Earth's gravity: Wikis

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Earth's gravity measured by NASA's GRACE mission, showing deviations from the theoretical gravity of an idealized smooth Earth.

Earth's gravity, denoted g, refers to the acceleration that the Earth imparts to objects on or near its surface. In SI units this acceleration is measured in m/s2 (metres per second per second, equivalently written as m·s−2 or N/kg). It has an approximate value of 9.81 m/s2, which means that, ignoring air resistance, the speed of an object falling freely near the Earth's surface increases by about 9.81 metres per second every second. This quantity is informally known as little g (contrasted with G, the gravitational constant, known as big G).

There is a direct relationship between gravitational acceleration and the downwards weight force experienced by objects on Earth (see Conversion between weight and mass). However, other factors such as the rotation of the Earth also contribute to the net acceleration and are usually considered part of Earth's gravity.

The precise strength of the Earth's gravity varies depending on location. The nominal "average" value at the Earth's surface, known as standard gravity is, by definition, 9.80665 m/s2 (32.1740 ft/s2). This quantity is denoted variously as gn, ge (though this sometimes means the normal equatorial value on Earth, 9.78033 m/s2), g0, gee, or simply g (which is also used for the variable local value). The symbol g should not be confused with g, the abbreviation for gram (which is not italicized).[1][2]

Variation in gravity and apparent gravity

Earth's gravitational force is often modeled as though the Earth were an inert sphere of uniform density. Such a body would produce a field of uniform magnitude and direction at all points on its surface. In reality there are slight deviations in both the magnitude and direction of gravity across the surface of the Earth because none of those qualities are exactly true of Earth.

Furthermore, the net force exerted on an object due to the Earth, called apparent gravity or effective gravity varies due to the presence of other forces. A scale or plumb bob measures only this effective gravity.

The strength of Earth's apparent gravity varies with latitude, altitude, local topography and geology:

Latitude

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

Apparent gravity is weaker nearer the equator because the Earth's rotation produces an apparent centrifugal force.

Gravity provides centripetal force, keeping objects on the surface (and indeed the surface itself) moving in a circular motion. Consider that if the gravity of the Earth were to shut off, objects would fly off into space in the direction of their motion in accordance with Newton's First Law of Motion. Alternatively, if Earth's gravity were weakened so as to match the centrifugal force (at, say, the equator where rotational speed is largest) then objects there would appear to float. At the poles the radius of curvature is zero, so only this weakened gravity would contribute to weight and objects would not float. In this sense, local gravity (gravity at a particular point on the surface of the Earth) felt as weight is gravity due to the Earth's mass minus the centrifugal force. Because rotational speed decreases as one moves towards the poles, local gravity, g, increases from 9.789 m·s−2 at the equator to 9.832 m·s−2 at the poles.[3]

The second major cause for the difference in gravity at different latitudes is that the Earth's equatorial bulge (itself also caused by centrifugal force) causes objects at the equator to be farther from the planet's centre than objects at the poles. Because the force due to gravitational attraction between two bodies (the Earth and the object being weighed) varies inversely with the square of the distance between them, objects at the equator experience a weaker gravitational pull than objects at the poles.

In combination, the equatorial bulge and the effects of centrifugal force mean that sea-level gravitational acceleration increases from about 9.780 m·s−2 at the equator to about 9.832 m·s−2 at the poles, so an object will weigh about 0.5% more at the poles than at the equator.[4]

The same two factors influence the direction of the effective gravity. Anywhere on Earth away from the equator or poles, effective gravity points not exactly toward the centre of the Earth, but rather perpendicular to the surface of the geoid, which, due to the flattened shape of the Earth, is somewhat toward the opposite pole. About half of the deflection is due to centrifugal force, and half because the extra mass around the equator causes a change in the direction of the true gravitational force relative to what it would be on a spherical Earth.

Altitude

Gravitational field strength within the Earth

Gravity decreases with altitude, since greater altitude means greater distance from the Earth's centre. All other things being equal, an increase in altitude from sea level to the top of Mount Everest (8,850 metres) causes a weight decrease of about 0.28%. (An additional factor affecting apparent weight is the decrease in air density at altitude, which lessens an object's buoyancy.[5]) 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 miles), equivalent to a typical orbit of the Space Shuttle, gravity is still nearly 90% as strong as at the Earth's surface, and weightlessness actually occurs because orbiting objects are in free-fall.

The following formula approximates the Earth's gravity variation with altitude:

$g_h=g_0(\frac{r_e}{r_e+h})^2$

Where

• $g_h\,$ is the gravity measure at height $h\,$ above sea level.
• $r_e\,$ is the Earth's mean radius.
• $g_0\,$ is the standard gravity.

