Gravitational accelerationFrom Wikipedia, the free encyclopedia

In physics, gravitational acceleration is the acceleration on an object caused by force of gravitation.Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the samerate relative to the center of mass.[1] This equality is true regardless of the masses or compositions of thebodies.

At different points on Earth, objects fall with an acceleration between 9.78 and 9.83 m/s2 depending onaltitude and latitude, with a conventional standard value of exactly 9.80665 m/s2 (approximately32.174 ft/s2). Objects with low densities do not accelerate as rapidly due to buoyancy and air resistance.

Contents

1 For point masses

2 Gravity model for Earth

3 General relativity

4 See also

5 References

For point masses

Newton's law of universal gravitation states that there is a gravitational force between any two massesthat is equal in magnitude for each mass, and is aligned to draw the two masses toward each other. Theformula is:

where and are the two masses, is the gravitational constant, and is the distance between thetwo masses. The formula was derived for planetary motion where the distances between the planets andthe Sun made it reasonable to consider the bodies to be point masses. (For a satellite in orbit, the'distance' refers to the distance from the mass centers rather than, say, the altitude above a planet'ssurface.)

If one of the masses is much larger than the other, it is convenient to define a gravitational field aroundthe larger mass as follows:[2]

where is the mass of the larger body, and is a unit vector directed from the large mass to thesmaller mass. The negative sign indicates that the force is an attractive force.

In that way, the force acting upon the smaller mass can be calculated as:

where is the force vector, is the smaller mass, and is a vector pointed toward the larger body.Note that has units of acceleration and is a vector function of location relative to the large body,independent of the magnitude (or even the presence) of the smaller mass.

This model represents the "far­field" gravitational acceleration associated with a massive body. Whenthe dimensions of a body are not trivial compared to the distances of interest, the principle ofsuperposition can be used for differential masses for an assumed density distribution throughout thebody in order to get a more detailed model of the "near­field" gravitational acceleration. For satellites inorbit, the far­field model is sufficient for rough calculations of altitude versus period, but not forprecision estimation of future location after multiple orbits.

The more detailed models include (among other things) the bulging at the equator for the Earth, andirregular mass concentrations (due to meteor impacts) for the Moon. The Gravity Recovery And ClimateExperiment (GRACE) mission launched in 2002 consists of two probes, nicknamed "Tom" and "Jerry",in polar orbit around the Earth measuring differences in the distance between the two probes in order tomore precisely determine the gravitational field around the Earth, and to track changes that occur overtime. Similarly, the Gravity Recovery and Interior Laboratory (GRAIL) mission from 2011­2012consisted of two probes ("Ebb" and "Flow") in polar orbit around the Moon to more precisely determinethe gravitational field for future navigational purposes, and to infer information about the Moon'sphysical makeup.

Gravity model for Earth

The type of gravity model used for the Earth depends upon the degree of fidelity required for a givenproblem. For many problems such as aircraft simulation, it may be sufficient to consider gravity to be aconstant, defined as:[3]

9.80665 metres (32.1740 ft) per s²

based upon data from World Geodetic System 1984 (WGS­84), where is understood to be pointing'down' in the local frame of reference.

If it is desirable to model an object's weight on Earth as a function of latitude, one could use thefollowing ([3] p. 41):

where

= 9.832 metres (32.26 ft) per s² = 9.806 metres (32.17 ft) per s²

= 9.780 metres (32.09 ft) per s²lat = latitude, between −90 and 90 degrees

Both these models take into account the centrifugal relief that is produced by the rotation of the Earth,and neither accounts for changes in gravity with changes in altitude. It is worth noting that for the massattraction effect by itself, the gravitational acceleration at the equator is about 0.18% less than that at thepoles due to being located farther from the mass center. When the rotational component is included (asabove), the gravity at the equator is about 0.53% less than that at the poles, with gravity at the polesbeing unaffected by the rotation. So the rotational component change due to latitude (0.35%) is abouttwice as significant as the mass attraction change due to latitude (0.18%), but both reduce strength ofgravity at the equator as compared to gravity at the poles.

Note that for satellites, orbits are decoupled from the rotation of the Earth so the orbital period is notnecessarily one day, but also that errors can accumulate over multiple orbits so that accuracy isimportant. For such problems, the rotation of the Earth would be immaterial unless variations withlongitude are modeled. Also, the variation in gravity with altitude becomes important, especially forhighly elliptical orbits.

The Earth Gravitational Model 1996 (EGM96) contains 130,676 coefficients that refine the model ofthe Earth's gravitational field ([3] p. 40). The most significant correction term is about two orders ofmagnitude more significant than the next largest term ([3] p. 40). That coefficient is referred to as the term, and accounts for the flattening of the poles, or the oblateness, of the Earth. (A shape elongated onits axis­of­symmetry, like an American football, would be called prolate.) A gravitational potentialfunction can be written for the change in potential energy for a unit mass that is brought from infinityinto proximity to the Earth. Taking partial derivatives of that function with respect to a coordinatesystem will then resolve the directional components of the gravitational acceleration vector, as afunction of location. The component due to the Earth's rotation can then be included, if appropriate,based on a sidereal day relative to the stars (≈366.24 days/year) rather than on a solar day (≈365.24days/year). That component is perpendicular to the axis of rotation rather than to the surface of theEarth.

A similar model adjusted for the geometry and gravitational field for Mars can be found in publicationNASA SP­8010.[4]

The barycentric gravitational acceleration at a point in space is given by:

where:

M is the mass of the attracting object, is the unit vector from center­of­mass of the attracting object tothe center­of­mass of the object being accelerated, r is the distance between the two objects, and G is thegravitational constant.

When this calculation is done for objects on the surface of the Earth, or aircraft that rotate with theEarth, one has to account that the Earth is rotating and the centrifugal acceleration has to be subtractedfrom this. For example, the equation above gives the acceleration at 9.820 m/s², whenGM = 3.986×1014 m³/s², and R=6.371×106 m. The centripetal radius is r = R cos(latitude), and thecentripetal time unit is approximately (day / 2π), reduces this, for r = 5×106 metres, to 9.79379 m/s²,which is closer to the observed value.

General relativity

In Einstein's theory of general relativity, gravitation is an attribute of curved spacetime instead of beingdue to a force propagated between bodies. In Einstein's theory, masses distort spacetime in their vicinity,and other particles move in trajectories determined by the geometry of spacetime. The gravitationalforce is a fictitious force. There is no gravitational acceleration, in that the proper acceleration and hencefour­acceleration of objects in free fall are zero. Rather than undergoing an acceleration, objects in freefall travel along straight lines (geodesics) on the curved spacetime.

See also

Air trackGravimetryGravity of EarthNewton's law of universal gravitationStandard gravity

References1. Gerald James Holton and Stephen G. Brush (2001). Physics, the human adventure: from Copernicus toEinstein and beyond (3rd ed.). Rutgers University Press. p. 113. ISBN 978­0­8135­2908­0.

2. Fredrick J. Bueche (1975). Introduction to Physics for Scientists and Engineers, 2nd Ed. USA: VonHoffmann Press. ISBN 0­07­008836­5.

3. Brian L. Stevens; Frank L. Lewis (2003). Aircraft Control And Simulation, 2nd Ed. Hoboken, New Jersey:John Wiley & Sons, Inc. ISBN 0­471­37145­9.

4. Richard B. Noll; Michael B. McElroy (1974), Models of Mars' Atmosphere [1974], Greenbelt, Maryland:NASA Goddard Space Flight Center, SP­8010.

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