Earth mass

Earth Mass
Common symbols	$M_\oplus$, $M_\mathrm{T}$, $M_\mathrm{E}$ or $E$
SI unit	kilogram (kg)
Other units	gram (g) [CGS] Solar Mass (M⊙) [IAU]
In SI base units	(5.9722±0.0006)×10^24 kg
SI dimension	$[M_\oplus]=\mathrm{M}$ (mass)
Derivations from other quantities	M⊕ = g · R2 ⊕ ∕ G M⊕ = ρ · V M⊕ = μ⊕ ∕ G

Earth mass (M_⊕, where ⊕ is the standard astronomical symbol for planet Earth) is the unit of mass equal to that of Earth. This value includes the atmosphere but excludes the moon. The current best estimate for Earth mass is M_⊕ = (5.9722±0.0006)×10²⁴ kg^[1][2] Earth mass is a standard unit of mass in astronomy that is used to indicate the masses of other planets, including rocky terrestrial planets and exoplanets.

Value

The mass of Earth is estimated to be:

$M_\oplus=(5.9722\;\pm\;0.0006)\times10^{24}\;\mathrm{kg}$,

or:

$M_\oplus=\frac{1}{332\;946.0487\;\pm\;0.0007}\;\mathrm{M_\odot}$.

Masses of noteworthy astronomical objects relative to the mass of Earth

Object	Earth mass M⊕	Ref
Moon	0.0123000371(4)	[3]
Sun	332946.0487±0.0007	[2]
Mercury	0.0553	[4]
Venus	0.815	[4]
Earth	1	By definition
Mars	0.107	[4]
Jupiter	317.8	[4]
Saturn	95.2	[4]
Uranus	14.5	[4]
Neptune	17.1	[4]
Gliese 667 Cc	3.8	[5]
Kepler-442b	1.0 – 8.2	[6]

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

$M_\oplus/M_L=81.300570\;\pm\;0.000005$^[7]

History of measurement

The mass of Earth is measured indirectly by determining other quantities such as Earth's density, gravity, or gravitational constant.

Using the GM_⊕ product

Modern methods of determining the mass of Earth involve calculating the gravitational coefficient of the Earth and dividing by the Newtonian constant of gravitation,

$M_\oplus =\frac{ GM_\oplus}{ G }.$

The GM_⊕ product is determined using laser ranging data from Earth-orbiting satellites.^[8] The GM_⊕ product can also be calculated by observing the motion of the Moon^[9] or the period of a pendulum at various elevations. These methods are less precise than observations of artificial satellites.

Using the gravitational constant

Earlier efforts (after 1798) to determine Earth's mass involved measuring G directly as in the Cavendish experiment. Earth's mass could be then found by combining two equations; Newton's second law, and Newton's law of universal gravitation.

$F = ma, \quad F = G\frac{mM_\oplus}{r^2}$

Substituting earth's gravity, g for the acceleration term, and combining the two equations gives

$mg = G\frac{mM_\oplus}{r^2}$.

The equation can then be solved for M_⊕.

$M_\oplus = \frac{gr^2}{G}$

With this method, the values for Earth's surface gravity, Earth's radius, and G were measured empirically.

Using the deflection of a pendulum

Before the Cavendish Experiment, attempts to "weigh" Earth involved estimating the mean density of Earth and its volume. The volume was well understood through surveying techniques, and the density was measured by observing the slight deflection of a pendulum near a mountain, as in the Schiehallion experiment. The Earth mass could then be calculated as

$M_\oplus = \rho V$.

This technique resulted in a mass estimate that is 20% lower than today's accepted value.

Using the period of a pendulum

An expedition from 1737 to 1740 by French scientist Pierre Bouguer attempted to determine the density of Earth by measuring the period of a pendulum (and therefor the strength of gravity) as a function of elevation. The experiments were carried out in Ecuador and Peru, on Pichincha Volcano and mount Chimborazo. Bouguer's work led to an estimate that is two to three times larger than the true mass of Earth. However, this historical determination showed that the Earth was not hollow nor filled with water, as some had argued at the time.^[10] Modern gravitometers are now used for measuring the local gravitational field. They surpass the accuracy limitations of pendulums.