This formula treats the Earth as a non-rotating perfect sphere with a radially symmetric distribution of mass; a more accurate mathematical treatment is discussed below.

If the Earth were of perfectly uniform composition, during a descent to the centre of the Earth, gravity would decrease linearly with distance, reaching zero at the centre. In reality, the gravitational field peaks within the Earth at the core-mantle boundary where it has a value of 10.7 m/s², because of the marked increase in density at that boundary.[citation needed]

Local topography and geology

Local variations in topography (such as the presence of mountains) and geology (such as the density of rocks in the vicinity) cause fluctuations in the Earth's gravitational field, known as gravitational anomalies. Some of these anomalies can be very extensive, resulting in bulges in sea level, and throwing pendulum clocks out of synchronisation.

The study of these anomalies forms the basis of gravitational geophysics. The fluctuations are measured with highly sensitive gravimeters, 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 oil and mineral deposits. Denser rocks (often containing mineral ores) cause higher than normal local gravitational fields on the Earth's surface. Less dense sedimentary rocks cause the opposite.

Other factors

In air, objects experience a supporting buoyancy force which reduces the apparent strength of gravity (as measured by an object's weight). The magnitude of the effect depends on air density (and hence air pressure); see Apparent weight for details.

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

Comparative gravities in various cities around the world

The table below shows the gravitational acceleration in various cities around the world; amongst these cities, it is lowest in Mexico City (9.779 m/s²) and highest in Oslo (Norway) and Helsinki (Finland) (9.819 m/s²).

 Amsterdam 9.813 m/s² Istanbul 9.808 m/s² Paris 9.809 m/s² Athens 9.800 m/s² Havana 9.788 m/s² Rio de Janeiro 9.788 m/s² Auckland 9.799 m/s² Helsinki 9.819 m/s² Rome 9.803 m/s² Bangkok 9.783 m/s² Kuwait 9.793 m/s² San Francisco 9.800 m/s² Brussels 9.811 m/s² Lisbon 9.801 m/s² Singapore 9.781 m/s² Buenos Aires 9.797 m/s² London 9.812 m/s² Stockholm 9.818 m/s² Calcutta 9.788 m/s² Los Angeles 9.796 m/s² Sydney 9.797 m/s² Cape Town 9.796 m/s² Madrid 9.800 m/s² Taipei 9.790 m/s² Chicago 9.803 m/s² Manila 9.784 m/s² Tokyo 9.798 m/s² Copenhagen 9.815 m/s² Mexico City 9.779 m/s² Vancouver 9.809 m/s² Nicosia 9.797 m/s² New York City 9.802 m/s² Washington, D.C. 9.801 m/s² Jakarta 9.781 m/s² Oslo 9.819 m/s² Wellington 9.803 m/s² Frankfurt 9.810 m/s² Ottawa 9.806 m/s² Zurich 9.807 m/s²

Mathematical models

The graph shows the difference of gravity relative to the height of an object

If the terrain is at sea level, we can estimate g:

$g_{\phi}=9.780 31846 \left( 1+0.0053024\sin^2 \phi-0.0000058\sin^2 2\phi \right) \ m/s^2$

where

$\ g_{\phi}$ = acceleration in m·s−2 at latitude :$\ \phi$

This is the International Gravity Formula 1967, the 1967 Geodetic Reference System Formula, Helmert's equation or Clairault's formula.[6]

The first correction to this formula is the free air correction (FAC), which accounts for heights above sea level. Gravity decreases with height at a rate which near the surface of the Earth is such that linear extrapolation would give zero gravity at a height of one half the Earth's radius, or about 9.8 m·s−2 per 3 200 km. Thus:

$g_{\phi}=\left[9.780 327 \left( 1+0.0053024\sin^2 \phi-0.0000058\sin^2 2\phi \right) - 3.086 \times 10^{-6}h \right] \ m/s^2$

where

h = height in metres above sea level

For flat terrain above sea level a second term is added, for the gravity due to the extra mass; for this purpose the extra mass can be approximated by an infinite horizontal slab, and we get 2πG times the mass per unit area, i.e. 4.2 × 10−10 m3·s−2·kg−1 (0.042 μGal·kg−1·m²)) (the Bouguer correction). For a mean rock density of 2.67 g·cm−3 this gives 1.1 × 10−6 s−2 (0.11 mGal·m−1). Combined with the free-air correction this means a reduction of gravity at the surface of ca. 2 µm·s−2 (0.20 mGal) for every metre of elevation of the terrain. (The two effects would cancel at a surface rock density of 4/3 times the average density of the whole Earth.)

