what happens to gravity at the center of the earth

Acceleration that the Earth imparts to objects on or well-nigh its surface

Earth's gravity measured by NASA GRACE mission, showing deviations from the theoretical gravity of an idealized, shine 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. (Blithe version.)^[1]

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

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 (North/kg or N·kg⁻¹). Near Earth's surface, the gravity dispatch is approximately ix.81 k/s² (32.2 ft/southwardⁱⁱ), which means that, ignoring the furnishings of air resistance, the speed of an object falling freely will increment by about ix.81 metres (32.ii ft) per 2d every second. This quantity is sometimes referred to informally equally little m (in contrast, the gravitational constant G is referred to as large Grand ).

The precise forcefulness of Earth's gravity varies depending on location. The nominal "average" value at Globe's surface, known as standard gravity is, past definition, 9.80665 m/s² (32.1740 ft/south²).^[4] This quantity is denoted variously as g _north , g _e (though this sometimes ways the normal equatorial value on Earth, 9.78033 m/sⁱⁱ (32.0877 ft/south²)), grand ₀ , gee, or simply grand (which is also used for the variable local value).

The weight of an object on World'southward surface is the downwardly force on that object, given past Newton's second police force of motion, or F = grand(a) ( forcefulness = mass × dispatch ). Gravitational acceleration contributes to the total gravity acceleration, but other factors, such as the rotation of World, also contribute, and, therefore, impact the weight of the object. Gravity does not unremarkably 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 middle (spherical symmetry), would produce a gravitational field of uniform magnitude at all points on its surface. The World 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 nine.7639 k/southward² on the Nevado Huascarán mountain in Peru to ix.8337 k/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 ix.825 in Oslo and Helsinki.

Conventional value [edit]

In 1901 the 3rd General Conference on Weights and Measures defined a standard gravitational acceleration for the surface of the Earth: one thousand _n = ix.80665 grand/s². Information technology was based on measurements washed at the Pavillon de Breteuil nearly Paris in 1888, with a theoretical correction applied in club to catechumen to a breadth of 45° at ocean level.^[7] This definition is thus non a value of any particular identify or advisedly 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 strength and pound strength.

Latitude [edit]

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

The surface of the Earth is rotating, so it is non an inertial frame of reference. At latitudes nearer the Equator, the outward centrifugal forcefulness produced by Earth's rotation is larger than at polar latitudes. This counteracts the Earth'due south gravity to a small degree – up to a maximum of 0.3% at the Equator – and reduces the apparent downwards acceleration of falling objects.

The 2nd major reason for the departure in gravity at dissimilar latitudes is that the World'southward equatorial bulge (itself also acquired past centrifugal forcefulness from rotation) causes objects at the Equator to be farther from the planet's centre than objects at the poles. Considering the force due to gravitational attraction between two bodies (the Earth and the object existence weighed) varies inversely with the foursquare of the distance betwixt 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 strength due to rotation mean that body of water-level gravity increases from nearly 9.780 m/due south² at the Equator to about 9.832 m/southⁱⁱ at the poles, then an object volition weigh approximately 0.five% more at the poles than at the Equator.^{[two] [9]}

Altitude [edit]

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

Gravity decreases with altitude every bit one rises above the Earth's surface because greater altitude means greater distance from the Earth'southward centre. All other things beingness equal, an increase in altitude from sea level to nine,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.^[10] This would increase a person's credible 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 distance of 400 kilometres (250 mi), equivalent to a typical orbit of the ISS, gravity is notwithstanding nearly ninety% as strong equally at the Earth'south surface. Weightlessness actually occurs because orbiting objects are in free-autumn.^[xi]

The result of ground meridian depends on the density of the ground (meet Slab correction section). A person flight at 9,100 m (xxx,000 ft) to a higher place ocean level over mountains will experience more gravity than someone at the same elevation but over the body of water. However, a person standing on the Earth's surface feels less gravity when the elevation is college.

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

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

Where

• g_h is the gravitational acceleration at pinnacle h above ocean level.
• R _{due east} is the Earth'due south mean radius.
• g ₀ 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 handling is discussed below.

Depth [edit]

Earth'south gravity according to the Preliminary Reference Earth Model (PREM).^[12] 2 models for a spherically symmetric Earth are included for comparison. The dark dark-green direct line is for a abiding density equal to the Globe's average density. The light green curved line is for a density that decreases linearly from middle to surface. The density at the center is the aforementioned equally in the PREM, but the surface density is called so that the mass of the sphere equals the mass of the real Earth.

