Gravity of Earth

In SI units, 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 acceleration due to gravity, accurate to 2 significant figures, is 9.8 m/s² (32 ft/s²). This means that, ignoring the effects of air resistance, the vertical component of velocity of an object falling freely will increase in the downwards direction by about 9.8 metres per second (32 ft/s) every second.

The precise strength of Earth's gravity varies with location. The conventional value for standard gravity is 9.80665 m⋅s⁻²‍ by definition, originally adopted by the CGPM in 1901. This quantity is denoted variously as g_n, g_e, g₀, or simply g (which is also used for the variable local value).

The weight of an object on Earth's surface is the downwards force on that object, given by Newton's second law of motion, or F = m a (force = mass × acceleration). Gravitational acceleration 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.

Magnitude

A non-rotating perfect sphere of uniform mass density, or whose density varies solely with distance from the centre (spherical symmetry), would produce a gravitational field of uniform magnitude at all points on its surface. 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/s² on the Nevado Huascarán mountain in Peru to 9.8337 m/s² at the surface of the Arctic Ocean. In large cities, it ranges from 9.7806 m/s² in Kuala Lumpur, Mexico City, and Singapore to 9.825 m/s² in Oslo and Helsinki.

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 at the Pavillon de Breteuil 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 and pound force.

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 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 (itself also caused by centrifugal force from rotation) causes objects at the Equator to be further from the planet's center than objects at the poles. The force due to gravitational attraction between two masses (a piece of the Earth and the object being weighed) varies inversely with the square of the distance between them. The distribution of mass is also different below someone on the equator and below someone at a pole. The net result is that an object at the Equator experiences a weaker gravitational pull than an object on one of the poles.

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 9,000 metres (30,000 ft) causes a weight decrease of about 0.29%.

For heights ${\displaystyle h>0}$ above the Earth's surface, gravitational acceleration can be computed by an inverse-square formula:

${\displaystyle g_h=g_0{\left(\frac{R_{\mathrm{e}}}{R_{\mathrm{e}}+h}\right)}^2}$

where

• g_h is the gravitational acceleration at height h above sea level.
• R_e is the Earth's mean radius.
• g₀/ is the standard gravitational acceleration. The formula treats the Earth as a perfect sphere with a radially symmetric distribution of mass.

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 mi), equivalent to a typical orbit of the ISS, gravity is still nearly 90% as strong as at the Earth's surface. Weightlessness actually occurs because orbiting objects are in free-fall.

Depth

An approximate value for gravity at a distance r from the center of the Earth can be obtained by assuming that the Earth's density is spherically symmetric. The force of gravity at a radius r depends only on the mass inside the sphere of that 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

${\displaystyle g(r)=-\frac{GM}{r^2}.}$

where G is the gravitational constant and M is the total mass enclosed within radius r. This result is known as the Shell theorem; it took Isaac Newton 20 years to prove this result, delaying his work on gravity.

If the Earth had a constant density ρ, the mass would be M(r) = (4/3)πρr³ and the dependence of gravity on depth would be

${\displaystyle g(r)=\frac{4\pi }{3}G\rho r.}$

The gravity g′ at depth d is given by g′ = g(1 − d/R) where g is acceleration due to gravity on the surface of the Earth, d is depth and R is the radius of the Earth. If the density decreased linearly with increasing radius from a density ρ₀ at the center to ρ₁ at the surface, then ρ(r) = ρ₀ − (ρ₀ − ρ₁) r / R, and the dependence would be

${\displaystyle g(r)=\frac{4\pi }{3}G{\rho }_{0}r-\pi G\left({\rho }_0-{\rho }_1\right)\frac{r^2}{R}.}$

The actual depth dependence of density and gravity, inferred from seismic travel times (see Adams–Williamson equation), is shown in the graphs below.

Local topography and geology

Local differences in topography (such as the presence of mountains), geology (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 gravity 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.