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Gravity of Earth

From Wikipedia, the free encyclopedia

Earth's gravity measured by NASAGRACE mission, showing deviations from thetheoretical gravity of an idealized, smooth Earth, the so-calledEarth ellipsoid. Red shows the areas where gravity is stronger than the smooth, standard value, and blue reveals areas where gravity is weaker.[1]
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Geophysics

Thegravity of Earth, denoted byg, is thenetacceleration that is imparted to objects due to the combined effect ofgravitation (frommass distribution withinEarth) and thecentrifugal force (from theEarth's rotation).[2][3]It is avector quantity, whose direction coincides with aplumb bob and strength or magnitude is given by thenormg=‖g‖{\displaystyle g=\|{\mathit {\mathbf {g} }}\|}.

InSI units, this acceleration is expressed inmetres per second squared (in symbols,m/s2 or m·s−2) or equivalently innewtons perkilogram (N/kg or N·kg−1). Near Earth's surface, the acceleration due to gravity, accurate to 2significant figures, is 9.8 m/s2 (32 ft/s2). This means that, ignoring the effects ofair resistance, the vertical component ofvelocity of an objectfalling freely will increase in the downwards direction by about 9.8 metres per second (32 ft/s) every second.

The precise strength of Earth's gravity varies with location. The conventional value forstandard gravity is9.80665 m⋅s−2‍[4] by definition, originally adopted by theCGPM in 1901.[5]: 159  This quantity is denoted variously asgn,ge,g0, or simplyg (which is also used for the variable local value).

Theweight of an object on Earth's surface is the downwards force on that object, given byNewton's second law of motion, orF =ma (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 oftidal effects.

Variation in magnitude

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A non-rotating perfectsphere of uniform mass density, or whose density varies solely with distance from the centre (spherical symmetry), would produce agravitational field of uniform magnitude at all points on itssurface. The Earth is rotating and is also not spherically symmetric; rather, it is slightly flatter at the poles while bulging at the Equator: anoblate 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 theNevado Huascarán mountain in Peru to 9.8337 m/s2 at the surface of theArctic Ocean.[6]In large cities, it ranges from 9.7806 m/s2[7] inKuala Lumpur,Mexico City, andSingapore to 9.825 m/s2 inOslo andHelsinki.

Conventional value

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In 1901, the thirdGeneral 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 at thePavillon de Breteuil near Paris in 1888, with a theoretical correction applied in order to convert to a latitude of 45° at sea level.[8] 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.[9] It is also used to define the unitskilogram force andpound force.

Latitude

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

The surface of the Earth is rotating, so it isnot an inertial frame of reference. At latitudes nearer the Equator, the outwardcentrifugal 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'sequatorial 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 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][10]

Altitude

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The gravity of a body is inversely proportional to the distance form that body. This graph displays this relation as the distance changes from the surface (0 km) to 30000 km.
Earth's gravity vs. distance from it, from the surface to30000 km
Earth vsMars vsMoongravity atelevation

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

The effect of ground elevation depends on the density of the ground (seeLocal geology). 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:

Calculator
Re6,371.00877 km
g09.80665 m/s2
h0 km
gh9.80665 m/s2
gh=g0(ReRe+h)2{\displaystyle g_{h}=g_{0}\left({\frac {R_{\mathrm {e} }}{R_{\mathrm {e} }+h}}\right)^{2}}

where

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

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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)
Earth's radial density distribution according to thePreliminary Reference Earth Model (PREM).[13]
Earth's gravity according to the Preliminary Reference Earth Model (PREM).[13] 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 distancer from the center of the Earth can be obtained by assuming that the Earth's density is spherically symmetric. The force of gravity at a radiusr depends only on the mass inside the sphere of that radius. All the contributions from outside cancel out as a consequence of theinverse-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[14]

g(r)=−GM(r)r2.{\displaystyle g(r)=-{\frac {GM(r)}{r^{2}}}.}

whereG is thegravitational constant andM(r) is the total mass enclosed within radiusr. This result is known as theShell theorem; it tookIsaac Newton 20 years to prove this result, delaying his work on gravity.[15]: 13 

If the Earth had a constant densityρ, the mass would beM(r) = (4/3)πρr3 and the dependence of gravity on depth would be

g(r)=4π3Gρr.{\displaystyle g(r)={\frac {4\pi }{3}}G\rho r.}

The gravityg′ at depthd is given byg′ =g(1 −d/R) whereg is acceleration due to gravity on the surface of the Earth,d is depth andR is the radius of theEarth.If the density decreased linearly with increasing radius from a densityρ0 at the center toρ1 at the surface, thenρ(r) =ρ0 − (ρ0 −ρ1)r /R, and the dependence would be

g(r)=4π3Gρ0r−πG(ρ0−ρ1)r2R.{\displaystyle g(r)={\frac {4\pi }{3}}G\rho _{0}r-\pi G\left(\rho _{0}-\rho _{1}\right){\frac {r^{2}}{R}}.}

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

Local topography and geology

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See also:Physical geodesy
Further information:Gravity anomaly § Computation

Local differences intopography (such as the presence of mountains),geology (such as the density of rocks in the vicinity), and deepertectonic structure cause local and regional differences in the Earth's gravitational field, known asgravity anomalies.[16] Some of these anomalies can be very extensive, resulting in bulges insea level, and throwingpendulum clocks out of synchronisation.

