What is the rate of acceleration due to gravity on Earth?

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

World'due south gravity measured by NASA GRACE mission, showing deviations from the theoretical gravity of an arcadian, shine Earth, the so-called Globe ellipsoid. Red shows the areas where gravity is stronger than the smooth, standard value, and blueish reveals areas where gravity is weaker. (Animated version.)^[1]

The gravity of Earth, denoted by g , is the internet acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution inside World) and the centrifugal force (from the World'south rotation).^{[2] [iii]} It is a vector (physics) quantity, whose management coincides with a plumb bob and strength or magnitude is given by the norm ${\displaystyle g=\|{\mathit {\mathbf {k} }}\|}$ .

In SI units this acceleration is expressed in metres per second squared (in symbols, thou/southward² or m·s⁻²) or equivalently in newtons per kilogram (N/kg or N·kg⁻ⁱ). Virtually World'southward surface, the gravity acceleration is approximately ix.81 1000/s² (32.2 ft/due south²), which means that, ignoring the effects of air resistance, the speed of an object falling freely will increment by nigh 9.81 metres (32.two ft) per second every 2d. This quantity is sometimes referred to informally as trivial g (in contrast, the gravitational abiding G is referred to as big G ).

The precise strength of Earth's gravity varies depending on location. The nominal "boilerplate" value at World's surface, known as standard gravity is, past definition, 9.80665 thou/s^two (32.1740 ft/due southⁱⁱ).^[4] This quantity is denoted variously equally k _n , g _e (though this sometimes means the normal equatorial value on Earth, 9.78033 m/sⁱⁱ (32.0877 ft/due south^two)), g ₀ , gee, or simply grand (which is besides used for the variable local value).

The weight of an object on Globe's surface is the downwards strength on that object, given by Newton's 2nd law of move, or F = m(a) ( force = mass × acceleration ). Gravitational dispatch contributes to the total gravity dispatch, but other factors, such every bit the rotation of World, 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 [edit]

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, information technology is slightly flatter at the poles while bulging at the Equator: an oblate spheroid. There are consequently slight deviations in the magnitude of gravity beyond its surface.

Gravity on the Earth's surface varies by effectually 0.7%, from 9.7639 thou/s² on the Nevado Huascarán mountain in Peru to 9.8337 grand/southward^two at the surface of the Chill Ocean.^[5] In big cities, information technology ranges from 9.7806^[half-dozen] in Kuala Lumpur, United mexican states City, and Singapore to nine.825 in Oslo and Helsinki.

Conventional value [edit]

In 1901 the tertiary General Briefing on Weights and Measures defined a standard gravitational dispatch for the surface of the Earth: g _n = nine.80665 1000/s². It was based on measurements done at the Pavillon de Breteuil near Paris in 1888, with a theoretical correction practical in order to catechumen to a breadth of 45° at sea level.^[7] This definition is thus not a value of whatever particular identify or carefully worked out average, but an agreement for a value to use if a better bodily local value is not known or not important.^[8] It is besides used to define the units kilogram force and pound force.

Latitude [edit]

The differences of Earth'southward gravity around the Antarctic continent.

The surface of the Globe is rotating, and so information technology 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 down dispatch of falling objects.

The second major reason for the divergence in gravity at different latitudes is that the Earth's equatorial burl (itself also acquired by centrifugal force from rotation) causes objects at the Equator to exist further from the planet'south center than objects at the poles. Considering the strength 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 burl and the furnishings of the surface centrifugal force due to rotation hateful that bounding main-level gravity increases from virtually 9.780 g/s² at the Equator to about 9.832 thousand/s² at the poles, so an object volition weigh approximately 0.five% more than at the poles than at the Equator.^{[2] [9]}

Altitude [edit]

The graph shows the variation in gravity relative to the height of an object above the surface

Gravity decreases with altitude as one rises in a higher place the Earth's surface because greater altitude means greater distance from the Earth's centre. All other things being equal, an increment in altitude from sea level to 9,000 metres (30,000 ft) causes a weight decrease of nearly 0.29%. (An additional cistron affecting apparent weight is the decrease in air density at altitude, which lessens an object'south buoyancy.^[x] This would increment a person's apparent weight at an altitude of 9,000 metres by most 0.08%)

Information technology is a common misconception that astronauts in orbit are weightless because they accept flown high plenty to escape the Globe'southward gravity. In fact, at an distance of 400 kilometres (250 mi), equivalent to a typical orbit of the ISS, gravity is yet nearly xc% as strong as at the Earth'south surface. Weightlessness actually occurs because orbiting objects are in free-fall.^[11]

The effect of footing elevation depends on the density of the ground (run across Slab correction department). A person flying at 9,100 1000 (thirty,000 ft) above sea level over mountains will feel more gravity than someone at the same superlative but over the bounding main. However, a person standing on the Globe'southward surface feels less gravity when the acme is higher.

