View source for Physics:Gravity of Earth

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[[File:Gravity anomalies on Earth.jpg|thumb|upright=1.35|Earth's gravity measured by NASA [[Physics:Gravity Recovery and Climate Experiment|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''').<ref>{{cite web|last=NASA/JPL/University of Texas Center for Space Research|title=PIA12146: GRACE Global Gravity Animation|url=http://photojournal.jpl.nasa.gov/catalog/PIA12146|work=Photojournal|publisher=NASA Jet Propulsion Laboratory|access-date=30 December 2013}}</ref>]]

The '''gravity of Earth''', denoted by '''{{mvar|g}}''', is the [[Physics:Net force|net]] [[Physics:Acceleration|acceleration]] that is imparted to objects due to the combined effect of gravitation (from [[Physics:Mass distribution|mass distribution]] within [[Astronomy:Earth|Earth]]) and the [[Physics:Centrifugal force|centrifugal force]] (from the [[Earth's rotation]]).<ref name="Boynton"/><ref>{{cite book |last1=Hofmann-Wellenhof |first1=B. |last2=Moritz |first2=H. |year=2006 |title=Physical Geodesy |publisher=Springer |edition=2nd |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."</ref>
It is a [[Euclidean vector|vector]] quantity, whose direction coincides with a [[Engineering:Plumb bob|plumb bob]] and strength or magnitude is given by the norm <math>g=\|\mathit{\mathbf{g}}\|</math>.

In [[International System of Units|SI units]], this acceleration is expressed in [[Metre per second squared|metres per second squared]] (in symbols, [[Metre|m]]/[[Second|s]]<sup>2</sup> or m·s<sup>−2</sup>) or equivalently in [[Newton (unit)|newtons]] per [[Kilogram|kilogram]] (N/kg or N·kg<sup>−1</sup>). Near Earth's surface, the acceleration due to gravity, accurate to 2 [[Significant figures|significant figures]], is {{convert|9.8|m/s2|abbr=on}}. This means that, ignoring the effects of [[Physics:Drag|air resistance]], the [[Physics:Speed|speed]] of an object [[Physics:Free fall|falling freely]] will increase by about {{convert|9.8|m|ft}} per second every second. This quantity is sometimes referred to informally as ''little {{mvar|g}}'' (in contrast, the [[Physics:Gravitational constant|gravitational constant]] {{mvar|G}} is referred to as ''big {{mvar|G}}'').

The precise strength of Earth's gravity varies with location. The agreed upon value for {{nowrap|[[Physics:Standard gravity|standard gravity]]}} is {{convert|9.80665|m/s2|abbr=on}} by definition.<ref>{{cite book
| last = Bureau International des Poids et Mesures
| date = 1901
| title = Comptes Rendus des Séances de la Troisième Conférence· Générale des Poids et Mesures
| section = Déclaration relative à l'unité de masse et à la définition du poids; valeur conventionnelle de {{math|''g''<sub>n</sub>}}
| url = https://www.bipm.org/fr/committees/cg/cgpm/3-1901/resolution-2
| location = Paris
| publisher = Gauthier-Villars
| page = 68
| language = FR
| quote = Le nombre adopté dans le Service international des Poids et Mesures pour la valeur de l'accélération normale de la pesanteur est 980,665 cm/sec², nombre sanctionné déjà par quelques législations. Déclaration relative à l'unité de masse et à la définition du poids; valeur conventionnelle de {{math|''g''<sub>n</sub>}}.
}}</ref> This quantity is denoted variously as {{math|''g''<sub>n</sub>}}, {{math|''g''<sub>e</sub>}} (though this sometimes means the normal gravity at the equator, {{convert|9.7803267715|m/s2|abbr=on}}),<ref>{{cite journal
| last1 = Moritz
| first1 = Helmut
| date = 2000
| title = Geodetic Reference System 1980
| journal = [[Earth:Journal of Geodesy|Journal of Geodesy]]
| volume = 74
| issue = 1
| pages = 128–133
| doi = 10.1007/s001900050278
| url = https://link.springer.com/article/10.1007/s001900050278
| access-date = 2023-07-26
| quote = γe = 9.780 326 7715 m/s² normal gravity at equator
}}</ref> {{math|''g''<sub>0</sub>}}, or simply {{mvar|g}} (which is also used for the variable local value).

