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Chapter: 2 General physics
    Section: 2.7 Astronomy and geophysics
        SubSection: 2.7.5 Gravity

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2.7.5 Gravity

The gravity field of the Earth

The potential of the external gravity field of the Earth is usually expressed as a series of spherical harmonic terms:

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V = − 

GM

1 −

Jn

a

n

 Pn (cos θ) + terms depending on longitude

m2 s-2

r

r


where

 r

 

= distance from the centre of the Earth

a

 

= Earth’s equatorial radius, θ = geocentric co-latitude,

M

 

= mass of the Earth (see below)

Pn(cos θ)

 

= Legendre function of degree n

106J2

 

= 1082.63

106J3

 

= −2.5

106J4

 

= −2.37


       The values of Jn are determined from the behaviour of artificial satellites about the Earth.
       (See section 2.7.4)
       The corresponding expression for the variation of the acceleration due to gravity over the surface of the (spinning) Earth is

              g = ge(1 + β1 sin2 − β2 sin2 2) − 3.088 × 10−6 H m s−2

where is the geographical latitude, H is the height above sea level (in metres) and ge is the value of gravity at the equator.
       The values recommended by the International Union of Geodesy and Geophysics are

             ge = 9.780 327 m s−2

             β1 = 0.005 302 4

             β2 = 0.000 005 8

     The above formula gives the best simple method of calculating g at a place where it has not been measured. It will almost always give results within 10−3 and usually within 5 × 10−4 m s−2. The agreement with observa­tion is usually made worse by the application of a correction for the attraction of the land above sea level.

   The standard acceleration of gravity is defined as 9.80665 m s−2 exactly.




Absolute value of the acceleration due to gravity, g.

Values of the acceleration due to gravity in terms of the fundamental units of length and time were, until recently, measured at very few sites and values elsewhere were found from measurements of differences. In recent years absolute apparatus using laser interferometer measurements of positions of falling reflectors has become highly developed and easy to use. The constant term in the gravity above is now derived from a number of absolute measurements and it is no longer necessary to give an absolute value at one or two or three preferred sites.




References

Geodetic Reference System (1980), IUGG.
W. Torge, (1989) Gravimetry, de Gruyter, Berlin, New York.
W. Torge, (1991), Geodesy, 2nd edn, de Gruyter, Berlin, New York.




Sir Alan Cook

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