The following formula approximates the Earth's gravity variation with altitude:

${\displaystyle g_{h}=g_{0}\left({\frac {R_{\mathrm {e} }}{R_{\mathrm {e} }+h}}\right)^{2}}$

Where

• g_h is the gravitational acceleration at height h above sea level.
• R _e is the World'southward hateful radius.
• k ₀ is the standard gravitational acceleration.

The formula treats the Earth every bit a perfect sphere with a radially symmetric distribution of mass; a more authentic mathematical treatment is discussed beneath.

Depth [edit]

Earth's gravity according to the Preliminary Reference Earth Model (PREM).^[12] Two models for a spherically symmetric Earth are included for comparison. The dark greenish straight line is for a constant density equal to the Earth's boilerplate density. The light dark-green curved line is for a density that decreases linearly from centre to surface. The density at the centre is the aforementioned as in the PREM, only the surface density is chosen so that the mass of the sphere equals the mass of the real Earth.

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 equally a consequence of the inverse-square constabulary of gravitation. Another issue is that the gravity is the aforementioned as if all the mass were concentrated at the eye. Thus, the gravitational dispatch at this radius is^[13]

${\displaystyle g(r)=-{\frac {GM(r)}{r^{2}}}.}$

where G is the gravitational abiding and Thou(r) is the total mass enclosed within radius r. If the Earth had a constant density ρ, the mass would be One thousand(r) = (4/three)πρr ³ and the dependence of gravity on depth would be

${\displaystyle yard(r)={\frac {iv\pi }{3}}1000\rho r.}$

The gravity grand′ at depth d is given past chiliad′ = k(1 − d/R) where thou is acceleration due to gravity on the surface of the Earth, d is depth and R is the radius of the World. If the density decreased linearly with increasing radius from a density ρ ₀ at the eye to ρ _one at the surface, and so ρ(r) = ρ ₀ − (ρ ₀ − ρ ₁) r / r _e , and the dependence would exist

${\displaystyle g(r)={\frac {4\pi }{3}}M\rho _{0}r-\pi K\left(\rho _{0}-\rho _{1}\correct){\frac {r^{2}}{r_{\mathrm {e} }}}.}$

The bodily depth dependence of density and gravity, inferred from seismic travel times (run across Adams–Williamson equation), is shown in the graphs below.

Local topography and geology [edit]

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 World's gravitational field, known every bit gravitational anomalies.^[14] Some of these anomalies can exist 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 issue of topography and other known factors is subtracted, and from the resulting information conclusions are drawn. Such techniques are now used by prospectors to find oil and mineral deposits. Denser rocks (oftentimes containing mineral ores) cause college than normal local gravitational fields on the Earth's surface. Less dense sedimentary rocks cause the opposite.

Other factors [edit]

In air or water, objects experience a supporting buoyancy force which reduces the apparent strength of gravity (equally 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 Credible weight for details.

The gravitational furnishings of the Moon and the Sunday (also the cause of the tides) take a very pocket-sized effect on the credible strength of Earth's gravity, depending on their relative positions; typical variations are 2 µm/south² (0.ii mGal) over the form of a solar day.

Direction [edit]

A plumb bob determines the local vertical direction

Gravity dispatch is a vector quantity, with direction in addition to magnitude. In a spherically symmetric Earth, gravity would bespeak straight towards the sphere'south centre. Every bit the Earth's figure is slightly flatter, at that place are consequently pregnant deviations in the direction of gravity: essentially the difference between geodetic latitude and geocentric breadth. Smaller deviations, chosen vertical deflection, are caused by local mass anomalies, such as mountains.