For the gravity below the surface we have to apply the free-air correction as well as a double Bouguer correction. With the infinite slab model this is because moving the point of observation below the slab changes the gravity due to it to its opposite. Alternatively, we can consider a spherically symmetrical Earth and subtract from the mass of the Earth that of the shell outside the point of observation, because that does not cause gravity inside. This gives the same result.

Helmert's equation may be written equivalently to the version above as either:

$\ g_{\phi}= \left(9.8061999 - 0.0259296\cos(2\phi) + 0.0000567\cos^2(2\phi)\right) \ m/s^2$

or

$\ g_{\phi}= \left( 9.780327 + 0.0516323\sin^2(\phi) + 0.0002269\sin^4(\phi) \right) \ m/s^2$

An alternate formula for g as a function of latitude is the WGS (World Geodetic System) 84 Ellipsoidal Gravity Formula:

$\ g_{\phi}= \left(9.7803267714 ~ \frac {1 + 0.00193185138639\sin^2\phi}{\sqrt{1 - 0.00669437999013\sin^2\phi}} \right) \ m/s^2$

A spot check comparing results from the WGS-84 formula with those from Helmert's equation (using increments 10 degrees of latitude starting with zero) indicated that they produce values which differ by less than 10−6 m·s−2.

Estimating g from the law of universal gravitation

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

$F = G \frac{m_1 m_2}{r^2}=\left(G \frac{m_1}{r^2}\right) m_2$

where r is the distance between the centre of the Earth and the body (see below), and here we take m1 to be the mass of the Earth and m2 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

$F = m_2g\,$

Comparing the two formulas it is seen that:

$g=G \frac {m_1}{r^2}$

So, to find the acceleration due to gravity at sea level, substitute the values of the gravitational constant, G, the Earth's mass (in kilograms), m1, and the Earth's radius (in metres), r, to obtain the value of g:

$g=G \frac {m_1}{r^2}=(6.6742 \times 10^{-11}) \frac{5.9736 \times 10^{24}}{(6.37101 \times 10^6)^2}=9.822 \mbox{ m} \cdot \mbox{s}^{-2}$

Note that 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 does not include the centrifugal pseudoforce effects that are found in practice due to the rotation of the Earth

There are significant uncertainties in the values of r and m1 as used in this calculation, and the value of 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 Henry Cavendish.

Comparative gravities of the Earth, Sun, Moon, planets and Pluto

The table below shows comparative gravitational accelerations at the surface of the Sun, the Earth's moon, each of the planets in the solar system and their major moons, Pluto and Eris. The “surface” is taken to mean the cloud tops of the gas giants (Jupiter, Saturn, Uranus and Neptune). For the Sun, the surface is taken to mean the photosphere. The values in the table have not been de-rated for the centrifugal effect of planet rotation (and cloud-top wind speeds for the gas giants) and therefore, generally speaking, are similar to the actual gravity that would be experienced near the poles.

Body Multiple of
Earth gravity
m/s2
Sun 27.90 274.1
Mercury 0.3770 3.703
Venus 0.9032 8.872
Earth 1 (by definition) 9.8226[7]
Moon 0.1655 1.625
Mars 0.3895 3.728
Jupiter 2.640 25.93
Io 0.182 1.789
Europa 0.134 1.314
Ganymede 0.145 1.426
Callisto 0.126 1.24
Saturn 1.139 11.19
Titan 0.138 1.358
Uranus 0.917 9.01
Titania 0.039 0.379
Oberon 0.035 0.347
Neptune 1.148 11.28
Triton 0.079 0.779
Pluto 0.0621 0.610
Eris ~0.0814 ~0.8

References

1. ^ Bureau International des Poids et Mesures (2006). "Chapter 5". The International System of Units (SI). 8th ed.. Retrieved 2009-11-25. "Unit names are normally printed in roman (upright) type .... Symbols for quantities are generally single letters set in an italic font, although they may be qualified by further information in subscripts or superscripts or in brackets.".
2. ^ "SI Unit rules and style conventions". National Institute For Standards and Technology (USA). September 2004. Retrieved 2009-11-25. "Variables and quantity symbols are in italic type. Unit symbols are in roman type."
3. ^ Boynton, Richard (2001). "Precise Measurement of Mass". SAWE PAPER No. 3147. Arlington, Texas: S.A.W.E., Inc.. Retrieved 2007-01-21.
4. ^ "Curious About Astronomy?", Cornell University, retrieved June 2007
5. ^ "I feel 'lighter' when up a mountain but am I?", National Physical Laboratory FAQ
6. ^ International Gravity formula
7. ^ This value excludes the adjustment for centrifugal force due to Earth’s rotation and is therefore greater than the 9.80665 m/s² value of standard gravity.