The [[Physics:Weight|weight]] of an object on Earth's surface is the downwards force on that object, given by [[Physics:Newton's laws of motion#Newton.27s second law|Newton's second law of motion]], or {{nowrap|{{math|''F'' {{=}} ''m'' ''a''}}}} ({{nowrap|''force'' {{=}} ''mass'' × ''acceleration''}}). [[Physics:Gravitational 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 [[Astronomy:Tide|tidal effects]].

==Variation in magnitude==
A non-rotating perfect [[Sphere|sphere]] of uniform mass density, or whose density varies solely with distance from the centre (spherical symmetry), would produce a [[Physics:Gravitational field|gravitational field]] of uniform magnitude at all points on its [[Planetary surface|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&nbsp;m/s<sup>2</sup> on the Nevado Huascarán mountain in Peru to 9.8337&nbsp;m/s<sup>2</sup> at the surface of the [[Earth:Arctic Ocean|Arctic Ocean]].<ref>{{cite journal |last1=Hirt |first1=Christian |last2=Claessens |first2=Sten |last3=Fecher |first3=Thomas |last4=Kuhn |first4=Michael |last5=Pail |first5=Roland |last6=Rexer |first6=Moritz |title=New ultrahigh-resolution picture of Earth's gravity field |journal=Geophysical Research Letters |date=August 28, 2013 |volume=40 |issue=16 |pages=4279–4283 |doi=10.1002/grl.50838 |bibcode=2013GeoRL..40.4279H |hdl=20.500.11937/46786 |s2cid=54867946 |hdl-access=free}}</ref> In large cities, it ranges from 9.7806&nbsp;m/s<sup>2 </sup><ref name="Wolfram Alpha">[https://www.wolframalpha.com/input/?i=gravity+in+kuala+lampur "Wolfram|Alpha Gravity in Kuala Lumpur", Wolfram Alpha, accessed November 2020]</ref> in [[Place:Kuala Lumpur|Kuala Lumpur]], [[Place:Mexico City|Mexico City]], and [[Place:Singapore|Singapore]] to 9.825&nbsp;m/s<sup>2</sup> in [[Place:Oslo|Oslo]] and [[Place:Helsinki|Helsinki]].

===Conventional value===
In 1901, the third [[General Conference on Weights and Measures]] defined a standard gravitational acceleration for the surface of the Earth: ''g''<sub>n</sub>&nbsp;=&nbsp;9.80665&nbsp;m/s<sup>2</sup>. 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.<ref>{{cite book |title=From Artefacts to Atoms: The BIPM and the Search for Ultimate Measurement Standards |author=Terry Quinn |page=127 |publisher=[[Organization:Oxford University Press|Oxford University Press]] |year=2011 |isbn=978-0-19-530786-3}}</ref> 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.<ref>Resolution of the 3rd CGPM (1901), page 70 (in cm/s<sup>2</sup>). [http://www.bipm.org/en/CGPM/db/3/2/ BIPM – Resolution of the 3rd CGPM]</ref> It is also used to define the units kilogram force and pound force.

===Latitude===
[[Image:Southern ocean gravity hg.png|thumb|upright=1.4|The differences of Earth's gravity around the Antarctic continent.]]
The surface of the Earth is rotating, so it is [[Physics:Non-inertial reference frame|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 [[Astronomy:Equatorial bulge|equatorial bulge]] (itself also caused by centrifugal force from rotation) causes objects at the Equator to be further from the planet's center than objects at the poles. The force due to gravitational attraction between two masses (a piece of the Earth and the object being weighed) varies inversely with the square of the distance between them. The distribution of mass is also different below someone on the equator and below someone at a pole. The net result is that an object at the Equator experiences a weaker gravitational pull than an object on one of the poles.