Comparative values worldwide [edit]

Tools exist for calculating the force of gravity at various cities effectually the world.^[15] The event of breadth can exist clearly seen with gravity in high-latitude cities: Anchorage (9.826 one thousand/south²), Helsinki (9.825 m/sⁱⁱ), being almost 0.five% greater than that in cities nearly the equator: Kuala Lumpur (9.776 g/south²). The upshot of altitude can be seen in Mexico City (9.776 1000/s²; distance 2,240 metres (7,350 ft)), and by comparing Denver (9.798 m/s^two; 1,616 metres (5,302 ft)) with Washington, D.C. (ix.801 one thousand/s²; 30 metres (98 ft)), both of which are most 39° N. Measured values can be obtained from Physical and Mathematical Tables by T.M. Yarwood and F. Castle, Macmillan, revised edition 1970.^[xvi]

Acceleration due to gravity in various cities

Location	m/south2	ft/due south2		Location	chiliad/s2	ft/sii		Location	m/sii	ft/s2
Amsterdam	9.817	32.21		Dki jakarta	ix.777	32.08		Ottawa	ix.806	32.17
Anchorage	nine.826	32.24		Kandy	9.775	32.07		Paris	ix.809	32.xviii
Athens	9.800	32.15		Kolkata	9.785	32.10		Perth	9.794	32.13
Auckland	9.799	32.fifteen		Kuala Lumpur	9.776	32.07		Rio de Janeiro	9.788	32.11
Bangkok	nine.780	32.09		Kuwait City	nine.792	32.13		Rome	9.803	32.16
Birmingham	9.817	32.21		Lisbon	nine.801	32.sixteen		Seattle	9.811	32.19
Brussels	9.815	32.20		London	9.816	32.20		Singapore	ix.776	32.07
Buenos Aires	9.797	32.14		Los Angeles	9.796	32.fourteen		Skopje	9.804	32.17
Greatcoat Town	9.796	32.fourteen		Madrid	9.800	32.15		Stockholm	9.818	32.21
Chicago	9.804	32.17		Manchester	9.818	32.21		Sydney	9.797	32.14
Copenhagen	9.821	32.22		Manila	9.780	32.09		Taipei	9.790	32.12
Denver	9.798	32.15		Melbourne	9.800	32.15		Tokyo	9.798	32.15
Frankfurt	9.814	32.twenty		Mexico City	nine.776	32.07		Toronto	9.807	32.18
Havana	9.786	32.11		Montréal	ix.809	32.18		Vancouver	9.809	32.xviii
Helsinki	ix.825	32.23		New York City	ix.802	32.16		Washington, D.C.	ix.801	32.xvi
Hong Kong	9.785	32.10		Nicosia	nine.797	32.xiv		Wellington	9.803	32.16
Istanbul	nine.808	32.18		Oslo	ix.825	32.23		Zurich	9.807	32.18

Mathematical models [edit]

If the terrain is at sea level, we can gauge, for the Geodetic Reference Organisation 1980, ${\displaystyle g\{\phi \}}$ , the acceleration at latitude ${\displaystyle \phi }$ :

${\displaystyle {\begin{aligned}thou\{\phi \}&=9.780327\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}\,\,\left(1+0.0053024\,\sin ^{2}\phi -0.0000058\,\sin ^{2}ii\phi \right),\\&=ix.780327\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}\,\,\left(ane+0.0052792\,\sin ^{ii}\phi +0.0000232\,\sin ^{four}\phi \right),\\&=9.780327\,\,\mathrm {one thousand} \cdot \mathrm {s} ^{-two}\,\,\left(1.0053024-0.0053256\,\cos ^{2}\phi +0.0000232\,\cos ^{iv}\phi \right),\\&=9.780327\,\,\mathrm {m} \cdot \mathrm {south} ^{-2}\,\,\left(1.0026454-0.0026512\,\cos two\phi +0.0000058\,\cos ^{2}2\phi \right)\cease{aligned}}}$

This is the International Gravity Formula 1967, the 1967 Geodetic Reference Organization Formula, Helmert's equation or Clairaut's formula.^[17]

An culling formula for g equally a function of latitude is the WGS (World Geodetic System) 84 Oblong Gravity Formula:^[18]

