# Earth's gravity

Earth's gravity, denoted by g, refers to the attractive force that the Earth exerts on objects on or near its surface (or, more generally, objects anywhere in the Earth's vicinity). Its strength is usually quoted as an acceleration, which in SI units is measured in m/sÂ² (metres per second per second, equivalently written as mÂ·s−2). It has an approximate value of 9.8 m/sÂ², which means that, ignoring air resistance, the speed of an object falling freely near the Earth's surface increases by about 9.8 metres per second every second.

There is a direct relationship between gravitational acceleration and the downwards weight force experienced by objects on Earth. This is explained at weight; see also apparent weight.

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/sÂ² (32.1740 ft/sÂ²). This quantity is denoted variously as gn, ge (though this sometimes means the normal equatorial value on Earth, 9.78033 m/sÂ²), g0, gee, or simply g (which is also used for the variable local value). The symbol g should not be confused with G, the gravitational constant, or g, the abbreviation for gram (which is not italicized).

The change in gravitational strength per unit distance (in a given direction) is known as the gravitational gradient. In the SI system this is measured in m/sÂ² (strength) per metre (distance), which resolves to simply s−2 (inverse seconds squared). In the cgs system, gravitational gradient is measured in eotvoses.

## Variations on Earth

The strength (or apparent strength) of Earth's gravity varies with latitude, altitude, and local topography and geology. For most purposes the gravitational force is assumed to act in a line directly towards a point at the centre of the Earth, but for very precise work the direction can also vary slightly because the Earth is not a perfectly uniform sphere.

### Latitude

The differences of Earth's gravity around the Antarctic continent

Gravity is weaker at lower latitudes (nearer the equator), for two reasons. The first is that in a rotating non-inertial or accelerated reference frame, as is the case on the surface of the Earth, there appears a 'fictitious' centrifugal force acting in a direction perpendicular to the axis of rotation. The gravitational force on a body is partially offset by this centrifugal force, reducing its weight. This effect is smallest at the poles, where the gravitational force and the centrifugal force are orthogonal, and largest at the equator. This effect on its own would result in a range of values of g from 9.789 mÂ·s−2 at the equator to 9.832 mÂ·s−2 at the poles.[1]

The second reason 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/sÂ² at the equator to about 9.832 m/sÂ² at the poles, so an object will weigh about 0.5% more at the poles than at the equator.[2]

### Altitude

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.[3]) 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), equivilant 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.

If the Earth was of perfectly uniform composition then, 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/sÂ², because of the marked increase in density at that boundary.

### 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 of up to 200 metres in the Pacific ocean, 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. Paris, France has been shown by this method to almost certainly be sitting on a huge, untouchable oilfield .

### 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 Âµm/sÂ² (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.[4]

 Amsterdam 9.813 m/sÂ² Istanbul 9.808 m/sÂ² Paris 9.809 m/sÂ² Athens 9.807 m/sÂ² Havana 9.788 m/sÂ² Rio de Janeiro 9.788 m/sÂ² Auckland, NZ 9.799 m/sÂ² Helsinki 9.819 m/sÂ² Rome 9.803 m/sÂ² Bangkok 9.783 m/sÂ² Kuwait 9.793 m/sÂ² San Francisco 9.800 m/sÂ² Brussels 9.811 m/sÂ² Lisbon 9.801 m/sÂ² Singapore 9.781 m/sÂ² Buenos Aires 9.797 m/sÂ² London 9.812 m/sÂ² Stockholm 9.818 m/sÂ² Calcutta 9.788 m/sÂ² Los Angeles 9.796 m/sÂ² Sydney 9.797 m/sÂ² Cape Town 9.796 m/sÂ² Madrid 9.800 m/sÂ² Taipei 9.790 m/sÂ² Chicago 9.803 m/sÂ² Manila 9.784 m/sÂ² Tokyo 9.798 m/sÂ² Copenhagen 9.815 m/sÂ² Mexico City 9.779 m/sÂ² Vancouver, BC 9.809 m/sÂ² Nicosia 9.797 m/sÂ² New York 9.802 m/sÂ² Washington, DC 9.801 m/sÂ² Jakarta 9.781 m/sÂ² Oslo 9.819 m/sÂ² Wellington, NZ 9.803 m/sÂ² Frankfurt 9.810 m/sÂ² Ottawa 9.806 m/sÂ² Zurich 9.807 m/sÂ²

### Mathematical models

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

where
= acceleration in mÂ·s−2 at latitude f

This is the International Gravity Formula 1967, the 1967 Geodetic Reference System Formula, Helmert's equation or Clairault's formula.[5]

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 radius of the Earth, i.e. the rate is 9.8 mÂ·s−2 per 3200 km. Thus:

where
h = height in meters 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. 4.210-10 mÂ³Â·s−2Â·kg−1 (0.042 μGalÂ·kg−1Â·mÂ²)) (the Bouguer correction). For a mean rock density of 2.67 gÂ·cm−3 this gives 1.110-6 s−2 (0.11 mGalÂ·m−1). Combined with the free-air correction this means a reduction of gravity at the surface of ca. 2 ÂµmÂ·s−2 (0.20 mGal) for every meter 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:
or

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

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 1e-6 m/sÂ².

## Estimating g from the law of universal gravitation

Given the law of universal gravitation, the acceleration due to gravity, g, is merely a collection of factors in that equation:
where g is the bracketed factor, and thus:
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 meters), r, to obtain the value of g:

Note that this formula only works because of the pleasant (but non-obvious) 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 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 force 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 and planets

The table below shows gravitational accelerations (in multiples of g) at the surface of the Sun, the Earth's moon, and each of the planets in the solar system. The "surface" is taken to mean the cloud tops of the gas giants (Jupiter, Saturn, Uranus and Neptune). It is usually specified as the location where the pressure is equal to a certain value (normally 75 kPa). For the Sun, the "surface" is taken to mean the photosphere. It is also normally specified at the equator, though the value given here for Earth is not the equatorial value.

 Body Multiple of g m/sÂ² Sun 27.935 273.95 Mercury 0.3774 3.701 Venus 0.904 8.87 Earth 1 (by definition) 9.80665 Moon 0.1654 1.622 Mars 0.376 3.69 Jupiter 2.528 24.79 Saturn 0.917 8.96 Uranus 0.886 8.69 Neptune 1.137 11.15 Pluto 0.059 0.58

For spherical bodies, surface gravity in m/sÂ² is 2.8 Ã— 10−10 times the radius in metres times the average density in kg/mÂ³ (kilograms per cubic metre).

## References

1. ^ Boynton, Richard (2001). "Precise Measurement of Mass". SAWE PAPER No. 3147, Arlington, Texas: S.A.W.E., Inc.. Retrieved on 2007-01-21.
2. ^ "Curious About Astronomy?", Cornell University, retrieved June 2007
3. ^ "I feel 'lighter' when up a mountain but am I?", National Physical Laboratory FAQ
4. ^ (2002) AND EKi/EW-i Instruction Manual. A&D, 41.
5. ^ International Gravity formula

EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001. Their greatest hit, their debut single "time after time", peaked at #13 in the Oricon singles chart.
International System of Units (abbreviated SI from the French Le SystÃ¨me international d'unitÃ©s) is the modern form of the metric system.
weight is a measurement of the gravitational force acting on an object. Near the surface of the Earth, the acceleration due to gravity is approximately constant; this means that an object's weight is roughly proportional to its mass.
apparent weight is the (usually upward) force (the normal force, or reaction force), typically transmitted through the ground, that opposes the (usually downward) acceleration of a supported object, preventing it from falling.
Standard gravity, usually denoted by g0 or gn, is the nominal acceleration due to gravity at the Earth's surface at sea level. By definition it is equal to exactly 9.80665  mÂ·s−2 (approx. 32.174 ftÂ·s−2).
gravitational constant, the universal gravitational constant, Newton's constant, and colloquially Big G. The gravitational constant is a physical constant which appears in Newton's law of universal gravitation and in Einstein's theory of general
gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
centimetre-gram-second system (CGS) is a system of physical units. It is always the same for mechanical units, but there are several variants of electric additions. It was replaced by the MKS, or metre-kilogram-second system, which in turn was replaced by the International
The eotvos is a unit of acceleration divided by distance that was used in conjunction with the older centimeter-gram-second system of units. The eotvos is defined as 1/1,000,000,000 galileo per centimetre. The symbol of the eotvos unit is E.
equator divides the planet into a Northern Hemisphere and a Southern Hemisphere, and has a latitude of 0. Latitude, usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator.
Topography (Greek topos, "place", and graphia, "writing") is the study of Earth's surface features or those of planets, moons, and asteroids.

In a broader sense, topography is concerned with local detail in general, including not only relief but also
Oceanic crust      0-20 Ma
In theoretical physics, an accelerated reference frame is usually a coordinate system or frame of reference, that undergoes a constant and continual change in velocity over time as judged from an inertial frame.
Centrifugal force (from Latin centrum "centre" and fugere "to flee") is a term which may refer to two different forces which are related to rotation.
In mathematics, orthogonal, as a simple adjective, not part of a longer phrase, is a generalization of perpendicular. It means at right angles, from the Greek ὀρθός orthos
An equatorial bulge is a planetological term which describes a bulge which a planet may have around its equator, distorting it into an oblate spheroid. The Earth has an equatorial bulge of 42.72 km (26.5 miles) due to its rotation.
Space Shuttle

Space Shuttle Atlantis on the launch pad prior to the STS-115 mission.
Fact sheet
Function Manned partially re-usable launch and reentry system
Manufacturer United Space Alliance:
Thiokol/Boeing (SRBs)
Free fall is motion with no acceleration other than that provided by gravity. This also applies to objects in orbit, even though these objects are not "falling" in the usual sense of the word.
Physical geodesy is the study of the physical properties of the gravity field of the Earth, the geopotential, with a view to their application in geodesy.

## Measurement procedure

Topography (Greek topos, "place", and graphia, "writing") is the study of Earth's surface features or those of planets, moons, and asteroids.

In a broader sense, topography is concerned with local detail in general, including not only relief but also
Oceanic crust      0-20 Ma
free-air anomaly (or Faye's anomaly): application of the normal gradient 0.3086, but no terrain model. This anomaly means a downward shift of the point, together with the whole shape of the terrain. This simple method is ideal for many geodetic applications.
Earth's oceans
(World Ocean)
• Arctic Ocean
• Atlantic Ocean
• Indian Ocean
• Pacific Ocean
• Southern Ocean

The Pacific Ocean (from the Latin name Mare Pacificum
v and A).]] A pendulum is an object that is attached to a pivot point so it can swing freely. This object is subject to a restoring force that will accelerate it toward an equilibrium position.
Geophysics, a branch of Earth sciences, is the study of the Earth by quantitative physical methods, especially by seismic, electromagnetic, and radioactivity methods. The theories and techniques of geophysics are employed extensively in the planetary sciences in general.

A gravimeter or gravitometer, is an instrument used in gravimetry for measuring the local gravitational field.
Prospecting is the physical search for minerals, fossils, precious metals or mineral specimens, and is also known as fossicking.

Prospecting is synonymous in some ways with mineral exploration which is an organised, large scale and at least semi-scientific effort undertaken