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&nbsp;m/s<sup>2</sup> at the Equator to about 9.832&nbsp;m/s<sup>2</sup> at the poles, so an object will weigh approximately 0.5% more at the poles than at the Equator.<ref name="Boynton">{{cite conference |last=Boynton |first=Richard |date=2001 |title=''Precise Measurement of Mass'' |book-title=Sawe Paper No. 3147 |publisher=S.A.W.E., Inc. |location=Arlington, Texas |url=http://www.space-electronics.com/Literature/Precise_Measurement_of_Mass.PDF |access-date=22 December 2023 |archive-date=27 February 2007 |archive-url=https://web.archive.org/web/20070227132140/http://www.space-electronics.com/Literature/Precise_Measurement_of_Mass.PDF |url-status=dead }}</ref><ref>{{cite web |url=http://curious.astro.cornell.edu/question.php?number=310 |title=Curious About Astronomy? |website= Cornell University |accessdate=22 December 2023 |archive-date=28 July 2013 |archiveurl=https://web.archive.org/web/20130728125707/http://curious.astro.cornell.edu/question.php?number=310}} </ref>

===Altitude===
[[Image:Erdgvarp.png|thumb|upright=1.25|The graph shows the variation in gravity relative to the height of an object above the surface]]
[[File:Earth vs Mars gravity at elevation.webp|thumb|300px|Earth vs Mars vs Moon gravity at [[Elevation|elevation]]]]
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 {{convert|30000|ft|m|sigfig=1|order=flip}} 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.<ref>[http://www.npl.co.uk/reference/faqs/i-feel-'lighter'-when-up-a-mountain-but-am-i-(faq-mass-and-density) "I feel 'lighter' when up a mountain but am I?"], National Physical Laboratory FAQ</ref> 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 {{convert|400|km|mi}}, equivalent to a typical orbit of the [[Astronomy:International Space Station|ISS]], gravity is still nearly 90% as strong as at the Earth's surface. Weightlessness actually occurs because orbiting objects are in free-fall.<ref>[https://science.nasa.gov/science-news/science-at-nasa/2003/24jan_micro-g/ "The G's in the Machine"], NASA, see "Editor's note #2"</ref>

The effect of ground elevation depends on the density of the ground (see Slab correction section). A person flying at {{cvt|30000|ft|order=flip}} 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:
:<math>g_h=g_0\left(\frac{R_\mathrm{e}}{R_\mathrm{e}+h}\right)^2</math>

Where
*{{math|''g<sub>h</sub>''}} is the gravitational acceleration at height {{mvar|h}} above sea level.
*{{math|''R''<sub>e</sub>}} is the [[Astronomy:Earth radius|Earth's mean radius]].
*{{math|''g''<sub>0</sub>}} 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===
[[File:Earth-G-force.png|thumb|upright=1.35|Gravity at different internal layers of Earth (1 = continental crust, 2 = oceanic crust, 3 = upper mantle, 4 = lower mantle, 5+6 = core, A = crust-mantle boundary)]]
[[Image:RadialDensityPREM.jpg|thumb|upright=1.25|Earth's radial density distribution according to the Preliminary Reference Earth Model (PREM).<ref name="prem">{{cite journal|author=A. M. Dziewonski, D. L. Anderson|title=Preliminary reference Earth model |journal=Physics of the Earth and Planetary Interiors|date= 1981|volume=25|issue=4|pages=297–356|url=http://www.openseismo.org/contributors/Lee/MoWorking_Backups/Mo2012_0414backup/MoWorking/Paper_Pending/Dziewonski-Anderson_PEPI1981_p297.pdf|doi=10.1016/0031-9201(81)90046-7|issn=0031-9201|bibcode = 1981PEPI...25..297D}}</ref>]]
[[Image:EarthGravityPREM.svg|thumb|Earth's gravity according to the Preliminary Reference Earth Model (PREM).<ref name=prem/> 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 center to surface. The density at the center 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.]]

