What is the magnitude of earths gravitational force on a 1 kg body at earths surface

Gravity is a pretty awesome fundamental force. If it wasn't for the Earth's comfortable 1 g, which causes objects to fall towards the Earth at a speed of 9.8 m/s², we'd all float off into space. And without it, all us terrestrial species would slowly wither and die as our muscles degenerated, our bones became brittle and weak, and our organs ceased to function properly.

So one can say without exaggerations that gravity is not only a fact of life here on Earth, but a prerequisite for it. However, since human beings seem intent on getting off this rock – escaping the "surly bonds of Earth", as it were – understanding Earth's gravity and what it takes to escape it is necessary. So just how strong is Earth's gravity?

Definition

To break it down, gravity is a natural phenomena in which all things that possess mass are brought towards one another – i.e. asteroids, planets, stars, galaxies, super clusters, etc. The more mass an object has, the more gravity it will exert on objects around it. The gravitational force of an object is also dependent on distance – i.e. the amount it exerts on an object decreases with increased distance.

Gravity is also one of the four fundamental forces which govern all interactions in nature (along with weak nuclear force, strong nuclear force, and electromagnetism). Of these forces, gravity is the weakest, being approximately 1038 times weaker than the strong nuclear force, 1036 times weaker than the electromagnetic force and 1029 times weaker than the weak nuclear force.

As a consequence, gravity has a negligible influence on matter at the smallest of scales (i.e. subatomic particles). However, at the macroscopic level – that of planets, stars, galaxies, etc. – gravity is the dominant force affecting the interactions of matter. It causes the formation, shape and trajectory of astronomical bodies, and governs astronomical behavior. It also played a major role in the evolution of the early universe.

It was responsible for matter clumping together to form clouds of gas that underwent gravitational collapse, forming the first stars – which were then drawn together to form the first galaxies. And within individual star systems, it caused dust and gas to coalesce to form the planets. It also governs the orbits of the planets around stars, of moons around planets, the rotation of stars around their galaxy's center, and the merging of galaxies.

Universal Gravitation and Relativity

Since energy and mass are equivalent, all forms of energy, including light, also cause gravitation and are under the influence of it. This is consistent with Einstein's General Theory of Relativity, which remains the best means of describing gravity's behavior. According to this theory, gravity is not a force, but a consequence of the curvature of spacetime caused by the uneven distribution of mass/energy.

The most extreme example of this curvature of spacetime is a black hole, from which nothing can escape. Black holes are usually the product of a supermassive star that has gone supernova, leaving behind a white dwarf remnant that has so much mass, it's escape velocity is greater than the speed of light. An increase in gravity also results in gravitational time dilation, where the passage of time occurs more slowly.

For most applications though, gravity is best explained by Newton's Law of Universal Gravitation, which states that gravity exists as an attraction between two bodies. The strength of this attraction can calculated mathematically, where the attractive force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Earth's Gravity

On Earth, gravity gives weight to physical objects and causes the ocean tides. The force of Earth's gravity is the result of the planets mass and density – 5.97237 × 1024 kg (1.31668×1025 lbs) and 5.514 g/cm3, respectively. This results in Earth having a gravitational strength of 9.8 m/s² close to the surface (also known as 1 g), which naturally decreases the farther away one is from the surface.

In addition, the force of gravity on Earth actually changes depending on where you're standing on it. The first reason is because the Earth is rotating. This means that the gravity of Earth at the equator is 9.789 m/s2, while the force of gravity at the poles is 9.832 m/s2. In other words, you weigh more at the poles than you do at the equator because of this centripetal force, but only slightly more.

Finally, the force of gravity can change depending on what's under the Earth beneath you. Higher concentrations of mass, like high-density rocks or minerals can change the force of gravity that you feel. But of course, this amount is too slight to be noticeable. NASA missions have mapped the Earth's gravity field with incredible accuracy, showing variations in its strength, depending on location.

Gravity also decreases with altitude, since you're further away from the Earth's center. The decrease in force from climbing to the top of a mountain is pretty minimal (0.28% less gravity at the top of Mount Everest), but if you're high enough to reach the International Space Station (ISS), you would experience 90% of the force of gravity you'd feel on the surface.

However, since the station is in a state of free fall (and also in the vacuum of space) objects and astronauts aboard the ISS are capable of floating around. Basically, since everything aboard the station is falling at the same rate towards the Earth, those aboard the ISS have the feeling of being weightless – even though they still weight about 90% of what they would on Earth's surface.

Earth's gravity is also responsible for our planet having an "escape velocity" of 11.186 km/s (or 6.951 mi/s). Essentially, this means that a rocket needs to achieve this speed before it can hope to break free of Earth's gravity and reach space. And with most rocket launches, the majority of their thrust is dedicated to this task alone.

Because of the difference between Earth's gravity and the gravitational force on other bodies – like the moon (1.62 m/s²; 0.1654 g) and Mars (3.711 m/s²; 0.376 g) – scientists are uncertain what the effects would be to astronauts who went on long-term missions to these bodies.

While studies have shown that long-duration missions in microgravity (i.e. on the ISS) have a detrimental effect on astronaut health (including loss of bone density, muscle degeneration, damage to organs and to eyesight) no studies have been conducted regarding the effects of lower-gravity environments. But given the multiple proposals made to return to the moon, and NASA's proposed "Journey to Mars", that information should be forthcoming!

As terrestrial beings, we humans are both blessed and cursed by the force of Earth's gravity. On the one hand, it makes getting into space rather difficult and expensive. On the other, it ensures our health, since our species is the product of billions of years of species evolution that took place in a 1 g environment.

If we ever hope to become a truly space-faring and interplanetary species, we better figure out how we're going to deal with microgravity and lower-gravity. Otherwise, none of us are likely to get off-world for very long!

Citation: How strong is the force of gravity on Earth? (2016, December 7) retrieved 18 November 2022 from https://phys.org/news/2016-12-strong-gravity-earth.html

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Acceleration that the Earth imparts to objects on or near its surface

The gravity of Earth, denoted by g, is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).[2][3] It is a vector quantity, whose direction coincides with a plumb bob and strength or magnitude is given by the norm g = ‖ g ‖ {\displaystyle g=\|{\mathit {\mathbf {g} }}\|}

In SI units this acceleration is expressed 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 Earth's surface, the gravity acceleration is approximately 9.81 m/s2 (32.2 ft/s2), which means that, ignoring the effects of air resistance, the speed of an object falling freely will increase by about 9.81 metres (32.2 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 Earth's surface, known as standard gravity is, by definition, 9.80665 m/s2 (32.1740 ft/s2).[4] This quantity is denoted variously as gn, ge (though this sometimes means the normal equatorial value on Earth, 9.78033 m/s2 (32.0877 ft/s2)), g0, gee, or simply g (which is also used for the variable local value).

The weight of an object on Earth's surface is the downwards force on that object, given by Newton's second law of motion, or F = m(a) (force = mass × acceleration). Gravitational acceleration contributes to the total gravity acceleration, but other factors, such as the rotation of Earth, also contribute, and, therefore, affect the weight of the object. Gravity does not normally include the gravitational pull of the Moon and Sun, which are accounted for in terms of tidal effects.

Variation in magnitude

A non-rotating 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. The Earth is rotating and is also not spherically symmetric; rather, it is slightly flatter at the poles while bulging at the Equator: an oblate spheroid. There are consequently slight deviations in the magnitude of gravity across its surface.

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.[5] In large cities, it ranges from 9.7806[6] in Kuala Lumpur, Mexico City, and Singapore to 9.825 in Oslo and Helsinki.

Conventional value

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. It was based on measurements done at the Pavillon de Breteuil near Paris in 1888, with a theoretical correction applied in order to convert to a latitude of 45° at sea level.[7] This definition is thus 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.[8] It is also used to define the units kilogram force and pound force.

Calculating the gravity at Earth's surface using the average radius of Earth (6,371 kilometres (3,959 mi)),[9] the experimentally determined value of the gravitational constant, and the Earth mass of 5.9722 ×1024 kg gives an acceleration of 9.8203 m/s2,[10] slightly greater than the standard gravity of 9.80665 m/s2. The value of standard gravity corresponds to the gravity on Earth at a radius of 6,375.4 kilometres (3,961.5 mi).[10]

Latitude

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 center 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 on the pole.

In combination, the equatorial bulge and the effects of the surface centrifugal force due to rotation mean that sea-level 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 approximately 0.5% more at the poles than at the Equator.[2][11]

Altitude

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.[12] 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 ISS, gravity is still nearly 90% as strong as at the Earth's surface. Weightlessness actually occurs because orbiting objects are in free-fall.[13]

The effect of ground elevation depends on the density of the ground (see Slab correction section). A person flying at 9,100 m (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

See also: Shell theorem

An approximate value for gravity at a distance r from the center 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 center. Thus, the gravitational acceleration at this radius is[15]

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.}

The gravity g′ at depth d is given by g′ = g(1 − d/R) where g is acceleration due to gravity on the surface of the Earth, d is depth and R is the radius of the Earth. If the density decreased linearly with increasing radius from a density ρ0 at the center 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.

Local topography and geology

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.[16] 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.

There is a strong correlation between the gravity derivation map of earth from NASA GRACE with positions of recent volcanic activity, ridge spreading and volcanos: these regions have a stronger gravitation than theoretical predictions.

Other factors

In air or water, 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 the air density (and hence air pressure) or the water density respectively; 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.

Direction

Main article: Vertical direction

Gravity acceleration is a vector quantity, with direction in addition to magnitude. In a spherically symmetric Earth, gravity would point directly towards the sphere's centre. As the Earth's figure is slightly flatter, there are consequently significant deviations in the direction of gravity: essentially the difference between geodetic latitude and geocentric latitude. Smaller deviations, called vertical deflection, are caused by local mass anomalies, such as mountains.

Comparative values worldwide

Tools exist for calculating the strength of gravity at various cities around the world.[17] 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). 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.[18]

Mathematical models

Main article: Theoretical gravity

If the terrain is at sea level, we can estimate, for the Geodetic Reference System 1980, g { ϕ } {\displaystyle g\{\phi \}}

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.[19]

An alternative formula for g as a function of latitude is the WGS (World Geodetic System) 84 Ellipsoidal Gravity Formula:[20]

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}}

g { ϕ } = 9.7803253359 m ⋅ s − 2 [ 1 + 0.001931852652 sin 2 ⁡ ϕ 1 − 0.0066943799901 sin 2 ⁡ ϕ ] {\displaystyle g\{\phi \}=9.7803253359\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}\left[{\frac {1+0.001931852652\,\sin ^{2}\phi }{\sqrt {1-0.0066943799901\,\sin ^{2}\phi }}}\right]}

where the semi-axes of the earth are:

a = 6378137.0 m {\displaystyle a=6378137.0\,\,{\mbox{m}}}

b = 6356752.314245 m {\displaystyle b=6356752.314245\,\,{\mbox{m}}}

The difference between the WGS-84 formula and Helmert's equation is less than 0.68 μm·s−2.

Further reductions are applied to obtain gravity anomalies (see: Gravity anomaly#Computation).

Estimating g from the law of universal gravitation

From the law of universal gravitation, the force on a body acted upon by Earth's gravitational force is given by

F = G m 1 m 2 r 2 = ( G M ⊕ r 2 ) m {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}=(G{\frac {M_{\oplus }}{r^{2}}})m}

where r is the distance between the centre of the Earth and the body (see below), and here we take M ⊕ {\displaystyle M_{\oplus }}

Additionally, Newton's second law, F = ma, where m is mass and a is acceleration, here tells us that

F = m g {\displaystyle F=mg}

Comparing the two formulas it is seen that:

g = G M ⊕ r 2 {\displaystyle g=G{\frac {M_{\oplus }}{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 ⊕ r 2 = 6.67 ⋅ 10 − 11 m 3 k g − 1 s − 2 × 6 × 10 24 k g ( 6.4 × 10 6 m ) 2 = 9.77 m . s − 2 {\displaystyle g=G{\frac {M_{\oplus }}{r^{2}}}=6.67\cdot 10^{-11}{m}^{3}{kg}^{-1}{s}^{-2}\times {\frac {6\times 10^{24}{kg}}{(6.4\times 10^{6}{m})^{2}}}=9.77{m}.{s}^{-2}}

[21]

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.

Measurement

Main article: Gravimetry

The measurement of Earth's gravity is called gravimetry.

Satellite measurements

This section is an excerpt from Gravimetry § Satellite gravimetry.[edit]

Currently, the static and time-variable Earth's gravity field parameters are being determined using modern satellite missions, such as GOCE, CHAMP, Swarm, GRACE and GRACE-FO.[22][23] The lowest-degree parameters, including the Earth's oblateness and geocenter motion are best determined from Satellite laser ranging.[24]

Large-scale gravity anomalies can be detected from space, as a by-product of satellite gravity missions, e.g., GOCE. These satellite missions aim at the recovery of a detailed gravity field model of the Earth, typically presented in the form of a spherical-harmonic expansion of the Earth's gravitational potential, but alternative presentations, such as maps of geoid undulations or gravity anomalies, are also produced.