The study of these anomalies forms the basis of gravitationalgeophysics. The fluctuations are measured with highly sensitivegravimeters, the effect of topography and other known factors is subtracted, and from the resulting data conclusions are drawn. Such techniques are now used byprospectors to findoil andmineral deposits. Denser rocks (often containing mineralores) cause higher than normal local gravitational fields on the Earth's surface. Less densesedimentary rocks cause the opposite.

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

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In air or water, objects experience a supportingbuoyancy 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; seeApparent weight for details.

The gravitational effects of theMoon and theSun (also the cause of thetides) have a very small effect on the apparent strength of Earth's gravity, depending on their relative positions; typical variations are 2 μm/s2 (0.2mGal) over the course of a day.

Direction

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Main article:Vertical direction
A plumb bob determines the local vertical direction

Gravity acceleration is avector quantity, withdirection in addition tomagnitude. In a spherically symmetric Earth, gravity would point directly towards the sphere's centre. As theEarth's figure is slightly flatter, there are consequently significant deviations in the direction of gravity: essentially the difference betweengeodetic latitude andgeocentric latitude. Smaller deviations, calledvertical deflection, are caused by local mass anomalies, such as mountains.

Comparative values worldwide

[edit]

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]

Acceleration due to gravity in various cities
Locationm/s2ft/s2Locationm/s2ft/s2Locationm/s2ft/s2Locationm/s2ft/s2
Anchorage9.82632.24Helsinki9.82532.23Oslo9.82532.23Copenhagen9.82132.22
Stockholm9.81832.21Manchester9.81832.21Amsterdam9.81732.21Kotagiri9.81732.21
Birmingham9.81732.21London9.81632.20Brussels9.81532.20Frankfurt9.81432.20
Seattle9.81132.19Paris9.80932.18Montréal9.80932.18Vancouver9.80932.18
Istanbul9.80832.18Toronto9.80732.18Zurich9.80732.18Ottawa9.80632.17
Skopje9.80432.17Chicago9.80432.17Rome9.80332.16Wellington9.80332.16
New York City9.80232.16Lisbon9.80132.16Washington, D.C.9.80132.16Athens9.80032.15
Madrid9.80032.15Melbourne9.80032.15Auckland9.79932.15Denver9.79832.15
Tokyo9.79832.15Buenos Aires9.79732.14Sydney9.79732.14Nicosia9.79732.14
Los Angeles9.79632.14Cape Town9.79632.14Perth9.79432.13Kuwait City9.79232.13
Taipei9.79032.12Rio de Janeiro9.78832.11Havana9.78632.11Kolkata9.78532.10
Hong Kong9.78532.10Bangkok9.78032.09Manila9.78032.09Jakarta9.77732.08
Kuala Lumpur9.77632.07Singapore9.77632.07Mexico City9.77632.07Kandy9.77532.07

Mathematical models

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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 \}}, the acceleration at latitudeϕ{\displaystyle \phi }:

g{ϕ}=9.780327m⋅s−2(1+0.0053024sin2⁡ϕ−0.0000058sin2⁡2ϕ),=9.780327m⋅s−2(1+0.0052792sin2⁡ϕ+0.0000232sin4⁡ϕ),=9.780327m⋅s−2(1.0053024−0.0053256cos2⁡ϕ+0.0000232cos4⁡ϕ),=9.780327m⋅s−2(1.0026454−0.0026512cos⁡2ϕ+0.0000058cos2⁡2ϕ){\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 theInternational Gravity Formula 1967, the 1967 Geodetic Reference System Formula, Helmert's equation orClairaut's formula.[19]

An alternative formula forg as a function of latitude is the WGS (World Geodetic System) 84 EllipsoidalGravity Formula:[20]

g{ϕ}=Ge[1+ksin2⁡ϕ1−e2sin2⁡ϕ],{\displaystyle g\{\phi \}=\mathbb {G} _{e}\left[{\frac {1+k\sin ^{2}\phi }{\sqrt {1-e^{2}\sin ^{2}\phi }}}\right],}

where

then, whereGp=9.8321849378m⋅s−2{\displaystyle \mathbb {G} _{p}=9.8321849378\,\,\mathrm {m} \cdot \mathrm {s} ^{-2}},[20]

g{ϕ}=9.7803253359m⋅s−2[1+0.001931852652sin2⁡ϕ1−0.0066943799901sin2⁡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=6378137.0m{\displaystyle a=6378137.0\,\,\mathrm {m} }
b=6356752.314245m{\displaystyle b=6356752.314245\,\,\mathrm {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).

