Gravity of Earth

The gravity of Earth, 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 meters per second squared (in symbols, m/s2 or m·s−2) or equivalently in newtons per kilogram (N/kg or N·kg−1). It has an approximate value of 9.81 m/s2, which means that, ignoring the effects of air resistance, the speed of an object falling freely near the Earth's surface will increase by about 9.81 metres (32.2 ft) per second every second. This quantity is sometimes referred to informally as little g (in contrast, the gravitational constant G is referred to as big G).
There is a direct relationship between gravitational acceleration and the downwards weight force experienced by objects on Earth, given by the equation F = ma (force = mass × acceleration). However, other factors such as the rotation of the Earth also contribute to the net acceleration.
Earth's gravity measured by NASA's GRACE mission, showing deviations from the theoretical gravity of an idealized smooth Earth, the so-called earth ellipsoid. Red shows the areas where gravity is stronger than the smooth, standard value, and blue reveals areas where gravity is weaker. (Animated version

Contents

  • 1 Variation in gravity and apparent gravity
    • 1.1 Latitude
    • 1.2 Altitude
    • 1.3 Depth
    • 1.4 Local topography and geology
    • 1.5 Other factors
    • 1.6 Comparative gravities in various cities around the world
    • 1.7 Mathematical models
      • 1.7.1 Latitude model
      • 1.7.2 Free air correction
      • 1.7.3 Slab correction
  • 2 Estimating g from the law of universal gravitation
  • 3 Comparative gravities of the Earth, Sun, Moon, and planets

    Variation in gravity and apparent gravity

    A perfect sphere of spherically uniform density (density varies solely with distance from centre) would produce a gravitational field of uniform magnitude at all points on its surface, always pointing directly towards the sphere's centre. However, the Earth deviates slightly from this ideal, and there are consequently slight deviations in both the magnitude and direction of gravity across its surface. Furthermore, the net force exerted on an object due to the Earth, called "effective gravity" or "apparent gravity", varies due to the presence of other factors, such as inertial response to the Earth's rotation. A scale or plumb bob measures only this effective gravity.
    Parameters affecting the apparent or actual strength of Earth's gravity include latitude, altitude, and the local topography and geology.
    Apparent 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 9.766 in Kuala Lumpur, Mexico City, and Singapore to 9.825 in Oslo and Helsinki.

    Latitude

    The differences of Earth's gravity around the Antarctic continent.
    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 inertia) 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, an object at the Equator experiences a weaker gravitational pull than an object at the poles.
    In combination, the equatorial bulge and the effects of the Earth's inertia 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.
    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 inertia, 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

    The graph shows the variation in gravity relative to the height of an object
    Gravity decreases with altitude as one rises above the earth's surface because greater altitude means greater distance from the Earth's center. 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%. (An additional factor affecting apparent weight is the decrease in air density at altitude, which lessens an object's buoya. 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 400 kilometres (250 mi), equivalent to a typical orbit of the Space Shuttle, gravity is still nearly 90% as strong as at the Earth's surface. Weightlessness actually occurs because orbiting objects are in free-fall.
    The effect of ground elevation depends on the density of the ground (see Slab correction section). A person flying at 30 000 ft 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:
    g_h=g_0\left(\frac{r_e}{r_e+h}\right)^2
    Where
  • gh is the gravitational acceleration at height h\, above sea level.
  • re is the Earth's mean radius.
  • g0 is the standard gravitational acceleration.
This formula treats the Earth as a perfect sphere with a radially symmetric distribution of mass; a more accurate mathematical treatment is discussed below.

Depth

See also: Shell theorem
An approximate depth dependence of density in the Earth can be obtained by assuming that the mass is spherically symmetric (it depends only on depth, not on latitude or longitude). In such a body, the gravitational acceleration is towards the center. The gravity at a radius r depends only on the mass inside the sphere of radius r; all the contributions from outside cancel out. This is 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 of the Earth. Thus, the gravitational acceleration at this radius is
g(r) = -\frac{GM(r)}{r^2}.
where G is the gravitational constant and M(r) is the total mass enclosed within radius r. 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
g(r) = \frac{4\pi}{3} G \rho r.
If the density decreased linearly with increasing radius from a density ρ0 at the centre to ρ1 at the surface, then ρ(r) = ρ0 − (ρ0 − ρ1) r / re, and the dependence would be
g(r) = \frac{4\pi}{3} G \rho_0 r - \pi G \left(\rho_0-\rho_1\right) \frac{r^2}{r_e}.
The actual depth dependence of density and gravity, inferred from seismic travel times (see Adams–Williamson equation), is shown in the graphs below.
Earth's radial density distribution according to the Preliminary Reference Earth Model
Earth's gravity according to the Preliminary Reference Earth Model (PREM). Two models for a spherically symmetric Earth are included for comparison. The straight dashed line is for a constant density equal to the Earth's average density. The curved dotted line is for a density that decreases linearly from center to surface. The density at the centre is the same as in the PREM, but the surface density is chosen so that the mass of the sphere equals the mass of the real Earth.