Experiments with pendulums in the nineteenth century

Much later, in 1821, Francesco Carlini determined a density value of ρ = 4.39 g/cm³ through measurements made with pendulums in the Milan area. This value was refined in 1827 by Edward Sabine to 4.77 g/cm³, and then in 1841 by Carlo Ignazio Giulio to 4.95 g/cm³. On the other hand, George Biddell Airy sought 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 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 value 6.6 g/cm³ by 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 values ρ between 5.0 and 6.3 g/cm³. This led to the concept of isostasy, which limits the ability to accurately measure ρ, 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 Mendenhall in 1880 realized a gravimetry experiment in Tokyo and at the top of Mount Fuji. The result was ρ = 5.77 g/cm³.

Variation

Earth's mass is constantly changing due to many contributors. Earth primarily gains mass from micrometeorites and cosmic dust, whereas it loses hydrogen and helium gas. The combined effect is a net loss of material, though the annual mass deficit represents an inconsequential fraction of its total mass,^{[lower-alpha 1]} or even the uncertainty in its mass. So its inclusion does not affect total mass calculations. A number of other mechanisms are responsible for mass adjustments, and can be classified into two categories; physical transfer of matter, and mass which is gained or lost through the absorption or release of energy due to the mass–energy equivalence principle. Several examples are provided for completeness, but their relative contribution is negligible.

Net gains

<div " style="padding-left: 3em; padding-right:0em; overflow: hidden; ">In-falling material

Cosmic dust, Cosmic Rays, meteors, comets, etc. are the most significant contributor to Earth's increase in mass. The sum of material is estimated to be 37000–78000 tons annually^[12][13]

Global Warming

Nasa has calculated that the Earth is gaining energy due to rising temperatures. It has been estimated that this added energy increases the mass of Earth by a tiny amount – 160 tonnes per year.^[14]

Solar energy conversion (minuscule)

Solar energy is converted to chemical energy by photosynthetic pigments as plants construct carbohydrate molecules. This stored chemical energy represents in increase in mass. Most of the chemical energy is reconverted into heat and then lost (radiated) through chemical processes, but some is sequestered and becomes biomass or fossil fuel.

Artificial photosynthesis (minuscule)

Can also theoretically add mass, assumed to be negligible but added for sake of completeness.

Heat conversion (probably minuscule)

Some outbound radiation is absorbed within the atmosphere by photosynthetic bacteria and archaea, including from chlorophyll f, which bind the energy into matter in the form of chemical bonds.

Net losses

<div " style="padding-left: 3em; padding-right:0em; overflow: hidden; ">Atmospheric escape of gases.

About 3 kg/s of hydrogen or 95,000 tons per year^[15] and 1,600 tons of helium per year^[16] are lost through atmospheric escape.

Spacecraft on escape trajectories (minuscule)

Spacecraft that are on escape trajectories represent an average mass loss at a rate of 65 tons per year.^[11] Earth lost about 3473 tons in the initial 53 years of the space age, but the trend is currently decreasing.

Human energy use (minuscule)

Human activities conversely reduce Earth's mass, by liberation of heat that is later radiated into space; solar photovoltaics generally do not add to the mass of Earth because the energy collected is merely transmitted (as electricity or heat) and subsequently radiated, which is generally not converted into chemical means to be stored on Earth. In 2010, the human world consumed 550 EJ of energy,^[17] or 6 tons of matter converted into heat, then almost entirely lost to space.

Deceleration of Earth's core (minuscule)

As the rotation rate of Earth's inner core decelerates, it loses rotational kinetic energy, which equates to a loss of 16 tons per year. However, this rotation speed has been shown to fluctuate over decades.^[18]

Non photosynthesizing life forms consume energy, and radiate as heat.