See also

• 
  Earth sciences portal

• Escape velocity
  • Atmospheric escape
• Figure of the Earth
• Geopotential
  • Geopotential model
• Gravity (Gravitation)
• Gravity anomaly, Bouguer anomaly
• Gravitation of the Moon
• Gravitational acceleration
• Gravity of Mars
• Newton's law of universal gravitation
• Vertical deflection

References

1. ^ NASA/JPL/University of Texas Center for Space Research. "PIA12146: GRACE Global Gravity Animation". Photojournal. NASA Jet Propulsion Laboratory. Retrieved 30 December 2013.
2. ^ 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.
3. ^ Hofmann-Wellenhof, B.; Moritz, H. (2006). Physical Geodesy (2nd ed.). Springer. ISBN 978-3-211-33544-4. § 2.1: "The total force acting on a body at rest on the earth’s surface is the resultant of gravitational force and the centrifugal force of the earth’s rotation and is called gravity."{{cite book}}: CS1 maint: postscript (link)
4. ^ Taylor, Barry N.; Thompson, Ambler, eds. (March 2008). The international system of units (SI) (PDF) (Report). National Institute of Standards and Technology. p. 52. NIST special publication 330, 2008 edition.
5. ^ 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. hdl:20.500.11937/46786.
6. ^ "Wolfram|Alpha Gravity in Kuala Lumpur", Wolfram Alpha, accessed November 2020
7. ^ Terry Quinn (2011). From Artefacts to Atoms: The BIPM and the Search for Ultimate Measurement Standards. Oxford University Press. p. 127. ISBN 978-0-19-530786-3.
8. ^ Resolution of the 3rd CGPM (1901), page 70 (in cm/s2). BIPM – Resolution of the 3rd CGPM
9. ^ "GEODETIC REFERENCE SYSTEM 1980" (PDF). International Association of Geodesy. Retrieved 2022-05-31.
10. ^ a b "Gravitational Acceleration Calculator". sanjaysplanet.com. Retrieved 2022-05-31.
11. ^ "Curious About Astronomy?", Cornell University, retrieved June 2007
12. ^ "I feel 'lighter' when up a mountain but am I?", National Physical Laboratory FAQ
13. ^ "The G's in the Machine", NASA, see "Editor's note #2"
14. ^ 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. doi:10.1016/0031-9201(81)90046-7. ISSN 0031-9201.
15. ^ Tipler, Paul A. (1999). Physics for scientists and engineers (4th ed.). New York: W.H. Freeman/Worth Publishers. pp. 336–337. ISBN 9781572594913.
16. ^ Watts, A. B.; Daly, S. F. (May 1981). "Long wavelength gravity and topography anomalies". Annual Review of Earth and Planetary Sciences. 9: 415–418. Bibcode:1981AREPS...9..415W. doi:10.1146/annurev.ea.09.050181.002215.
17. ^ Gravitational Fields Widget as of Oct 25th, 2012 – WolframAlpha
18. ^ 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.
19. ^ International Gravity formula Archived 2008-08-20 at the Wayback Machine
20. ^ 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
21. ^ "Gravitation". www.ncert.nic. Retrieved 2022-01-25.{{cite web}}: CS1 maint: url-status (link)
22. ^ Meyer, Ulrich; Sosnica, Krzysztof; Arnold, Daniel; Dahle, Christoph; Thaller, Daniela; Dach, Rolf; Jäggi, Adrian (22 April 2019). "SLR, GRACE and Swarm Gravity Field Determination and Combination". Remote Sensing. 11 (8): 956. Bibcode:2019RemS...11..956M. doi:10.3390/rs11080956.
23. ^ Tapley, Byron D.; Watkins, Michael M.; Flechtner, Frank; Reigber, Christoph; Bettadpur, Srinivas; Rodell, Matthew; Sasgen, Ingo; Famiglietti, James S.; Landerer, Felix W.; Chambers, Don P.; Reager, John T.; Gardner, Alex S.; Save, Himanshu; Ivins, Erik R.; Swenson, Sean C.; Boening, Carmen; Dahle, Christoph; Wiese, David N.; Dobslaw, Henryk; Tamisiea, Mark E.; Velicogna, Isabella (May 2019). "Contributions of GRACE to understanding climate change". Nature Climate Change. 9 (5): 358–369. Bibcode:2019NatCC...9..358T. doi:10.1038/s41558-019-0456-2. PMC 6750016. PMID 31534490.
24. ^ Sośnica, Krzysztof; Jäggi, Adrian; Meyer, Ulrich; Thaller, Daniela; Beutler, Gerhard; Arnold, Daniel; Dach, Rolf (October 2015). "Time variable Earth's gravity field from SLR satellites". Journal of Geodesy. 89 (10): 945–960. Bibcode:2015JGeod..89..945S. doi:10.1007/s00190-015-0825-1.

External links

• Altitude gravity calculator
• GRACE – Gravity Recovery and Climate Experiment
• GGMplus high resolution data (2013)
• Geoid 2011 model Potsdam Gravity Potato

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