Earth mass

Earth mass
19th-century illustration ofArchimedes' quip of "give me aleverlong enough and afulcrumon which to place it, and I will move the earth "
General information
Unit system	astronomy
Unit of	mass
Symbol	M🜨
Conversions
1M🜨in...	... is equal to...
SI base unit	(5.9722±0.0006)×1024kg
U.S. customary	≈ 1.3166×1025pounds

AnEarth mass(denoted asM_🜨,M_♁orM_E,where🜨and♁are 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.It is equivalent to anaverage densityof5515 kg/m³.Using the nearestmetric prefix,the Earth mass is approximately sixronnagrams,or 6.0 Rg.

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

Further information:Solar mass,Standard gravitational parameter,andGaussian gravitational constant

The mass of Earth is estimated to be:

$M_{\oplus }=(5.9722\;\pm \;0.0006)\times 10^{24}\;\mathrm {kg}$,

which can be expressed in terms ofsolar massas:

$M_{\oplus }={\frac {1}{332\;946.0487\;\pm \;0.0007}}\;M_{\odot }\approx 3.003\times 10^{-6}\;M_{\odot }$.

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

$M_{\oplus }/M_{L}=81.3005678\;\pm \;0.0000027$

Masses of noteworthyastronomical objectsrelative to the mass of Earth

Object	Earth massME	Ref
Moon	0.0123000371(4)
Sun	332946.0487±0.0007
Mercury	0.0553
Venus	0.815
Earth	1	by definition
Mars	0.107
Jupiter	317.8
Saturn	95.2
Uranus	14.5
Neptune	17.1
Pluto	0.0025
Eris	0.0027
Gliese 667 Cc	3.8
Kepler-442b	1.0 – 8.2

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

The relative uncertainty ofGM_Eis just2×10⁻⁹,considerably smaller than the relative uncertainty forM_Eitself.M_Ecan be found out only by dividingGM_EbyG,andGis known only to a relative uncertainty of2.2×10⁻⁵,soM_Ewill have the same uncertainty at best. For this reason and others, astronomers prefer to useGM_E,or mass ratios (masses expressed in units of Earth mass orSolar mass) rather than mass in kilograms when referencing and comparing planetary objects.

Composition

Further information:Internal structure of EarthandAbundance of elements in Earth's crust

Earth's density varies considerably, between less than2700 kg/m³in the uppercrustto as much as13000kg/m³in theinner core.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.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%),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.

History of measurement

Main article:Gravitational constant § History of measurement

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,to 0.1% by 1930.

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,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

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.

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,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.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.

Schiehallion experiment

Main article:Schiehallion experiment

That a further attempt should be made on the experiment was proposed to theRoyal Societyin 1772 byNevil Maskelyne,Astronomer Royal.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.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.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.This corresponds to a mean density about4+1⁄2higher 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%).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.

Cavendish experiment

Main article:Cavendish experiment

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

$M_{\oplus }={\frac {GM_{\oplus }}{G}}={\frac {gR_{\oplus }^{2}}{G}}.$

Wheregravity of Earth,"little g", is

$g=G{\frac {M_{\oplus }}{R_{\oplus }^{2}}}$.

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

19th century

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".

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).Beckett(1871) cites the "weight of the earth" as "5842quintilliontons"[5.936×10²⁴kg].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. 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.

Modern value

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.Gis notoriously difficult to measure, and some high-precision measurements during the 1980s to 2010s have yielded mutually exclusive results.Sagitov (1969) based on the measurement ofGby Heyl and Chrzanowski (1942) cited a value ofM_E=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,M_E5.9722(6)×10²⁴kg

Variation

Main article:Atmospheric escape

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.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(3 kg/s) and 1,600 tons of helium per yearare 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,although this can vary significantly; to take an extreme example, theChicxulub impactor,with a midpoint mass estimate of2.3×10¹⁷kg,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.

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.

See also

• Abundance of elements in Earth's crust
• Cavendish experiment
• Earth radius
• Orders of magnitude (mass)
• Planetary mass
• Solar mass
• Internal structure of Earth