Natural processes (probably minuscule)

Events including earthquakes and volcanoes can release energy as well as hydrogen, which may be lost as heat or atmospheric escape.

Radiation Losses(minuscule)

From radioisotopes either naturally or through human induced reactions such as nuclear fusion or nuclear fission amount to 16 tons per year.^[11]

Additional human impact by induced nuclear fission

Nuclear fission, both for civilian and military purposes, greatly speeds up natural process of radiodecay. Some 59,000 tons of uranium was supplied by mines in 2013.^[19] The mass of the uranium is reduced as it is converted to energy during the fission reaction. Also, the growing spent fuel stockpiles and environmental releases continues to produce heat (and therefore mass) largely lost to space.

See also

• Abundance of elements in Earth's crust
• Cavendish experiment
• Schiehallion experiment
• Earth radius
• Planetary mass
• Orders of magnitude (mass)
• Solar mass
• Structure of the Earth
• Gravitational constant
• Earth Similarity Index

Notes

1. The total estimated anual loss is 5.5×10⁷ kg,^[11] which constitutes a fraction of 5.5e7/5.97e24 ≈ 1/1e17 = 1/100 Quadrillion

References

1. "Solar System Exploration: Earth: Facts & Figures". NASA. 13 Dec 2012. Retrieved 2012-01-22.
2. "2016 Selected Astronomical Constants" in The Astronomical Almanac Online, USNO–UKHO.
3. Pitjeva, E.V.; Standish, E.M. (2009-04-01). "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. Retrieved 2016-02-12.
4. "Planetary Fact Sheet – Ratio to Earth". nssdc.gsfc.nasa.gov. Retrieved 2016-02-12.
5. "The Habitable Exoplanets Catalog – Planetary Habitability Laboratory @ UPR Arecibo".
6. "HEC: Data of Potential Habitable Worlds".
7. Konopliv, A (December 2002). "A Global Solution for the Gravity Field, Rotation, Landmarks, and Ephemeris of Eros" (PDF). Icarus 160 (2): 289–299. Bibcode:2002Icar..160..289K. doi:10.1006/icar.2002.6975.
8. 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). Bibcode:1992GeoRL..19..529R. doi:10.1029/92GL00259. Retrieved 5 February 2016.
9. 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 (American Radio Relay League): 25–30. Retrieved 28 February 2016.
10. N. Kollerstrom (1992). "The hollow world of Edmond Halley". Journal for History of Astronomy 23: 185–192. archive
11. Saxena, Shivam; Chandra, Mahesh (May 2013). "Loss in Earth Mass due to Extraterrestrial Space Exploration Missions". International Journal of Scientific and Research Publications 3 (5): 1. Retrieved 9 February 2016.
12. "Spacecraft Measurements of the Cosmic Dust Flux", Herbert A. Zook. doi:10.1007/978-1-4419-8694-8_5
13. Carter, Lynn. "How many meteorites hit Earth each year?". Ask an Astronomer. The Curious Team, Cornell University. Retrieved 6 February 2016.
14. McDonald, Charlotte (31 January 2012). "Who, What, Why: Is the Earth getting lighter?". BBC Magazine. BBC News. Retrieved 9 February 2016.
15. "Fantasy and Science Fiction: Science by Pat Murphy & Paul Doherty".
16. "Earth Loses 50,000 Tonnes of Mass Every Year". SciTech Daily.
17. "World energy consumption – beyond 500 exajoules". Resilience.
18. Tkalčić, Hrvoje; Young, Mallory; Bodin, Thomas; Ngo, Silvie; Sambridge, Malcolm (12 May 2013). "The shuffling rotation of the Earth’s inner core revealed by earthquake doublets". Nature Geoscience 6: 497–502. Bibcode:2013NatGe...6..497T. doi:10.1038/ngeo1813.
19. "Uranium Markets".