Local topography and geology

See also: Physical geodesy
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.

Comparative gravities in various cities around the world

The table below shows the gravitational acceleration in various cities around the world amongst these listed cities, it is lowest in Mexico City or Kuala Lumpur (9.776 m/s2) and highest in Anchorage, Alaska (9.826 m/s2).

Mathematical models

Latitude model

If the terrain is at sea level, we can estimate g:
g_{\phi}=9.780327 \left(1+0.0053024\sin^2 \phi-0.0000058\sin^2 2\phi \right) \frac{\mathrm{m}}{\mathrm{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 Clairaut's formula
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)\,\frac{\mathrm{m}}{\mathrm{s}^2}
or
\ g_{\phi}= \left( 9.780327 + 0.0516323\sin^2(\phi) + 0.0002269\sin^4(\phi) \right)\,\frac{\mathrm{m}}{\mathrm{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)\,\frac{\mathrm{m}}{\mathrm{s}^2}
The difference between the WGS-84 formula and Helmert's equation is less than 0.68·10−6 m·s−2.

Free air correction

The first correction to be applied to the model is the free air correction (FAC) that accounts for heights above sea level. Near the surface of the Earth (sea level), gravity decreases with height such that linear extrapolation would give zero gravity at a height of one half the radius is 9.8 m·s−2 per 3,200 km.
Using the mass and radius of the Earth:
r_\mathrm{Earth}= 6.371 \times 10^{6}\,\mathrm{m}
m_\mathrm{Earth}= 5.9736 \times 10^{24}\,\mathrm{kg}
The FAC correction factor (Δg) can be derived from the definition of the acceleration due to gravity in terms of G, the Gravitational Constant (see Estimating g from the law of universal gravitation, below):
g_0 = G \, m_\mathrm{Earth} / r_\mathrm{Earth}^2 = 9.8331\,\frac{\mathrm{m}}{\mathrm{s}^2}
where:
G = 6.67428 \times 10^{-11}\,\frac{\mathrm{m}^3}{\mathrm{kg}\cdot\mathrm{s}^2}.
At a height h above the nominal surface of the earth gh is given by:
g_h = G \, m_\mathrm{Earth} / \left( r_\mathrm{Earth} + h \right) ^2
So the FAC for a height h above the nominal earth radius can be expressed:
\Delta g_h = \left [ G \, m_\mathrm{Earth} / \left( r_\mathrm{Earth} + h \right) ^2 \right ] - \left[G \, m_\mathrm{Earth} / r_\mathrm{Earth}^2 \right]
This expression can be readily used for programming or inclusion in a spreadsheet. Collecting terms, simplifying and neglecting small terms (h<<rEarth), however yields the good approximation:
\Delta g_h \approx - \, \dfrac{ G \, m_\mathrm{Earth}}{ r_\mathrm{Earth} ^2} \times \dfrac{ 2 \,h}{r_\mathrm{Earth}}
Using the numerical values above and for a height h in metres:
\Delta g_h \approx - 3.084 \times 10^{-6}\, h
Grouping the latitude and FAC altitude factors the expression most commonly found in the literature is:
g_{\phi, h}=9.780 327 \left( 1+0.0053024\sin^2 \phi-0.0000058\sin^2 2\phi \right) - 3.086 \times 10^{-6}h
where \ g_{\phi, h} = acceleration in m·s−2 at latitude \ \phi and altitude h in metres. Alternatively (with the same units for h) the expression can be grouped as follows:
g_{\phi, h}=9.780327 \left[ \left( 1+0.0053024\sin^2 \phi-0.0000058\sin^2 2\phi \right) - 3.155 \times 10^{-7}h \right] \,\frac{\mathrm{m}}{\mathrm{s}^2}


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