${\displaystyle thou\{\phi \}=\mathbb {Thousand} _{e}\left[{\frac {1+k\sin ^{two}\phi }{\sqrt {one-eastward^{2}\sin ^{2}\phi }}}\right],\,\!}$

where,

• ${\displaystyle a,\,b}$ are the equatorial and polar semi-axes, respectively;
• ${\displaystyle e^{2}=1-(b/a)^{2}}$ is the spheroid's eccentricity, squared;
• ${\displaystyle \mathbb {One thousand} _{e},\,\mathbb {G} _{p}\,}$ is the defined gravity at the equator and poles, respectively;
• ${\displaystyle k={\frac {b\,\mathbb {G} _{p}-a\,\mathbb {G} _{due east}}{a\,\mathbb {G} _{e}}}}$ (formula abiding);

then, where ${\displaystyle \mathbb {G} _{p}=nine.8321849378\,\,\mathrm {m} \cdot \mathrm {due south} ^{-2}}$ ,^[18]

${\displaystyle grand\{\phi \}=9.7803253359\,\,\mathrm {thousand} \cdot \mathrm {s} ^{-two}\left[{\frac {1+0.001931852652\,\sin ^{2}\phi }{\sqrt {ane-0.0066943799901\,\sin ^{2}\phi }}}\right]}$ .

where the semi-axes of the earth are:

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

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

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

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

Estimating k from the law of universal gravitation [edit]

From the police of universal gravitation, the force on a body acted upon by Earth'due south gravitational forcefulness is given by

${\displaystyle F=G{\frac {m_{ane}m_{two}}{r^{2}}}=(Thou{\frac {M_{\oplus }}{r^{ii}}})chiliad}$

where r is the altitude between the centre of the Globe and the body (see below), and hither we take ${\displaystyle M_{\oplus }}$ to be the mass of the Earth and m to exist the mass of the body.

Additionally, Newton's 2d law, F = ma, where 1000 is mass and a is dispatch, here tells us that

${\displaystyle F=mg}$

Comparing the two formulas it is seen that:

${\displaystyle thousand=G{\frac {M_{\oplus }}{r^{2}}}}$

So, to find the acceleration due to gravity at sea level, substitute the values of the gravitational constant, Grand, the Earth's mass (in kilograms), 1000 ₁, and the Globe's radius (in metres), r, to obtain the value of thousand:

${\displaystyle g=G{\frac {M_{\oplus }}{r^{2}}}=six.67\cdot 10^{-11}{g}^{3}{kg}^{-1}{s}^{-2}\times {\frac {6\times 10^{24}{kg}}{(6.4\times x^{6}{thou})^{2}}}=9.77{m}.{s}^{-2}}$

^[19]

This formula only works because of the mathematical fact that the gravity of a compatible spherical body, every bit measured on or above its surface, is the same as if all its mass were concentrated at a bespeak at its centre. This is what allows us to employ the World'south radius for r.

The value obtained agrees approximately with the measured value of g. The difference may be attributed to several factors, mentioned in a higher place under "Variations":

• The Earth is not homogeneous
• The Earth is not a perfect sphere, and an boilerplate value must exist used for its radius
• This calculated value of g simply includes true gravity. It does not include the reduction of constraint force that nosotros perceive as a reduction of gravity due to the rotation of Globe, and some of gravity existence counteracted by centrifugal strength.

There are significant uncertainties in the values of r and one thousand ₁ every bit used in this calculation, and the value of G is also rather difficult to measure out precisely.

If G, yard and r are known and then a reverse adding will give an gauge of the mass of the Earth. This method was used by Henry Cavendish.

Measurement [edit]

The measurement of Earth's gravity is called gravimetry.

Satellite measurements [edit]

Gravity anomaly map from GRACE

Currently, the static and time-variable Earth's gravity field parameters are being adamant using modern satellite missions, such as GOCE, Gnaw, Swarm, GRACE and GRACE-FO.^{[xx] [21]} The lowest-degree parameters, including the World'due south oblateness and geocenter movement are best adamant from Satellite laser ranging.^[22]

Big-scale gravity anomalies can be detected from space, equally a past-production 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 grade 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.

The Gravity Recovery and Climate Experiment (GRACE) consists of 2 satellites that can notice gravitational changes across the Earth. Also these changes can be presented every bit gravity anomaly temporal variations. The Gravity Recovery and Interior Laboratory (GRAIL) also consisted of ii spacecraft orbiting the Moon, which orbited for 3 years earlier their deorbit in 2015.