An approximate value for gravity at a altitude r from the center of the Earth can be obtained by assuming that the Earth'south density is spherically symmetric. The gravity depends only on the mass inside the sphere of radius r. All the contributions from exterior cancel out as a consequence of the changed-square law of gravitation. Another consequence is that the gravity is the same as if all the mass were full-bodied at the centre. Thus, the gravitational acceleration at this radius is^[13]

${\displaystyle grand(r)=-{\frac {GM(r)}{r^{two}}}.}$

where Yard is the gravitational constant and M(r) is the total mass enclosed inside radius r. If the Earth had a constant density ρ, the mass would be M(r) = (4/3)πρr ³ and the dependence of gravity on depth would be

${\displaystyle g(r)={\frac {4\pi }{3}}G\rho r.}$

The gravity yard′ at depth d is given by k′ = one thousand(1 − d/R) where thou is dispatch due to gravity on the surface of the Globe, d is depth and R is the radius of the Earth. If the density decreased linearly with increasing radius from a density ρ ₀ at the center to ρ _ane at the surface, then ρ(r) = ρ ₀ − (ρ ₀ − ρ ₁) r / r _{due east} , and the dependence would be

${\displaystyle g(r)={\frac {4\pi }{3}}G\rho _{0}r-\pi G\left(\rho _{0}-\rho _{one}\right){\frac {r^{2}}{r_{\mathrm {e} }}}.}$

The actual 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 every bit the density of rocks in the vicinity), and deeper tectonic structure crusade local and regional differences in the World'due south gravitational field, known as gravitational anomalies.^[14] Some of these anomalies tin be very all-encompassing, resulting in bulges in sea level, and throwing pendulum clocks out of synchronisation.

The report of these anomalies forms the basis of gravitational geophysics. The fluctuations are measured with highly sensitive gravimeters, the consequence 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 dumbo sedimentary rocks crusade the opposite.

Other factors [edit]

In air or h2o, objects feel 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 Dominicus (also the crusade of the tides) have a very small effect on the apparent force of Globe'due south gravity, depending on their relative positions; typical variations are 2 µm/southward^two (0.2 mGal) over the course of a day.

Management [edit]

A plumb bob determines the local vertical management

Gravity acceleration is a vector quantity, with direction in improver to magnitude. In a spherically symmetric Earth, gravity would betoken directly towards the sphere's centre. As the Earth's effigy is slightly flatter, there are consequently significant deviations in the management of gravity: substantially the difference between geodetic breadth and geocentric latitude. Smaller deviations, chosen vertical deflection, are acquired by local mass anomalies, such equally mountains.

Comparative values worldwide [edit]

Tools exist for calculating the strength of gravity at diverse cities around the world.^[xv] The effect of latitude can be clearly seen with gravity in high-latitude cities: Anchorage (nine.826 m/s²), Helsinki (9.825 m/southward²), being about 0.5% greater than that in cities nigh the equator: Kuala Lumpur (9.776 m/s²). The event of distance tin can be seen in Mexico City (9.776 m/sⁱⁱ; altitude 2,240 metres (7,350 ft)), and by comparing Denver (ix.798 m/s²; i,616 metres (five,302 ft)) with Washington, D.C. (9.801 yard/s²; xxx metres (98 ft)), both of which are near 39° Northward. Measured values tin be obtained from Physical and Mathematical Tables past T.M. Yarwood and F. Castle, Macmillan, revised edition 1970.^[16]