Estimatingg from the law of universal gravitation

[edit]

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

F=Gm1m2r2=(GM⊕r2)m{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}=\left(G{\frac {M_{\oplus }}{r^{2}}}\right)m}

wherer is the distance between the centre of the Earth and the body (see below), and here we takeM⊕{\displaystyle M_{\oplus }} to be the mass of the Earth andm to be the mass of the body.

Additionally,Newton's second law,F =ma, wherem is mass anda is acceleration, here tells us that

F=mg{\displaystyle F=mg}

Comparing the two formulas it is seen that:

g=GM⊕r2{\displaystyle g=G{\frac {M_{\oplus }}{r^{2}}}}

So, to find the acceleration due to gravity at sea level, substitute the values of thegravitational constant,G, the Earth'smass (in kilograms),m1, and the Earth'sradius (in metres),r, to obtain the value ofg:[21]

g=GM⊕r2=6.674×10−11 m3⋅kg−1⋅s−2×6×1024 kg(6.4×106 m)2≈9.77637 m⋅s−2{\displaystyle g=G{\frac {M_{\oplus }}{r^{2}}}=6.674\times 10^{-11}\ \mathrm {{m}^{3}{\cdot }{kg}^{-1}{\cdot }{s}^{-2}} \times {\frac {6\times 10^{24}\ \mathrm {kg} }{(6.4\times 10^{6}\ \mathrm {m} )^{2}}}\approx 9.77637\ \mathrm {{m}{\cdot }{s}^{-2}} }

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 forr.

The value obtained agrees approximately with the measured value ofg. The difference may be attributed to several factors, mentioned above under "Variation in magnitude":

  • The Earth is nothomogeneous
  • The Earth is not a perfect sphere, and an average value must be used for its radius
  • This calculated value ofg 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 ofr andm1 as used in this calculation, and the value ofG is also rather difficult to measure precisely.

IfG,g andr are known then a reverse calculation will give an estimate of the mass of the Earth. This method was used byHenry Cavendish.

Measurement

[edit]
Main article:Gravimetry

The measurement of Earth's gravity is calledgravimetry.

Satellite measurements

[edit]
This section is an excerpt fromGravimetry § Satellite gravimetry.[edit]
Gravity anomaly map from GRACE

Currently, the static and time-variable Earth's gravity field parameters are determined using modern satellite missions, such asGOCE,CHAMP,Swarm,GRACE and GRACE-FO.[22][23] The lowest-degree parameters, including the Earth's oblateness and geocenter motion are best determined fromsatellite 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 aspherical-harmonic expansion of the Earth's gravitational potential, but alternative presentations, such as maps ofgeoid undulations or gravity anomalies, are also produced.

TheGravity Recovery and Climate Experiment (GRACE) consisted of two satellites that detected gravitational changes across the Earth. Also these changes could be presented as gravity anomaly temporal variations. TheGravity Recovery and Interior Laboratory (GRAIL) also consisted of two spacecraft orbiting the Moon, which orbited for three years before their deorbit in 2015.

See also

[edit]

References

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  1. ^NASA/JPL/University of Texas Center for Space Research."PIA12146: GRACE Global Gravity Animation".Photojournal. NASA Jet Propulsion Laboratory. Retrieved30 December 2013.
  2. ^abBoynton, Richard (2001)."Precise Measurement of Mass"(PDF).Sawe Paper No. 3147. Arlington, Texas: S.A.W.E., Inc. Archived fromthe original(PDF) on 27 February 2007. Retrieved22 December 2023.
  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."
  4. ^"2022 CODATA Value: standard acceleration of gravity".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
  5. ^International Bureau of Weights and Measures (Dec 2022),The International System of Units (SI)(PDF), vol. 2 (9th ed.),ISBN 978-92-822-2272-0,archived from the original on 18 October 2021
  6. ^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.S2CID 54867946.
  7. ^"Wolfram|Alpha Gravity in Kuala Lumpur", Wolfram Alpha, accessed November 2020
  8. ^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.
  9. ^Resolution of the 3rd CGPM (1901), page 70 (in cm/s2).BIPM – Resolution of the 3rd CGPM
  10. ^"Curious About Astronomy?".Cornell University. Archived fromthe original on 28 July 2013. Retrieved22 December 2023.
  11. ^"I feel 'lighter' when up a mountain but am I?", National Physical Laboratory FAQ
  12. ^"The G's in the Machine"Archived 2020-09-21 at theWayback Machine, NASA, see "Editor's note #2"
  13. ^abA. 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.
  14. ^Tipler, Paul A. (1999).Physics for scientists and engineers (4th ed.). New York: W.H. Freeman/Worth Publishers. pp. 336–337.ISBN 9781572594913.
  15. ^Weinberg, Steven (1972).Gravitation and cosmology. John Wiley & Sons.ISBN 9780471925675.
  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 formulaArchived 2008-08-20 at theWayback Machine
  20. ^ab"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"(PDF). Archived fromthe original(PDF) on 2014-04-11. Retrieved2015-10-15.
  21. ^"Gravitation".www.ncert.nic. Retrieved2022-01-25.
  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.hdl:10281/240694.
  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.

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