Meet also [edit]

• Figure of Globe
• Geopotential
  • Geopotential model
• Gravity (Gravitation)
• Gravity anomaly, Bouguer bibelot
• Gravitation of the Moon
• Gravitational acceleration
• Gravity of Mars
• Newton'southward constabulary of universal gravitation
• Vertical deflection

References [edit]

1. ^ NASA/JPL/University of Texas Eye for Infinite Research. "PIA12146: GRACE Global Gravity Blitheness". Photojournal. NASA Jet Propulsion Laboratory. Retrieved 30 December 2013.
2. ^ ^{a b} Boynton, Richard (2001). "Precise Measurement of Mass" (PDF). Sawe Newspaper No. 3147. Arlington, Texas: South.A.W.Due east., Inc. Retrieved 2007-01-21 .
3. ^ Hofmann-Wellenhof, B.; Moritz, H. (2006). Physical Geodesy (second ed.). Springer. ISBN978-3-211-33544-4. § two.1: "The total forcefulness acting on a body at residuum on the earth's surface is the resultant of gravitational force and the centrifugal force of the globe'south 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 Engineering science. 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 flick of Globe's gravity field". Geophysical Research Letters. forty (sixteen): 4279–4283. Bibcode:2013GeoRL..twoscore.4279H. doi:ten.1002/grl.50838. hdl:xx.500.11937/46786.
6. ^ "Wolfram|Alpha Gravity in Kuala Lumpur", Wolfram Alpha, accessed Nov 2020
7. ^ Terry Quinn (2011). From Artefacts to Atoms: The BIPM and the Search for Ultimate Measurement Standards. Oxford University Press. p. 127. ISBN978-0-xix-530786-iii.
8. ^ Resolution of the 3rd CGPM (1901), folio seventy (in cm/s²). BIPM – Resolution of the 3rd CGPM
9. ^ "Curious About Astronomy?", Cornell University, retrieved June 2007
10. ^ "I feel 'lighter' when up a mountain but am I?", National Concrete Laboratory FAQ
11. ^ "The G'south in the Machine", NASA, see "Editor's note #ii"
12. ^ ^{a b} A. M. Dziewonski, D. L. Anderson (1981). "Preliminary reference World 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.
13. ^ Tipler, Paul A. (1999). Physics for scientists and engineers (fourth ed.). New York: W.H. Freeman/Worth Publishers. pp. 336–337. ISBN9781572594913.
14. ^ Watts, A. B.; Daly, South. F. (May 1981). "Long wavelength gravity and topography anomalies". Annual Review of Globe and Planetary Sciences. 9: 415–418. Bibcode:1981AREPS...9..415W. doi:10.1146/annurev.ea.09.050181.002215.
15. ^ Gravitational Fields Widget as of Oct 25th, 2012 – WolframAlpha
16. ^ T.M. Yarwood and F. Castle, Physical and Mathematical Tables, revised edition, Macmillan and Co LTD, London and Basingstoke, Printed in Slap-up Uk by The Academy Press, Glasgow, 1970, pp 22 & 23.
17. ^ International Gravity formula Archived 2008-08-xx at the Wayback Auto
18. ^ ^{a b} Department of Defence force Earth Geodetic System 1984 ― Its Definition and Relationships with Local Geodetic Systems ,NIMA TR8350.ii, 3rd ed., Tbl. 3.4, Eq. 4-1
19. ^ "Gravitation". www.ncert.nic . Retrieved 2022-01-25 . {{cite web}}: CS1 maint: url-condition (link)
20. ^ 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. xi (8): 956. Bibcode:2019RemS...11..956M. doi:10.3390/rs11080956.
21. ^ Tapley, Byron D.; Watkins, Michael M.; Flechtner, Frank; Reigber, Christoph; Bettadpur, Srinivas; Rodell, Matthew; Sasgen, Ingo; Famiglietti, James Southward.; Landerer, Felix Due west.; 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, Marker E.; Velicogna, Isabella (May 2019). "Contributions of GRACE to agreement climate modify". Nature Climate Alter. 9 (v): 358–369. Bibcode:2019NatCC...9..358T. doi:10.1038/s41558-019-0456-2. PMC6750016. PMID 31534490.
22. ^ Sośnica, Krzysztof; Jäggi, Adrian; Meyer, Ulrich; Thaller, Daniela; Beutler, Gerhard; Arnold, Daniel; Dach, Rolf (October 2015). "Time variable Earth'south 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 [edit]

• Distance gravity reckoner
• GRACE – Gravity Recovery and Climate Experiment
• GGMplus loftier resolution data (2013)
• Geoid 2011 model Potsdam Gravity Irish potato

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Source: https://en.wikipedia.org/wiki/Gravity_of_Earth

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