Gravity of Earth
The gravity of Earth, denoted by g, is the
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 speed of an object falling freely will increase by about 9.8 metres per second (32 ft/s) every second. This quantity is sometimes referred to informally as little g (in contrast, the gravitational constant G is referred to as big G).
The precise strength of Earth's gravity varies with location. The agreed upon value for standard gravity is 9.80665 m/s2 (32.1740 ft/s2) by definition.[4] This quantity is denoted variously as gn, ge (though this sometimes means the normal gravity at the equator, 9.7803267715 m/s2 (32.087686258 ft/s2)),[5] 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.
Variation in magnitude
A non-rotating perfect
Gravity on the Earth's surface varies by around 0.7%, from 9.7639 m/s2 on the
Conventional value
In 1901, the third
Latitude
The surface of the Earth is rotating, so it is
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.[2][10]
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%. (An additional factor affecting apparent weight is the decrease in air density at altitude, which lessens an object's buoyancy.[11] 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
The effect of ground elevation depends on the density of the ground (see Slab correction section). A person flying at 9,100 m (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:
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.
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 r 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 r. 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[14]
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
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
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 gravitational anomalies.[15] 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
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 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 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.
Direction
Gravity acceleration is a
Comparative values worldwide
Tools exist for calculating the strength of gravity at various cities around the world.[16] 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 2,240 metres (7,350 ft)), and by comparing Denver (9.798 m/s2; 1,616 metres (5,302 ft)) with Washington, D.C. (9.801 m/s2; 30 metres (98 ft)), 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.[17]
Location | m/s2 | ft/s2 | Location | m/s2 | ft/s2 | Location | m/s2 | ft/s2 | Location | m/s2 | ft/s2 | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Anchorage |
9.826 | 32.24 | Helsinki | 9.825 | 32.23 | Oslo | 9.825 | 32.23 | Copenhagen | 9.821 | 32.22 | |||
Stockholm | 9.818 | 32.21 | Manchester | 9.818 | 32.21 | Amsterdam | 9.817 | 32.21 | Kotagiri | 9.817 | 32.21 | |||
Birmingham | 9.817 | 32.21 | London | 9.816 | 32.20 | Brussels | 9.815 | 32.20 | Frankfurt | 9.814 | 32.20 | |||
Seattle | 9.811 | 32.19 | Paris | 9.809 | 32.18 | Montréal |
9.809 | 32.18 | Vancouver | 9.809 | 32.18 | |||
Istanbul | 9.808 | 32.18 | Toronto | 9.807 | 32.18 | Zurich |
9.807 | 32.18 | Ottawa | 9.806 | 32.17 | |||
Skopje | 9.804 | 32.17 | Chicago | 9.804 | 32.17 | Rome | 9.803 | 32.16 | Wellington | 9.803 | 32.16 | |||
New York City | 9.802 | 32.16 | Lisbon | 9.801 | 32.16 | Washington, D.C. | 9.801 | 32.16 | Athens | 9.800 | 32.15 | |||
Madrid | 9.800 | 32.15 | Melbourne | 9.800 | 32.15 | Auckland | 9.799 | 32.15 | Denver | 9.798 | 32.15 | |||
Tokyo | 9.798 | 32.15 | Buenos Aires | 9.797 | 32.14 | Sydney | 9.797 | 32.14 | Nicosia | 9.797 | 32.14 | |||
Los Angeles | 9.796 | 32.14 | Cape Town | 9.796 | 32.14 | Perth | 9.794 | 32.13 | Kuwait City | 9.792 | 32.13 | |||
Taipei | 9.790 | 32.12 | Rio de Janeiro | 9.788 | 32.11 | Havana | 9.786 | 32.11 | Kolkata | 9.785 | 32.10 | |||
Hong Kong | 9.785 | 32.10 | Bangkok | 9.780 | 32.09 | Manila | 9.780 | 32.09 | Jakarta | 9.777 | 32.08 | |||
Kuala Lumpur | 9.776 | 32.07 | Singapore | 9.776 | 32.07 | Mexico City | 9.776 | 32.07 | Kandy | 9.775 | 32.07 |
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
An alternative formula for g as a function of latitude is the WGS (
where,
- are the equatorial and polar semi-axes, respectively;
- is the spheroid's eccentricity, squared;
- is the defined gravity at the equator and poles, respectively;
- (formula constant);
then, where ,[19]
- .
where the semi-axes of the earth are:
The difference between the WGS-84 formula and Helmert's equation is less than 0.68 μm·s−2.
Further reductions are applied to obtain gravity anomalies (see: Gravity anomaly#Computation).
