
International Gravity Formula

values:
/wiki/Gravity_formula
mw-redirect
Gravity formula

values: vertical-align: -4.671ex; margin-right: -0.387ex; width:31.21ex; height:10.509ex;

values: vertical-align: -0.671ex; width:3.68ex; height:2.509ex;

values: vertical-align: -0.838ex; width:15.601ex; height:3.176ex;

values: vertical-align: -1.005ex; width:7.528ex; height:2.843ex;

values: {\displaystyle \mathbb {G} _{e},\,\mathbb {G} _{p}\,}

values: vertical-align: -2.338ex; width:16.75ex; height:6.176ex;

values: vertical-align: -1.005ex; width:27.238ex; height:3.343ex;

values: vertical-align: -4.671ex; width:64.68ex; height:10.509ex;

values:
/wiki/Free-air_anomaly
mw-redirect
Free-air anomaly

values: vertical-align: -0.671ex; width:21.356ex; height:3.009ex;

values: vertical-align: -0.838ex; width:24.815ex; height:3.176ex;

values:
/wiki/Gravitational_Constant
mw-redirect
Gravitational Constant

values: vertical-align: -2.171ex; width:33.457ex; height:5.009ex;

values: vertical-align: -2.505ex; width:27.822ex; height:6.343ex;

values: vertical-align: -0.838ex; width:27.831ex; height:3.343ex;

values: vertical-align: -1.838ex; width:52.129ex; height:4.843ex;

values: vertical-align: -3.005ex; width:27.08ex; height:6.509ex;

values: vertical-align: -0.671ex; width:22.648ex; height:3.009ex;

values: vertical-align: -0.838ex; width:31.171ex; height:3.176ex;

values: vertical-align: -0.838ex; width:7.262ex; height:2.843ex;

values: vertical-align: -0.671ex; width:1.977ex; height:2.509ex;

values: vertical-align: -2.171ex; width:34.383ex; height:5.009ex;

values:
/wiki/Galileo_(unit)
mw-redirect
Galileo (unit)

values: Edit section: Estimating g from the law of universal gravitation

values:
/wiki/Law_of_universal_gravitation
mw-redirect
Law of universal gravitation

values: vertical-align: -2.505ex; width:30.369ex; height:6.176ex;

values:
/wiki/Newton%27s_second_law
mw-redirect
Newton's second law

values: vertical-align: -0.671ex; width:9.873ex; height:2.509ex;

values: vertical-align: -2.171ex; width:10.408ex; height:5.009ex;

values: vertical-align: -3.005ex; width:78.215ex; height:7.009ex;

values:
/wiki/Homogeneous_(chemistry)
mw-redirect
Homogeneous (chemistry)

values: display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle

values: margin:0.5em 0 0.5em 1em;border:solid #aaa 1px

values: National Institute of Standards and Technology

values:
nofollow
external text
//doi.org/10.1002%2Fgrl.50838

values:
nofollow
http://www.bipm.org/en/CGPM/db/3/2/
external text

values:
nofollow
external text
//www.worldcat.org/issn/0031-9201

values: -moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;

values:
nofollow
http://www.csr.utexas.edu/grace/
external text

values:
/wiki/Coriolis_effect
mw-redirect
Coriolis effect

values:
/wiki/Precession_of_the_equinoxes
mw-redirect
Precession of the equinoxes

values: Environmental and Engineering Geophysical Society

values: European Association of Geoscientists and Engineers

values: International Association of Geomagnetism and Aeronomy

values: International Union of Geodesy and Geophysics

values: nowraplinks collapsible autocollapse navbox-inner

values: navigation navbox Geophysics padding:3px

values:
catlinks mw-normal-catlinks
catlinks mw-normal-catlinks
interface

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values:
/wiki/Gravity_of_Earth
View the content page [ctrl-alt-c]
c

values: Discussion about the content page [ctrl-alt-t]

values: p-namespaces navigation vectorTabs p-namespaces-label

values:
p-variants
navigation
vectorMenu emptyPortlet
p-variants-label

values: Past revisions of this page [ctrl-alt-h]

values: p-views navigation vectorTabs p-views-label

values:
p-cactions
navigation
vectorMenu emptyPortlet
p-cactions-label

values:
mw-wiki-logo
/wiki/Main_Page
Visit the main page

values:
/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: Find background information on current events

values:
/wiki/Special:Random
Load a random article [ctrl-alt-x]
x

values: portal navigation p-navigation p-navigation-label

values: جاذبية أرضية – Arabic

values: অভিকর্ষজ বল – Bangla

values: Земно ускорение – Bulgarian

values: گرانش زمین – Persian

values: Ծանրության ուժ – Armenian

values: पृथ्वी का गुरुत्व – Hindi

values: Campo gravitazionale terrestre – Italian

values: ഭൂഗുരുത്വം – Malayalam

values:
https://oc.wikipedia.org/wiki/Gravitat
Gravitat – Occitan
oc
oc
interlanguage-link-target

values: ਗੁਰੂਤਾ-ਪ੍ਰਵੇਗ – Punjabi

values: Гравитационное поле Земли – Russian

values: Убрзање Земљине теже – Serbian

values: భూమి యొక్క గురుత్వాకర్షణ – Telugu

values: Прискорення вільного падіння – Ukrainian

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Gravity_of_Earth rootpage-Gravity_of_Earth skin-vector action-view