The gravity of Earth, denoted by , is the
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
that is imparted to objects due to the combined effect of
gravitation (from
mass distribution within
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
) and the
centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parall ...
(from the
Earth's rotation).
It is a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
quantity, whose direction coincides with a
plumb bob
A plumb bob, plumb bob level, or plummet, is a weight, usually with a pointed tip on the bottom, suspended from a string and used as a vertical reference line, or plumb-line. It is a precursor to the spirit level and used to establish a verti ...
and strength or magnitude is given by the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
.
In
SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
this acceleration is expressed in
metres per second squared (in symbols,
m/
s2 or m·s
−2) or equivalently in
newtons per
kilogram (N/kg or N·kg
−1). Near Earth's surface, the gravity acceleration is approximately , which means that, ignoring the effects of
air resistance
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
, the
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
of an object
falling freely will increase by about per second every second. This quantity is sometimes referred to informally as ''little '' (in contrast, the
gravitational constant is referred to as ''big '').
The precise strength of Earth's gravity varies depending on location. The nominal "average" value at Earth's surface, known as is, by definition, . This quantity is denoted variously as , (though this sometimes means the normal equatorial value on Earth, ), , gee, or simply (which is also used for the variable local value).
The
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
of an object on Earth's surface is the downwards force on that object, given by
Newton's second law of motion, or ().
Gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
contributes to the total gravity acceleration, but other factors, such as the rotation of Earth, also contribute, and, therefore, affect the weight of the object. Gravity does not normally include the gravitational pull of the Moon and Sun, which are accounted for in terms of
tidal effects.
Variation in magnitude
A non-rotating perfect
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
of uniform mass density, or whose density varies solely with distance from the centre (
spherical symmetry
In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the ...
), would produce a
gravitational field of uniform magnitude at all points on its
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
. The Earth is rotating and is also not spherically symmetric; rather, it is slightly flatter at the poles while bulging at the Equator: an
oblate spheroid. There are consequently slight deviations in the magnitude of gravity across its surface.
Gravity on the Earth's surface varies by around 0.7%, from 9.7639 m/s
2 on the
Nevado Huascarán mountain in Peru to 9.8337 m/s
2 at the surface of the
Arctic Ocean
The Arctic Ocean is the smallest and shallowest of the world's five major oceans. It spans an area of approximately and is known as the coldest of all the oceans. The International Hydrographic Organization (IHO) recognizes it as an ocean, a ...
.
In large cities, it ranges from 9.7806
["Wolfram, Alpha Gravity in Kuala Lumpur", Wolfram Alpha, accessed November 2020](_blank)
/ref> in Kuala Lumpur
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, image_map =
, map_caption =
, pushpin_map = Malaysia#Southeast Asia#Asia
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, coordinates =
, su ...
, Mexico City
Mexico City ( es, link=no, Ciudad de México, ; abbr.: CDMX; Nahuatl: ''Altepetl Mexico'') is the capital and largest city of Mexico, and the most populous city in North America. One of the world's alpha cities, it is located in the Valley o ...
, and Singapore
Singapore (), officially the Republic of Singapore, is a sovereign island country and city-state in maritime Southeast Asia. It lies about one degree of latitude () north of the equator, off the southern tip of the Malay Peninsula, borde ...
to 9.825 in Oslo
Oslo ( , , or ; sma, Oslove) is the capital and most populous city of Norway. It constitutes both a county and a municipality. The municipality of Oslo had a population of in 2022, while the city's greater urban area had a population of ...
and Helsinki
Helsinki ( or ; ; sv, Helsingfors, ) is the capital, primate, and most populous city of Finland. Located on the shore of the Gulf of Finland, it is the seat of the region of Uusimaa in southern Finland, and has a population of . The city ...
.
Conventional value
In 1901 the third General Conference on Weights and Measures defined a standard gravitational acceleration for the surface of the Earth: ''g''n = 9.80665 m/s2. It was based on measurements done at the Pavillon de Breteuil
The Pavillon de Breteuil is the headquarters of the International Bureau of Weights and Measures (BIPM). The building lies in the southeastern section of the Parc de Saint-Cloud in Saint-Cloud, France, to the west of Paris. It is listed in France ...
near Paris in 1888, with a theoretical correction applied in order to convert to a latitude of 45° at sea level. This definition is thus not a value of any particular place or carefully worked out average, but an agreement for a value to use if a better actual local value is not known or not important. It is also used to define the units kilogram force
The kilogram-force (kgf or kgF), or kilopond (kp, from la, pondus, lit=weight), is a non-standard gravitational metric unit of force. It does not comply with the International System of Units (SI) and is deprecated for most uses. The kilogram-f ...
and pound force
The pound of force or pound-force (symbol: lbf, sometimes lbf,) is a unit of force used in some systems of measurement, including English Engineering units and the foot–pound–second system.