Estimating g from the law of universal gravitation
From the
where r is the distance between the centre of the Earth and the body (see below), and here we take to be the mass of the Earth and m to be the mass of the body.
Additionally,
Comparing the two formulas it is seen that:
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:[20]
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 "Variation in magnitude":
- 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 only includes true gravity. It does not include the reduction of constraint force that we perceive as a reduction of gravity due to the rotation of Earth, and some of gravity being counteracted by centrifugal force.
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.
Measurement
The measurement of Earth's gravity is called gravimetry.
Satellite measurements
Currently, the static and time-variable Earth's gravity field parameters are being determined using modern satellite missions, such as
Large-scale gravity anomalies can be detected from space, as a by-product of satellite gravity missions, e.g., GOCE. These satellite missions aim at the recovery of a detailed gravity field model of the Earth, typically presented in the form of a spherical-harmonic expansion of the Earth's gravitational potential, but alternative presentations, such as maps of geoid undulations or gravity anomalies, are also produced.
TheSee also
- Escape velocity – Concept in celestial mechanics
- Atmospheric escape – Loss of planetary atmospheric gases to outer space
- Figure of the Earth – Size and shape used to model the Earth for geodesy
- Geopotential – Energy related to Earth's gravity
- Geopotential model – Theoretical description of Earth's gravimetric shape
- Bouguer anomaly – Type of gravity anomaly
- Gravitation of the Moon
- Gravitational acceleration – Change in speed due only to gravity
- Gravity – Attraction of masses and energy
- Gravity anomaly – Difference between ideal and observed gravitational acceleration at a location
- Gravity of Mars – Gravitational force exerted by the planet Mars
- Newton's law of universal gravitation – Classical statement of gravity as force
- Vertical deflection – Measure of the downward gravitational force's shift due to nearby mass
References
- ^ NASA/JPL/University of Texas Center for Space Research. "PIA12146: GRACE Global Gravity Animation". Photojournal. NASA Jet Propulsion Laboratory. Retrieved 30 December 2013.
- ^ a b Boynton, Richard (2001). "Precise Measurement of Mass" (PDF). Sawe Paper No. 3147. Arlington, Texas: S.A.W.E., Inc. Archived from the original (PDF) on 27 February 2007. Retrieved 22 December 2023.
- ISBN 978-3-211-33544-4. § 2.1: "The total force acting on a body at rest on the earth's surface is the resultant of gravitational force and the centrifugal force of the earth's rotation and is called gravity."
- ^ Bureau International des Poids et Mesures (1901). "Déclaration relative à l'unité de masse et à la définition du poids; valeur conventionnelle de gn". Comptes Rendus des Séances de la Troisième Conférence· Générale des Poids et Mesures (in French). Paris: Gauthier-Villars. p. 68.
Le nombre adopté dans le Service international des Poids et Mesures pour la valeur de l'accélération normale de la pesanteur est 980,665 cm/sec², nombre sanctionné déjà par quelques législations. Déclaration relative à l'unité de masse et à la définition du poids; valeur conventionnelle de gn.
- S2CID 195290884. Retrieved 2023-07-26.
γe = 9.780 326 7715 m/s² normal gravity at equator
- S2CID 54867946.
- ^ "Wolfram|Alpha Gravity in Kuala Lumpur", Wolfram Alpha, accessed November 2020
- ISBN 978-0-19-530786-3.
- ^ Resolution of the 3rd CGPM (1901), page 70 (in cm/s2). BIPM – Resolution of the 3rd CGPM
- ^ "Curious About Astronomy?". Cornell University. Archived from the original on 28 July 2013. Retrieved 22 December 2023.
- ^ "I feel 'lighter' when up a mountain but am I?", National Physical Laboratory FAQ
- ^ "The G's in the Machine", NASA, see "Editor's note #2"
- ^ ISSN 0031-9201.
- ISBN 9781572594913.
- .
- ^ Gravitational Fields Widget as of Oct 25th, 2012 – WolframAlpha
- ^ T.M. Yarwood and F. Castle, Physical and Mathematical Tables, revised edition, Macmillan and Co LTD, London and Basingstoke, Printed in Great Britain by The University Press, Glasgow, 1970, pp 22 & 23.
- ^ International Gravity formula Archived 2008-08-20 at the Wayback Machine
- ^ a b "Department of Defense World Geodetic System 1984 ― Its Definition and Relationships with Local Geodetic Systems,NIMA TR8350.2, 3rd ed., Tbl. 3.4, Eq. 4-1" (PDF). Archived from the original (PDF) on 2014-04-11. Retrieved 2015-10-15.
- ^ "Gravitation". www.ncert.nic. Retrieved 2022-01-25.
- hdl:10281/240694.
- PMID 31534490.
- .