Acceleration due to gravity in various cities

Location	m/stwo	ft/s2		Location	grand/s2	ft/s2		Location	thou/south2	ft/southwardtwo
Amsterdam	9.817	32.21		Jakarta	nine.777	32.08		Ottawa	9.806	32.17
Anchorage	9.826	32.24		Kandy	9.775	32.07		Paris	9.809	32.18
Athens	9.800	32.15		Kolkata	9.785	32.ten		Perth	9.794	32.thirteen
Auckland	9.799	32.15		Kuala Lumpur	9.776	32.07		Rio de Janeiro	9.788	32.11
Bangkok	9.780	32.09		Kuwait City	9.792	32.13		Rome	9.803	32.16
Birmingham	9.817	32.21		Lisbon	9.801	32.sixteen		Seattle	ix.811	32.xix
Brussels	ix.815	32.20		London	9.816	32.twenty		Singapore	nine.776	32.07
Buenos Aires	9.797	32.14		Los Angeles	ix.796	32.xiv		Skopje	9.804	32.17
Cape Boondocks	9.796	32.xiv		Madrid	9.800	32.fifteen		Stockholm	nine.818	32.21
Chicago	9.804	32.17		Manchester	9.818	32.21		Sydney	9.797	32.14
Copenhagen	9.821	32.22		Manila	ix.780	32.09		Taipei	9.790	32.12
Denver	9.798	32.xv		Melbourne	9.800	32.xv		Tokyo	9.798	32.15
Frankfurt	ix.814	32.20		Mexico City	9.776	32.07		Toronto	9.807	32.18
Havana	ix.786	32.11		Montréal	nine.809	32.18		Vancouver	9.809	32.18
Helsinki	9.825	32.23		New York City	9.802	32.16		Washington, D.C.	9.801	32.xvi
Hong Kong	nine.785	32.10		Nicosia	9.797	32.xiv		Wellington	9.803	32.sixteen
Istanbul	9.808	32.18		Oslo	ix.825	32.23		Zurich	9.807	32.18

Mathematical models [edit]

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

${\displaystyle {\begin{aligned}g\{\phi \}&=9.780327\,\,\mathrm {thou} \cdot \mathrm {s} ^{-2}\,\,\left(ane+0.0053024\,\sin ^{2}\phi -0.0000058\,\sin ^{ii}ii\phi \right),\\&=9.780327\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}\,\,\left(i+0.0052792\,\sin ^{2}\phi +0.0000232\,\sin ^{4}\phi \right),\\&=9.780327\,\,\mathrm {m} \cdot \mathrm {s} ^{-two}\,\,\left(1.0053024-0.0053256\,\cos ^{2}\phi +0.0000232\,\cos ^{four}\phi \correct),\\&=nine.780327\,\,\mathrm {yard} \cdot \mathrm {s} ^{-two}\,\,\left(1.0026454-0.0026512\,\cos 2\phi +0.0000058\,\cos ^{2}ii\phi \right)\end{aligned}}}$

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

An culling formula for thou every bit a office of latitude is the WGS (Globe Geodetic System) 84 Ellipsoidal Gravity Formula:^[eighteen]

${\displaystyle g\{\phi \}=\mathbb {1000} _{e}\left[{\frac {i+k\sin ^{2}\phi }{\sqrt {1-east^{2}\sin ^{2}\phi }}}\correct],\,\!}$

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;
• ${\displaystyle \mathbb {G} _{e},\,\mathbb {Grand} _{p}\,}$ is the divers gravity at the equator and poles, respectively;
• ${\displaystyle g={\frac {b\,\mathbb {1000} _{p}-a\,\mathbb {Grand} _{e}}{a\,\mathbb {G} _{e}}}}$ (formula constant);

then, where ${\displaystyle \mathbb {Thousand} _{p}=nine.8321849378\,\,\mathrm {m} \cdot \mathrm {s} ^{-ii}}$ ,^[18]

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

where the semi-axes of the earth are:

${\displaystyle a=6378137.0\,\,{\mbox{one thousand}}}$

${\displaystyle b=6356752.314245\,\,{\mbox{one thousand}}}$

The divergence betwixt 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 bibelot#Computation).

Estimating g from the law of universal gravitation [edit]

From the law of universal gravitation, the force on a body acted upon past World's gravitational forcefulness is given past

${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}=(G{\frac {M_{\oplus }}{r^{2}}})g}$

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

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

${\displaystyle F=mg}$

Comparing the two formulas information technology is seen that:

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

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

${\displaystyle thousand=G{\frac {M_{\oplus }}{r^{2}}}=6.67\cdot 10^{-11}{m}^{3}{kg}^{-1}{s}^{-2}\times {\frac {6\times 10^{24}{kg}}{(6.iv\times 10^{six}{m})^{2}}}=9.77{m}.{s}^{-ii}}$

^[19]

This formula merely works because of the mathematical fact that the gravity of a compatible spherical body, as measured on or to a higher place 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 deviation may be attributed to several factors, mentioned above under "Variations":

• The Earth is non homogeneous
• The Earth is not a perfect sphere, and an average value must be used for its radius
• This calculated value of k simply includes true gravity. It does non include the reduction of constraint force that nosotros 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 m ₁ as used in this calculation, and the value of One thousand is besides rather difficult to measure precisely.

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

Measurement [edit]

The measurement of World's gravity is called gravimetry.

Satellite measurements [edit]

Gravity anomaly map from GRACE

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

Big-scale gravity anomalies tin be detected from space, equally a past-production of satellite gravity missions, due east.1000., GOCE. These satellite missions aim at the recovery of a detailed gravity field model of the Earth, typically presented in the class of a spherical-harmonic expansion of the Earth'south gravitational potential, but culling presentations, such as maps of geoid undulations or gravity anomalies, are likewise produced.

The Gravity Recovery and Climate Experiment (GRACE) consists of two satellites that tin detect gravitational changes beyond the Earth. Also these changes can exist presented as gravity bibelot temporal variations. The Gravity Recovery and Interior Laboratory (GRAIL) as well consisted of ii spacecraft orbiting the Moon, which orbited for 3 years before their deorbit in 2015.

See also [edit]

• Figure of 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 [edit]

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). Concrete Geodesy (2nd ed.). Springer. ISBN978-iii-211-33544-4. § 2.one: "The full forcefulness acting on a body at rest on the earth's surface is the resultant of gravitational force and the centrifugal forcefulness of the world'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 show of Globe's gravity field". Geophysical Research Letters. 40 (16): 4279–4283. Bibcode:2013GeoRL..40.4279H. doi:10.1002/grl.50838. hdl:xx.500.11937/46786.
6. ^ "Wolfram|Blastoff 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 Academy Press. p. 127. ISBN978-0-19-530786-3.
8. ^ Resolution of the tertiary CGPM (1901), page lxx (in cm/sⁱⁱ). 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 Physical Laboratory FAQ
11. ^ "The One thousand's in the Machine", NASA, see "Editor's annotation #2"
12. ^ ^{a b} A. M. Dziewonski, D. Fifty. Anderson (1981). "Preliminary reference World model" (PDF). Physics of the World and Planetary Interiors. 25 (four): 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 (4th ed.). New York: W.H. Freeman/Worth Publishers. pp. 336–337. ISBN9781572594913.
14. ^ Watts, A. B.; Daly, S. F. (May 1981). "Long wavelength gravity and topography anomalies". Annual Review of Globe and Planetary Sciences. nine: 415–418. Bibcode:1981AREPS...ix..415W. doi:10.1146/annurev.ea.09.050181.002215.
15. ^ Gravitational Fields Widget every bit of October 25th, 2012 – WolframAlpha
16. ^ T.Chiliad. Yarwood and F. Castle, Physical and Mathematical Tables, revised edition, Macmillan and Co LTD, London and Basingstoke, Printed in Great United kingdom of great britain and northern ireland by The University Printing, Glasgow, 1970, pp 22 & 23.
17. ^ International Gravity formula Archived 2008-08-20 at the Wayback Machine
18. ^ ^{a b} Section of Defense force Earth Geodetic Organization 1984 ― Its Definition and Relationships with Local Geodetic Systems ,NIMA TR8350.2, 3rd ed., Tbl. three.4, Eq. 4-ane
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 Apr 2019). "SLR, GRACE and Swarm Gravity Field Determination and Combination". Remote Sensing. xi (8): 956. Bibcode:2019RemS...11..956M. doi:ten.3390/rs11080956.
21. ^ Tapley, Byron D.; Watkins, Michael One thousand.; 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, Marking E.; Velicogna, Isabella (May 2019). "Contributions of GRACE to understanding climate change". Nature Climate Modify. 9 (five): 358–369. Bibcode:2019NatCC...9..358T. doi:x.1038/s41558-019-0456-2. PMC6750016. PMID 31534490.
22. ^ Sośnica, Krzysztof; Jäggi, Adrian; Meyer, Ulrich; Thaller, Daniela; Beutler, Gerhard; Arnold, Daniel; Dach, Rolf (Oct 2015). "Fourth dimension variable Earth'south gravity field from SLR satellites". Periodical of Geodesy. 89 (x): 945–960. Bibcode:2015JGeod..89..945S. doi:10.1007/s00190-015-0825-ane.

External links [edit]

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

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

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