An approximate value for gravity at a distance {{mvar|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 {{mvar|r}}. All the contributions from outside cancel out as a consequence of the [[Inverse-square law|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<ref>{{cite book|last=Tipler|first=Paul A.|title=Physics for scientists and engineers.|date=1999|publisher=W.H. Freeman/Worth Publishers|location=New York|isbn=9781572594913|pages=336&ndash;337|edition=4th}}</ref>
:<math>g(r) = -\frac{GM(r)}{r^2}.</math>
where {{mvar|G}} is the [[Physics:Gravitational constant|gravitational constant]] and {{math|''M''(''r'')}} is the total mass enclosed within radius {{mvar|r}}. If the Earth had a constant density {{mvar|ρ}}, the mass would be {{math|1=''M''(''r'') = (4/3)''πρr''<sup>3</sup>}} and the dependence of gravity on depth would be
:<math>g(r) = \frac{4\pi}{3} G \rho r.</math>
The gravity {{math|''g′''}} at depth {{mvar|d}} is given by {{math|1=''g′'' = ''g''(1 − ''d''/''R'')}} where {{mvar|g}} is acceleration due to gravity on the surface of the Earth, {{mvar|d}} is depth and {{mvar|R}} is the radius of the [[Astronomy:Earth|Earth]].
If the density decreased linearly with increasing radius from a density {{math|''ρ''<sub>0</sub>}} at the center to {{math|''ρ''<sub>1</sub>}} at the surface, then {{math|1=''ρ''(''r'') = ''ρ''<sub>0</sub> − (''ρ''<sub>0</sub> − ''ρ''<sub>1</sub>) ''r'' / ''R''}}, and the dependence would be
:<math>g(r) = \frac{4\pi}{3} G \rho_0 r - \pi G \left(\rho_0-\rho_1\right) \frac{r^2}{R}.</math>

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===
Local differences in [[Earth:Topography|topography]] (such as the presence of mountains), [[Earth:Geology|geology]] (such as the density of rocks in the vicinity), and deeper [[Earth:Plate tectonics|tectonic structure]] cause local and regional differences in the Earth's gravitational field, known as [[Physics:Gravity anomaly|gravitational anomalies]].<ref>{{cite journal |first1=A. B. |last1=Watts |first2=S. F. |last2=Daly |title=Long wavelength gravity and topography anomalies |journal=Annual Review of Earth and Planetary Sciences |volume=9 |pages=415–418 |date=May 1981 |doi=10.1146/annurev.ea.09.050181.002215 |bibcode=1981AREPS...9..415W }}</ref> Some of these anomalies can be very extensive, resulting in bulges in [[Sea level|sea level]], and throwing [[Physics:Pendulum|pendulum]] clocks out of synchronisation.

The study of these anomalies forms the basis of gravitational [[Physics:Geophysics|geophysics]]. The fluctuations are measured with highly sensitive [[Physics:Gravimeter|gravimeter]]s, 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 [[Chemistry:Oil|oil]] and mineral deposits. Denser rocks (often containing mineral [[Earth:Ore|ore]]s) cause higher than normal local gravitational fields on the Earth's surface. Less dense [[Earth:Sedimentary rock|sedimentary rock]]s cause the opposite.

[[File:Spreading ridges volcanoes map-en.svg|thumb|A map of recent volcanic activity and ridge spreading. The areas where NASA GRACE measured gravity to be stronger than the theoretical gravity have a strong correlation with the positions of the volcanic activity and ridge spreading.]]
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 [[Physics:Buoyancy|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 [[Physics:Apparent weight|Apparent weight]] for details.

The gravitational effects of the [[Astronomy:Moon|Moon]] and the [[Astronomy:Sun|Sun]] (also the cause of the [[Astronomy:Tide|tide]]s) have a very small effect on the apparent strength of Earth's gravity, depending on their relative positions; typical variations are 2&nbsp;µm/s<sup>2</sup> (0.2 [[Physics:Gal (unit)|mGal]]) over the course of a day.

==Direction==
[[File:Plumb bob.jpg|thumb|upright|A plumb bob determines the local vertical direction]]

Gravity acceleration is a vector quantity, with direction in addition to [[Magnitude (mathematics)|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 [[Physics:Vertical deflection|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.<ref name=wolfram>[http://www.wolframalpha.com/widgets/view.jsp?id=d34e8683df527e3555153d979bcda9cf Gravitational Fields Widget as of Oct 25th, 2012]  – [[Physics:WolframAlpha|WolframAlpha]]</ref> The effect of latitude can be clearly seen with gravity in high-latitude cities: Anchorage (9.826&nbsp;m/s<sup>2</sup>), Helsinki (9.825&nbsp;m/s<sup>2</sup>), being about 0.5% greater than that in cities near the equator: Kuala Lumpur (9.776&nbsp;m/s<sup>2</sup>).<!-- (| V1 - V2 | / ((V1 + V2)/2)) * 100 = 0.52038% -->  The effect of altitude can be seen in Mexico City (9.776&nbsp;m/s<sup>2</sup>; altitude {{convert|2240|m|ft|}}), and by comparing Denver (9.798&nbsp;m/s<sup>2</sup>; {{convert|1616|m|ft}}) with Washington, D.C. (9.801&nbsp;m/s<sup>2</sup>; {{convert|30|m|ft}}), both of which are near 39°&nbsp;N. Measured values can be obtained from Physical and Mathematical Tables by T.M. Yarwood and F. Castle, Macmillan, revised edition 1970.<ref>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.</ref>

{|class="wikitable mw-collapsible"
|+ class="nowrap" | Acceleration due to gravity in various cities
|-
! Location !! m/s<sup>2</sup>!! ft/s<sup>2</sup>
|rowspan=18|
! Location !! m/s<sup>2</sup>!! ft/s<sup>2</sup>
|rowspan=18|
! Location !! m/s<sup>2</sup>!! ft/s<sup>2</sup>
|rowspan=18|
! Location !! m/s<sup>2</sup>!! ft/s<sup>2</sup>
|-
|Anchorage        || {{convert | 9.826 |m|ft||disp=tablecen}}
|[[Place:Helsinki|Helsinki]]         || {{convert | 9.825 |m|ft||disp=tablecen}}
|[[Place:Oslo|Oslo]]             || {{convert | 9.825 |m|ft||disp=tablecen}}
|[[Place:Copenhagen|Copenhagen]]       || {{convert | 9.821 |m|ft||disp=tablecen}}
|-
|[[Place:Stockholm|Stockholm]]        || {{convert | 9.818 |m|ft||disp=tablecen}}
|Manchester       || {{convert | 9.818 |m|ft||disp=tablecen}}
|[[Place:Amsterdam|Amsterdam]]        || {{convert | 9.817 |m|ft||disp=tablecen}}
|Kotagiri         || {{convert | 9.817 |m|ft||disp=tablecen}}
|-
|Birmingham       || {{convert | 9.817 |m|ft||disp=tablecen}}
|[[Place:London|London]]           || {{convert | 9.816 |m|ft||disp=tablecen}}
|Brussels         || {{convert | 9.815 |m|ft||disp=tablecen}}
|Frankfurt        || {{convert | 9.814 |m|ft||disp=tablecen}}
|-
|Seattle          || {{convert | 9.811 |m|ft||disp=tablecen}}
|[[Place:Paris|Paris]]            || {{convert | 9.809 |m|ft||disp=tablecen}}
|Montréal         || {{convert | 9.809 |m|ft||disp=tablecen}}
|{{wipe|Vancouver}}        || {{convert | 9.809 |m|ft||disp=tablecen}}
|-
|Istanbul         || {{convert | 9.808 |m|ft||disp=tablecen}}
|Toronto          || {{convert | 9.807 |m|ft||disp=tablecen}}
|Zurich           || {{convert | 9.807 |m|ft||disp=tablecen}}
|[[Place:Ottawa|Ottawa]]           || {{convert | 9.806 |m|ft||disp=tablecen}}
|-
|[[Place:Skopje|Skopje]]           || {{convert | 9.804 |m|ft||disp=tablecen}}
|{{wipe|Chicago}}          || {{convert | 9.804 |m|ft||disp=tablecen}}
|[[Place:Rome|Rome]]             || {{convert | 9.803 |m|ft||disp=tablecen}}
|[[Place:Wellington|Wellington]]       || {{convert | 9.803 |m|ft||disp=tablecen}}
|-
|{{wipe|New York City}}    || {{convert | 9.802 |m|ft||disp=tablecen}}
|[[Place:Lisbon|Lisbon]]           || {{convert | 9.801 |m|ft||disp=tablecen}}
|[[Place:Washington, D.C.|Washington, D.C.]] || {{convert | 9.801 |m|ft||disp=tablecen}}
|[[Place:Athens|Athens]]           || {{convert | 9.800 |m|ft||disp=tablecen}}
|-
|[[Place:Madrid|Madrid]]           || {{convert | 9.800 |m|ft||disp=tablecen}}
|[[Place:Melbourne|Melbourne]]        || {{convert | 9.800 |m|ft||disp=tablecen}}
|[[Place:Auckland|Auckland]]         || {{convert | 9.799 |m|ft||disp=tablecen}}
|Denver           || {{convert | 9.798 |m|ft||disp=tablecen}}
|-
|[[Place:Tokyo|Tokyo]]            || {{convert | 9.798 |m|ft||disp=tablecen}}
|[[Place:Buenos Aires|Buenos Aires]]     || {{convert | 9.797 |m|ft||disp=tablecen}}
|[[Place:Sydney|Sydney]]           || {{convert | 9.797 |m|ft||disp=tablecen}}
|[[Place:Nicosia|Nicosia]]          || {{convert | 9.797 |m|ft||disp=tablecen}}
|-
|{{wipe|Los Angeles}}      || {{convert | 9.796 |m|ft||disp=tablecen}}
|[[Place:Cape Town|Cape Town]]        || {{convert | 9.796 |m|ft||disp=tablecen}}
|Perth            || {{convert | 9.794 |m|ft||disp=tablecen}}
|[[Place:Kuwait City|Kuwait City]]      || {{convert | 9.792 |m|ft||disp=tablecen}}
|-
|[[Place:Taipei|Taipei]]           || {{convert | 9.790 |m|ft||disp=tablecen}}
|Rio de Janeiro   || {{convert | 9.788 |m|ft||disp=tablecen}}
|[[Place:Havana|Havana]]           || {{convert | 9.786 |m|ft||disp=tablecen}}
|[[Place:Kolkata|Kolkata]]          || {{convert | 9.785 |m|ft||disp=tablecen}}
|-
|[[Place:Hong Kong|Hong Kong]]        || {{convert | 9.785 |m|ft||disp=tablecen}}
|[[Place:Bangkok|Bangkok]]          || {{convert | 9.780 |m|ft||disp=tablecen}}
|[[Place:Manila|Manila]]           || {{convert | 9.780 |m|ft||disp=tablecen}}
|[[Place:Jakarta|Jakarta]]          || {{convert | 9.777 |m|ft||disp=tablecen}}
|-
|[[Place:Kuala Lumpur|Kuala Lumpur]]     || {{convert | 9.776 |m|ft||disp=tablecen}}
|[[Place:Singapore|Singapore]]        || {{convert | 9.776 |m|ft||disp=tablecen}}
|[[Place:Mexico City|Mexico City]]      || {{convert | 9.776 |m|ft||disp=tablecen}}
|Kandy            || {{convert | 9.775 |m|ft||disp=tablecen}}
|-
|}

==Mathematical models==
{{Main|Physics:Theoretical gravity}}

If the terrain is at sea level, we can estimate, for the Geodetic Reference System 1980, <math>g\{\phi\}</math>, the acceleration at latitude <math>\phi</math>:
:<math>\begin{align}
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{align}</math>

This is the International Gravity Formula 1967, the 1967 Geodetic Reference System Formula, Helmert's equation or [[Physics:Clairaut's theorem (gravity)#Formula|Clairaut's formula]].<ref name="IGF">[http://geophysics.ou.edu/solid_earth/notes/potential/igf.htm International Gravity formula] {{webarchive|url=https://web.archive.org/web/20080820093812/http://geophysics.ou.edu/solid_earth/notes/potential/igf.htm |date=2008-08-20 }}</ref>

An alternative formula for ''g'' as a function of latitude is the WGS ([[World Geodetic System]]) 84 Ellipsoidal Gravity Formula:<ref name="DoD-WGS84">{{Cite web |url=http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf |title='''''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 |access-date=2015-10-15 |archive-date=2014-04-11 |archive-url=https://web.archive.org/web/20140411101805/http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf |url-status=dead }}</ref>
:<math>g\{\phi\}= \mathbb{G}_e\left[\frac{1+k\sin^2\phi}{\sqrt{1-e^2\sin^2\phi}}\right],\,\!</math>
where,

