Jump to content

Earth mass

From Wikipedia, the free encyclopedia
(Redirected fromMass of Earth)

Earth mass
19th-century illustration ofArchimedes' quip of "give me aleverlong enough and afulcrumon which to place it, and I will move the earth "[1]
General information
Unit systemastronomy
Unit ofmass
SymbolM🜨
Conversions
1M🜨in...... is equal to...
SI base unit(5.9722±0.0006)×1024kg
U.S. customary1.3166×1025pounds

AnEarth mass(denoted asM🜨,MorME,where🜨andare the astronomicalsymbols for Earth), is a unit ofmassequal to the mass of the planetEarth.The current best estimate for the mass of Earth isM🜨=5.9722×1024kg,with a relative uncertainty of 10−4.[2]It is equivalent to anaverage densityof5515 kg/m3.Using the nearestmetric prefix,the Earth mass is approximately sixronnagrams,or 6.0 Rg.[3]

The Earth mass is a standardunit of massinastronomythat is used to indicate the masses of otherplanets,including rockyterrestrial planetsandexoplanets.OneSolar massis close to333000Earth masses. The Earth mass excludes the mass of theMoon.The mass of the Moon is about 1.2% of that of the Earth, so that the mass of the Earth–Moon system is close to6.0457×1024kg.

Most of the mass is accounted for byironandoxygen(c. 32% each),magnesiumandsilicon(c. 15% each),calcium,aluminiumandnickel(c. 1.5% each).

Precise measurement of the Earth mass is difficult, as it is equivalent to measuring thegravitational constant,which is the fundamentalphysical constantknown with least accuracy, due to the relative weakness of thegravitational force.The mass of the Earth was first measured with any accuracy (within about 20% of the correct value) in theSchiehallion experimentin the 1770s, and within 1% of the modern value in theCavendish experimentof 1798.

Unit of mass in astronomy

[edit]

The mass of Earth is estimated to be:

,

which can be expressed in terms ofsolar massas:

.

The ratio of Earth mass to lunar mass has been measured to great accuracy. The current best estimate is:[4][5]

Masses of noteworthyastronomical objectsrelative to the mass of Earth
Object Earth massME Ref
Moon 0.0123000371(4) [4]
Sun 332946.0487±0.0007 [2]
Mercury 0.0553 [6]
Venus 0.815 [6]
Earth 1 by definition
Mars 0.107 [6]
Jupiter 317.8 [6]
Saturn 95.2 [6]
Uranus 14.5 [6]
Neptune 17.1 [6]
Pluto 0.0025 [6]
Eris 0.0027
Gliese 667 Cc 3.8 [7]
Kepler-442b 1.0 – 8.2 [8]

The product ofMEand theuniversal gravitational constant(G) is known as thegeocentric gravitational constant(GME) and equals(398600441.8±0.8)×106m3s−2.It is determined using laser ranging data from Earth-orbiting satellites, such asLAGEOS-1.[9][10]GMEcan also be calculated by observing the motion of the Moon[11]or the period of a pendulum at various elevations, although these methods are less precise than observations of artificial satellites.

The relative uncertainty ofGMEis just2×10−9,considerably smaller than the relative uncertainty forMEitself.MEcan be found out only by dividingGMEbyG,andGis known only to a relative uncertainty of2.2×10−5,[12]soMEwill have the same uncertainty at best. For this reason and others, astronomers prefer to useGME,or mass ratios (masses expressed in units of Earth mass orSolar mass) rather than mass in kilograms when referencing and comparing planetary objects.

Composition

[edit]

Earth's density varies considerably, between less than2700 kg/m3in the uppercrustto as much as13000kg/m3in theinner core.[13]TheEarth's coreaccounts for 15% of Earth's volume but more than 30% of the mass, themantlefor 84% of the volume and close to 70% of the mass, while thecrustaccounts for less than 1% of the mass.[13]About 90% of the mass of the Earth is composed of theiron–nickel alloy (95% iron)in the core (30%), and thesilicon dioxides(c. 33%) andmagnesium oxide(c. 27%) in the mantle and crust. Minor contributions are fromiron(II) oxide(5%),aluminium oxide(3%) andcalcium oxide(2%),[14]besides numerous trace elements (inelementaryterms:ironandoxygenc. 32% each,magnesiumandsiliconc. 15% each,calcium,aluminiumandnickelc. 1.5% each).Carbonaccounts for 0.03%,waterfor 0.02%, and theatmospherefor about onepart per million.[15]

History of measurement

[edit]
Pendulums used in Mendenhallgravimeterapparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 byThomas C. Mendenhallprovided the most accurate relative measurements of the local gravitational field of the Earth.

The mass of Earth is measured indirectly by determining other quantities such as Earth's density, gravity, or gravitational constant. The first measurement in the 1770sSchiehallion experimentresulted in a value about 20% too low. TheCavendish experimentof 1798 found the correct value within 1%. Uncertainty was reduced to about 0.2% by the 1890s,[16]to 0.1% by 1930.[17]

Thefigure of the Earthhas been known to better than four significant digits since the 1960s (WGS66), so that since that time, the uncertainty of the Earth mass is determined essentially by the uncertainty in measuring thegravitational constant.Relative uncertainty was cited at 0.06% in the 1970s,[18]and at 0.01% (10−4) by the 2000s. The current relative uncertainty of 10−4amounts to6×1020kgin absolute terms, of the order of the mass of aminor planet(70% of the mass ofCeres).

Early estimates

[edit]

Before the direct measurement of thegravitational constant,estimates of the Earth mass were limited to estimating Earth's mean density from observation of thecrustand estimates on Earth's volume. Estimates on the volume of the Earth in the 17th century were based on a circumference estimate of 60 miles (97 km) to the degree of latitude, corresponding to a radius of5500 km(86% of theEarth's actual radiusof about6371 km), resulting in an estimated volume of about one third smaller than the correct value.[19]

The average density of the Earth was not accurately known. Earth was assumed to consist either mostly of water (Neptunism) or mostly ofigneous rock(Plutonism), both suggesting average densities far too low, consistent with a total mass of the order of1024kg.Isaac Newtonestimated, without access to reliable measurement, that the density of Earth would be five or six times as great as the density of water,[20]which is surprisingly accurate (the modern value is 5.515). Newton under-estimated the Earth's volume by about 30%, so that his estimate would be roughly equivalent to(4.2±0.5)×1024kg.

In the 18th century, knowledge ofNewton's law of universal gravitationpermitted indirect estimates on the mean density of the Earth, via estimates of (what in modern terminology is known as) thegravitational constant.Early estimates on the mean density of the Earth were made by observing the slight deflection of a pendulum near a mountain, as in theSchiehallion experiment.Newton considered the experiment inPrincipia,but pessimistically concluded that the effect would be too small to be measurable.

An expedition from 1737 to 1740 byPierre BouguerandCharles Marie de La Condamineattempted to determine the density of Earth by measuring the period of a pendulum (and therefore the strength of gravity) as a function of elevation. The experiments were carried out in Ecuador and Peru, onPichincha Volcanoand mountChimborazo.[21]Bouguer wrote in a 1749 paper that they had been able to detect a deflection of 8seconds of arc,the accuracy was not enough for a definite estimate on the mean density of the Earth, but Bouguer stated that it was at least sufficient to prove that the Earth was nothollow.[16]

Schiehallion experiment

[edit]

That a further attempt should be made on the experiment was proposed to theRoyal Societyin 1772 byNevil Maskelyne,Astronomer Royal.[22]He suggested that the experiment would "do honour to the nation where it was made" and proposedWhernsideinYorkshire,or theBlencathra-Skiddawmassif inCumberlandas suitable targets. The Royal Society formed the Committee of Attraction to consider the matter, appointing Maskelyne,Joseph BanksandBenjamin Franklinamongst its members.[23]The Committee despatched the astronomer and surveyorCharles Masonto find a suitable mountain.

After a lengthy search over the summer of 1773, Mason reported that the best candidate wasSchiehallion,a peak in the centralScottish Highlands.[23]The mountain stood in isolation from any nearby hills, which would reduce their gravitational influence, and its symmetrical east–west ridge would simplify the calculations. Its steep northern and southern slopes would allow the experiment to be sited close to itscentre of mass,maximising the deflection effect.Nevil Maskelyne,Charles HuttonandReuben Burrowperformed the experiment, completed by 1776. Hutton (1778) reported that the mean density of the Earth was estimated at9/5that of Schiehallion mountain.[24]This corresponds to a mean density about4+12higher than that of water (i.e., about4.5 g/cm3), about 20% below the modern value, but still significantly larger than the mean density of normal rock, suggesting for the first time that the interior of the Earth might be substantially composed of metal. Hutton estimated this metallic portion to occupy some20/31(or 65%) of the diameter of the Earth (modern value 55%).[25]With a value for the mean density of the Earth, Hutton was able to set some values toJérôme Lalande's planetary tables, which had previously only been able to express the densities of the major Solar System objects in relative terms.[24]

Cavendish experiment

[edit]

Henry Cavendish (1798) was the first to attempt to measure the gravitational attraction between two bodies directly in the laboratory. Earth's mass could be then found by combining two equations;Newton's second law,andNewton's law of universal gravitation.

In modern notation, the mass of the Earth is derived from thegravitational constantand the meanEarth radiusby

Wheregravity of Earth,"little g", is

.

Cavendish found a mean density of5.45 g/cm3,about 1% below the modern value.

19th century

[edit]
Experimental setup byFrancis BailyandHenry Fosterto determine the density of Earth using the Cavendish method.

While the mass of the Earth is implied by stating the Earth's radius and density, it was not usual to state the absolute mass explicitly prior to the introduction ofscientific notationusingpowers of 10in the later 19th century, because the absolute numbers would have been too awkward. Ritchie (1850) gives the mass of theEarth's atmosphereas "11,456,688,186,392,473,000 lbs". (1.1×1019lb=5.0×1018kg,modern value is5.15×1018kg) and states that "compared with the weight of the globe this mighty sum dwindles to insignificance".[26]

Absolute figures for the mass of the Earth are cited only beginning in the second half of the 19th century, mostly in popular rather than expert literature. An early such figure was given as "14septillionpounds "(14 Quadrillionen Pfund) [6.5×1024kg] in Masius (1859).[27]Beckett(1871) cites the "weight of the earth" as "5842quintilliontons"[5.936×1024kg].[28]The "mass of the earth in gravitational measure" is stated as "9.81996×63709802"inThe New Volumes of the Encyclopaedia Britannica(Vol. 25, 1902) with a "logarithm of earth's mass" given as "14.600522" [3.98586×1014]. This is thegravitational parameterin m3·s−2(modern value3.98600×1014) and not the absolute mass.

Experiments involving pendulums continued to be performed in the first half of the 19th century. By the second half of the century, these were outperformed by repetitions of the Cavendish experiment, and the modern value ofG(and hence, of the Earth mass) is still derived from high-precision repetitions of the Cavendish experiment.

In 1821,Francesco Carlinidetermined a density value ofρ=4.39 g/cm3through measurements made with pendulums in theMilanarea. This value was refined in 1827 byEdward Sabineto4.77 g/cm3,and then in 1841 by Carlo Ignazio Giulio to4.95 g/cm3.On the other hand,George Biddell Airysought to determine ρ by measuring the difference in the period of a pendulum between the surface and the bottom of a mine.[29] The first tests and experiments took place in Cornwall between 1826 and 1828. The experiment was a failure due to a fire and a flood. Finally, in 1854, Airy got the value6.6 g/cm3by measurements in a coal mine in Harton, Sunderland. Airy's method assumed that the Earth had a spherical stratification. Later, in 1883, the experiments conducted by Robert von Sterneck (1839 to 1910) at different depths in mines of Saxony and Bohemia provided the average density valuesρbetween 5.0 and6.3 g/cm3.This led to the concept of isostasy, which limits the ability to accurately measureρ,by either the deviation from vertical of a plumb line or using pendulums. Despite the little chance of an accurate estimate of the average density of the Earth in this way,Thomas Corwin Mendenhallin 1880 realized a gravimetry experiment in Tokyo and at the top ofMount Fuji.The result wasρ=5.77 g/cm3.[citation needed]

Modern value

[edit]

The uncertainty in the modern value for the Earth's mass has been entirely due to the uncertainty in thegravitational constantGsince at least the 1960s.[30]Gis notoriously difficult to measure, and some high-precision measurements during the 1980s to 2010s have yielded mutually exclusive results.[31]Sagitov (1969) based on the measurement ofGby Heyl and Chrzanowski (1942) cited a value ofME=5.973(3)×1024kg(relative uncertainty5×10−4).

Accuracy has improved only slightly since then. Most modern measurements are repetitions of the Cavendish experiment, with results (within standard uncertainty) ranging between 6.672 and6.676×10−11m3/kg/s2(relative uncertainty3×10−4) in results reported since the 1980s, although the 2014CODATArecommended value is close to6.674×10−11m3/kg/s2with a relative uncertainty below 10−4.TheAstronomical Almanach Onlineas of 2016 recommends a standard uncertainty of1×10−4for Earth mass,ME5.9722(6)×1024kg[2]

Variation

[edit]

Earth's mass is variable, subject to both gain and loss due to the accretion of in-falling material, including micrometeorites and cosmic dust and the loss of hydrogen and helium gas, respectively. The combined effect is a net loss of material, estimated at 5.5×107kg (5.4×104long tons) per year. This amount is 10−17of the total earth mass.[citation needed]The5.5×107kgannual net loss is essentially due to 100,000 tons lost due toatmospheric escape,and an average of 45,000 tons gained from in-falling dust and meteorites. This is well within the mass uncertainty of 0.01% (6×1020kg), so the estimated value of Earth's mass is unaffected by this factor.

Mass loss is due to atmospheric escape of gases. About 95,000 tons of hydrogen per year[32](3 kg/s) and 1,600 tons of helium per year[33]are lost through atmospheric escape. The main factor in mass gain is in-falling material,cosmic dust,meteors,etc. are the most significant contributors to Earth's increase in mass. The sum of material is estimated to be37000to 78000tonsannually,[34][35]although this can vary significantly; to take an extreme example, theChicxulub impactor,with a midpoint mass estimate of2.3×1017kg,[36]added 900 million times that annual dustfall amount to the Earth's mass in a single event.

Additional changes in mass are due to themass–energy equivalence principle,although these changes are relatively negligible. Mass loss due to the combination ofnuclear fissionand naturalradioactive decayis estimated to amount to 16 tons per year.[citation needed]

An additional loss due tospacecraftonescape trajectorieshas been estimated at65 tons per yearsince the mid-20th century. Earth lost about 3473 tons in the initial 53 years of the space age, but the trend is currently decreasing.[citation needed]

See also

[edit]

References

[edit]
  1. ^Attributed byPappus of Alexandria(Synagoge[Συναγωγή] VIII, 4th century), as« Δός μοί ποῦ στῶ, καὶ κινῶ τὴν Γῆν ».Engraving fromMechanic's Magazine(cover of bound Volume II, Knight & Lacey, London, 1824).
  2. ^abcThe cited value is the recommended value published by theInternational Astronomical Unionin 2009 (see2016 "Selected Astronomical Constants"Archived15 February 2016 at theWayback Machinein"The Astronomical Almanac Online"(PDF).USNO/UKHO.Archived fromthe originalon 24 December 2016.Retrieved8 February2016.).
  3. ^Lawler, Daniel."Earth now weighs six ronnagrams: New metric prefixes voted in".phys.org.Retrieved21 November2022.
  4. ^abPitjeva, E.V.; Standish, E.M. (1 April 2009)."Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit".Celestial Mechanics and Dynamical Astronomy.103(4): 365–372.Bibcode:2009CeMDA.103..365P.doi:10.1007/s10569-009-9203-8.S2CID121374703.
  5. ^Luzum, Brian; Capitaine, Nicole; Fienga, Agnès; et al. (10 July 2011)."The IAU 2009 system of astronomical constants: the report of the IAU working group on numerical standards for Fundamental Astronomy".Celestial Mechanics and Dynamical Astronomy.110(4): 293–304.Bibcode:2011CeMDA.110..293L.doi:10.1007/s10569-011-9352-4.
  6. ^abcdefgh"Planetary Fact Sheet – Ratio to Earth".nssdc.gsfc.nasa.gov.Retrieved12 February2016.
  7. ^"The Habitable Exoplanets Catalog".Planetary Habitability Laboratory @ UPR Arecibo.
  8. ^"HEC: Data of Potential Habitable Worlds".Archived fromthe originalon 1 June 2012.Retrieved17 February2016.
  9. ^Ries, J.C.; Eanes, R.J.; Shum, C.K.; Watkins, M.M. (20 March 1992). "Progress in the determination of the gravitational coefficient of the Earth".Geophysical Research Letters.19(6): 529.Bibcode:1992GeoRL..19..529R.doi:10.1029/92GL00259.
  10. ^Lerch, Francis J.; Laubscher, Roy E.; Klosko, Steven M.; Smith, David E.; Kolenkiewicz, Ronald; Putney, Barbara H.; Marsh, James G.; Brownd, Joseph E. (December 1978). "Determination of the geocentric gravitational constant from laser ranging on near-Earth satellites".Geophysical Research Letters.5(12): 1031–1034.Bibcode:1978GeoRL...5.1031L.doi:10.1029/GL005i012p01031.
  11. ^Shuch, H. Paul (July 1991)."Measuring the mass of the earth: the ultimate moonbounce experiment"(PDF).Proceedings, 25th Conference of the Central States VHF Society:25–30.Retrieved28 February2016.
  12. ^"2022 CODATA Value: Newtonian constant of gravitation".The NIST Reference on Constants, Units, and Uncertainty.NIST.May 2024.Retrieved18 May2024.
  13. ^abSeestructure of the Earth:inner corevolume 0.7%, density 12,600–13,000, mass c. 1.6%;outer corevol. 14.4%, density 9,900–12,200 mass c. 28.7–31.7%. Hazlett, James S.; Monroe, Reed; Wicander, Richard (2006).Physical Geology: Exploring the Earth(6. ed.). Belmont: Thomson. p. 346.