Pound-force should not be confused with pound-m ...
.
Calculating the gravity at Earth's surface using the average radius of Earth (), the experimentally determined value of the gravitational constant, and the Earth mass of 5.9722 kg gives an acceleration of 9.8203 m/s2, slightly greater than the standard gravity of 9.80665 m/s2. The value of standard gravity corresponds to the gravity on Earth at a radius of .[
]
Latitude
The surface of the Earth is rotating, so it is not an inertial frame of reference. At latitudes nearer the Equator, the outward centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parall ...
produced by Earth's rotation is larger than at polar latitudes. This counteracts the Earth's gravity to a small degree – up to a maximum of 0.3% at the Equator – and reduces the apparent downward acceleration of falling objects.
The second major reason for the difference in gravity at different latitudes is that the Earth's equatorial bulge
An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere.
On E ...
(itself also caused by centrifugal force from rotation) causes objects at the Equator to be farther from the planet's center 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, an object at the Equator experiences a weaker gravitational pull than an object on the pole.
In combination, the equatorial bulge and the effects of the surface centrifugal force due to rotation mean that sea-level gravity increases from about 9.780 m/s2 at the Equator to about 9.832 m/s2 at the poles, so an object will weigh approximately 0.5% more at the poles than at the Equator.
Altitude
Gravity decreases with altitude as one rises above the Earth's surface because greater altitude means greater distance from the Earth's centre. All other things being equal, an increase in altitude from sea level to causes a weight decrease of about 0.29%. (An additional factor affecting apparent weight is the decrease in air density at altitude, which lessens an object's buoyancy. This would increase a person's apparent weight at an altitude of 9,000 metres by about 0.08%)
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 , equivalent to a typical orbit of the ISS
The International Space Station (ISS) is the largest modular space station currently in low Earth orbit. It is a multinational collaborative project involving five participating space agencies: NASA (United States), Roscosmos (Russia), JAXA (J ...
, gravity is still nearly 90% as strong as at the Earth's surface. Weightlessness actually occurs because orbiting objects are in free-fall
In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on i ...
.
The effect of ground elevation depends on the density of the ground (see Slab correction section). A person flying at above sea level over mountains will feel more gravity than someone at the same elevation but over the sea. However, a person standing on the Earth's surface feels less gravity when the elevation is higher.
The following formula approximates the Earth's gravity variation with altitude:
:
Where
* is the gravitational acceleration at height above sea level.
* is the Earth's mean radius.
* is the standard gravitational acceleration.
The formula treats the Earth as a perfect sphere with a radially symmetric distribution of mass; a more accurate mathematical treatment is discussed below.
Depth
An approximate value for gravity at a distance from the center of the Earth can be obtained by assuming that the Earth's density is spherically symmetric. The gravity depends only on the mass inside the sphere of radius . All the contributions from outside cancel out as a consequence of the inverse-square law of gravitation. Another consequence is that the gravity is the same as if all the mass were concentrated at the center. Thus, the gravitational acceleration at this radius is
:
where is the gravitational constant and is the total mass enclosed within radius . If the Earth had a constant density , the mass would be and the dependence of gravity on depth would be
:
The gravity at depth is given by where is acceleration due to gravity on the surface of the Earth, is depth and is the radius of the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
.
If the density decreased linearly with increasing radius from a density at the center to at the surface, then , and the dependence would be
:
The actual depth dependence of density and gravity, inferred from seismic travel times (see Adams–Williamson equation
The Adams–Williamson equation, named after Leason H. Adams and E. D. Williamson, is an equation used to determine density as a function of radius, more commonly used to determine the relation between the velocities of seismic waves and the den ...
), is shown in the graphs below.
Local topography and geology
Local differences in topography
Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps.
Topography is a field of geoscience and planetary sc ...