*<math>a,\,b</math> are the equatorial and polar semi-axes, respectively;
*<math>e^2 = 1 - (b/a)^2</math> is the spheroid's [[Eccentricity (mathematics)|eccentricity]], squared;
*<math>\mathbb{G}_e,\,\mathbb{G}_p\,</math> is the defined gravity at the equator and poles, respectively;
*<math>k = \frac{b\,\mathbb{G}_p - a\,\mathbb{G}_e}{a\,\mathbb{G}_e}</math> (formula constant);

then, where <math>\mathbb{G}_p = 9.8321849378 \,\,\mathrm{m}\cdot\mathrm{s}^{-2}</math>,<ref name="DoD-WGS84"/>
:<math>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]</math>.

where the semi-axes of the earth are:
:<math>a = 6378137.0 \,\,\mbox{m} </math>
:<math>b = 6356752.314245 \,\,\mbox{m} </math>

The difference between the WGS-84 formula and Helmert's equation is less than 0.68&nbsp;&mu;m·s<sup>−2</sup>.

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

==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
:<math>F=G\frac{m_1m_2}{r^2} = \left(G\frac{M_\oplus}{r^2}\right)m</math>

where ''r'' is the distance between the centre of the Earth and the body (see below), and here we take <math>M_\oplus</math> to be the mass of the Earth and ''m'' 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
:<math>F=mg</math>

Comparing the two formulas it is seen that:
:<math>g=G\frac{M_\oplus}{r^2}</math>

So, to find the acceleration due to gravity at sea level, substitute the values of the [[Physics:Gravitational constant|gravitational constant]], ''G'', the Earth's [[Physics:Mass|mass]] (in kilograms), ''m''<sub>1</sub>, and the Earth's [[Radius|radius]] (in metres), ''r'', to obtain the value of ''g'':<ref>{{Cite web|title=Gravitation|url=https://ncert.nic.in/ncerts/l/iesc110.pdf&ved=2ahUKEwjd5Ifcss31AhX6S2wGHen0CFwQFnoECAgQAQ&usg=AOvVaw0td_Cs5UkX7ggc5yu-ms12|access-date=2022-01-25|website=www.ncert.nic}}</ref>
:<math>g=G\frac{M_\oplus}{r^2}=6.674 \cdot 10^{-11} \mathrm{{m}^3\,{kg}^{-1}{s}^{-2}} \times \frac{6\times 10^{24} \mathrm{kg}}{(6.4\times 10^6\mathrm{m})^2} = 9 + \frac{795}{2^{10}} \approx 9.77637 \, \mathrm{{m}\,{s}^{-2}}</math>
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 ''m''<sub>1</sub> as used in this calculation, and the value of ''[[Physics:Gravitational constant|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 [[Biography:Henry Cavendish|Henry Cavendish]].

==Measurement==
{{Main|Physics:Gravimetry}}

The measurement of Earth's gravity is called ''[[Physics:Gravimetry|gravimetry]]''.

===Satellite measurements===
{{excerpt|Gravimetry|Satellite gravimetry}}

==See also==
{{Div col}}
*{{annotated link|Astronomy:Escape velocity}}
**{{annotated link|Earth:Atmospheric escape}}
*{{annotated link|Earth:Figure of the Earth}}
*{{annotated link|Physics:Geopotential}}
**{{annotated link|Physics:Geopotential model}}
**{{annotated link|Physics:Bouguer anomaly}}
*{{annotated link|Physics:Gravitation of the Moon}}
*{{annotated link|Physics:Gravitational acceleration}}
*{{annotated link|Physics:Gravity}}
*{{annotated link|Physics:Gravity anomaly}}
*{{annotated link|Physics:Gravity of Mars}}
*{{annotated link|Physics:Newton's law of universal gravitation}}
*{{annotated link|Physics:Vertical deflection}}
{{Div col end}}

==References==
{{Reflist|30em}}

==External links==
* [http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html Altitude gravity calculator]
* [http://www.csr.utexas.edu/grace/ GRACE – Gravity Recovery and Climate Experiment]
* [https://web.archive.org/web/20140210113408/http://geodesy.curtin.edu.au/research/models/GGMplus/  GGMplus high resolution data (2013)]
* [http://www.universetoday.com/116801/the-potsdam-gravity-potato-shows-earths-gravity-variations/ Geoid 2011 model] Potsdam Gravity Potato

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