  14. ^Jackson, Ian (1998).The Earth's Mantle – Composition, Structure, and Evolution.Cambridge University Press. pp. 311–378.
  15. ^Thehydrosphere(Earth'soceans) account for about 0.02%2.3×10−4of total mass,Carbonfor about 0.03% of the crust, or3×10−6of total mass,Earth's atmospherefor about8.6×10−7of total mass.Biomassis estimated at 10−10(5.5×1014kg,see Bar-On, Yinon M.; Phillips, Rob; Milo, Ron. "The biomass distribution on Earth"Proc. Natl. Acad. Sci. USA,2018).
  16. ^abPoynting, J.H. (1913).The Earth: its shape, size, weight and spin.Cambridge. pp. 50–56.
  17. ^P. R. Heyl,A redetermination of the constant of gravitation,National Bureau of Standards Journal of Research5 (1930), 1243–1290.
  18. ^IAU (1976) System of Astronomical Constants
  19. ^Mackenzie, A. Stanley,The laws of gravitation; memoirs by Newton, Bouguer and Cavendish, together with abstracts of other important memoirs,American Book Company (1900 [1899]), p. 2.
  20. ^"Sir Isaac Newton thought it probable, that the mean density of the earth might be five or six times as great as the density of water; and we have now found, by experiment, that it is very little less than what he had thought it to be: so much justness was even in the surmises of this wonderful man!" Hutton (1778), p. 783
  21. ^Ferreiro, Larrie (2011).Measure of the Earth: The Enlightenment Expedition that Reshaped Our World.New York: Basic Books.ISBN978-0-465-01723-2.
  22. ^Maskelyne, N. (1772). "A proposal for measuring the attraction of some hill in this Kingdom".Philosophical Transactions of the Royal Society.65:495–499.Bibcode:1775RSPT...65..495M.doi:10.1098/rstl.1775.0049.
  23. ^abDanson, Edwin (2006).Weighing the World.Oxford University Press. pp. 115–116.ISBN978-0-19-518169-2.
  24. ^abHutton, C. (1778)."An Account of the Calculations Made from the Survey and Measures Taken at Schehallien".Philosophical Transactions of the Royal Society.68:689–788.doi:10.1098/rstl.1778.0034.
  25. ^Hutton (1778), p. 783.
  26. ^Archibald Tucker Ritchie,The Dynamical Theory of the Formation of the Earthvol. 2 (1850), Longman, Brown, Green and Longmans, 1850,p. 280.
  27. ^J.G.Mädler in: Masius, Hermann,Die gesammten Naturwissenschaften,vol. 3 (1859), p. 562.
  28. ^Edmund Beckett Baron Grimthorpe,Astronomy Without Mathematics(1871), p. 254. Max Eyth,Der Kampf um die Cheopspyramide: Erster Band(1906),p. 417cites the "weight of the globe" (Das Gewicht des Erdballs) as "5273 quintillion tons".
  29. ^ Poynting, John Henry (1894).The Mean Density of the Earth.London: Charles Griffin. pp.22–24.
  30. ^"Since the geocentric gravitational constant [...] is now determined to a relative accuracy of 10−6,our knowledge of the mass of the earth is entirely limited by the low accuracy of our knowledge of the Cavendish gravitational constant. "Sagitov (1970 [1969]), p. 718.
  31. ^Schlamminger, Stephan (18 June 2014). "Fundamental constants: A cool way to measure big G".Nature.510(7506): 478–480.Bibcode:2014Natur.510..478S.doi:10.1038/nature13507.PMID24965646.S2CID4396011.
  32. ^"Fantasy and Science Fiction: Science by Pat Murphy & Paul Doherty".
  33. ^"Earth Loses 50,000 Tonnes of Mass Every Year".SciTech Daily.5 February 2012.
  34. ^Zook, Herbert A. (2001), "Spacecraft Measurements of the Cosmic Dust Flux",Accretion of Extraterrestrial Matter Throughout Earth's History,pp. 75–92,doi:10.1007/978-1-4419-8694-8_5,ISBN978-1-4613-4668-5
  35. ^Carter, Lynn."How many meteorites hit Earth each year?".Ask an Astronomer.The Curious Team, Cornell University.Retrieved6 February2016.
  36. ^Durand-Manterola, H. J.; Cordero-Tercero, G. (2014). "Assessments of the energy, mass and size of the Chicxulub Impactor".arXiv:1403.6391[astro-ph.EP].