(such as the presence of mountains), geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
(such as the density of rocks in the vicinity), and deeper tectonic structure cause local and regional differences in the Earth's gravitational field, known as gravitational anomalies
In theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics — usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with som ...
. Some of these anomalies can be very extensive, resulting in bulges in sea level
Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardise ...
, and throwing pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
clocks out of synchronisation.
The study of these anomalies forms the basis of gravitational geophysics
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...
. The fluctuations are measured with highly sensitive gravimeter
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest.
Units of measurement
Gr ...
s, 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
An oil is any nonpolar chemical substance that is composed primarily of hydrocarbons and is hydrophobic (does not mix with water) & lipophilic (mixes with other oils). Oils are usually flammable and surface active. Most oils are unsaturated ...
and mineral deposits
In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. (2 ...
. Denser rocks (often containing mineral ore
Ore is natural rock or sediment that contains one or more valuable minerals, typically containing metals, that can be mined, treated and sold at a profit.Encyclopædia Britannica. "Ore". Encyclopædia Britannica Online. Retrieved 7 Apr ...
s) cause higher than normal local gravitational fields on the Earth's surface. Less dense sedimentary rock
Sedimentary rocks are types of rock that are formed by the accumulation or deposition of mineral or organic particles at Earth's surface, followed by cementation. Sedimentation is the collective name for processes that cause these particles ...
s cause the opposite.
There is a strong correlation between the gravity derivation map of earth from NASA GRACE with positions of recent volcanic activity, ridge spreading and volcanos: these regions have a stronger gravitation than theoretical predictions.
Other factors
In air or water, objects experience a supporting buoyancy
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the ...
force which reduces the apparent strength of gravity (as measured by an object's weight). The magnitude of the effect depends on the air density (and hence air pressure) or the water density respectively; see Apparent weight
In physics, apparent weight is a property of objects that corresponds to how heavy an object is. The apparent weight of an object will differ from the weight of an object whenever the force of gravity acting on the object is not balanced by an equa ...
for details.
The gravitational effects of the Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
and the Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
(also the cause of the tide
Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another.
Tide tables ...
s) 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.
Direction
Gravity acceleration is a vector quantity, with direction in addition to magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
. In a spherically symmetric Earth, gravity would point directly towards the sphere's centre. As the Earth's figure
Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A sphere is a well-known historical approxima ...
is slightly flatter, there are consequently significant deviations in the direction of gravity: essentially the difference between geodetic latitude
Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a '' reference ellipsoid''.
They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as ...
and geocentric latitude
In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
. Smaller deviations, called vertical deflection
The vertical deflection (VD) or deflection of the vertical (DoV), also known as deflection of the plumb line and astro-geodetic deflection, is a measure of how far the gravity direction at a given point of interest is rotated by local mass anom ...
, are caused by local mass anomalies, such as mountains.
Comparative values worldwide
Tools exist for calculating the strength of gravity at various cities around the world.[Gravitational Fields Widget as of Oct 25th, 2012]
– WolframAlpha
WolframAlpha ( ) is an answer engine developed by Wolfram Research. It answers factual queries by computing answers from externally sourced data.
WolframAlpha was released on May 18, 2009 and is based on Wolfram's earlier product Wolfram Mathe ...
The effect of latitude can be clearly seen with gravity in high-latitude cities: Anchorage (9.826 m/s2), Helsinki (9.825 m/s2), being about 0.5% greater than that in cities near the equator: Kuala Lumpur (9.776 m/s2). The effect of altitude can be seen in Mexico City (9.776 m/s2; altitude ), and by comparing Denver (9.798 m/s2; ) with Washington, D.C. (9.801 m/s2; ), both of which are near 39° N. Measured values can be obtained from Physical and Mathematical Tables by T.M. Yarwood and F. Castle, Macmillan, revised edition 1970.
Mathematical models
If the terrain is at sea level, we can estimate, for the Geodetic Reference System 1980, , the acceleration at latitude :
:
This is the International Gravity Formula 1967, the 1967 Geodetic Reference System Formula, Helmert's equation or Clairaut's formula.[International Gravity formula](_blank)
An alternative formula for ''g'' as a function of latitude is the WGS (World Geodetic System
The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...
) 84 Ellipsoidal Gravity Formula:
:
where,
* are the equatorial and polar semi-axes, respectively;
* is the spheroid's eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
, squared;
* is the defined gravity at the equator and poles, respectively;
* (formula constant);
then, where ,
: