218 Tables 49.51 (continued) 



RELATION BETWEEN GEOPOTENTIAL AND GEOMETRIC HEIGHT 



Integration of equation (2) after (3) is substituted therein yields 



*=(ff)(*T2> ingpm - <4) 



or 



Z= — in meters (5) 



m- 



$ 



Equations (4) and (5) are strictly valid for a nonrotating sphere composed of spherical 

 shells of equal density. Since these conditions are not fulfilled for the earth, and since 

 centrifugal acceleration does not diminish according to the inverse-square law but rather 

 increases with the distance from the center, it is, strictly speaking, necessary to make 

 some allowance for the deviations from the simple conditions assumed. 



This is done by following a suggestion of W. D. Lambert ° of the U. S. Coast and 

 Geodetic Survey. Taking the partial derivative of g with respect to Z in equation (3), 



and evaluating it for Z = 0, i.e., obtaining l S= I , we find for the corresponding 



\oZ /z = o 

 value of R, without giving it a special symbol, 



R = *2L (6) 



"" \dz)z = o 



The quantity in the denominator is a function of latitude, given by 



— (!&) = 3:085462 X 10- 9 + 2.27 X10- 9 cos 2^-2 XIO" 12 cos 40 (7) 



\dZ /Z=zO 



An equation expressing g$ as a function of latitude is given in Table 167. 



When equation (7) is substituted in equation (6) and the resulting value of R used in 

 equations (4) and (5), these expressions for <£ and Z are made to satisfy two boundary 

 conditions, that is, (a) they are in harmony with the value of g$ for the given latitude 

 (neglecting local anomalies), and (b) they are in harmony with the vertical gradient of 

 g at the given latitude at sea level (neglecting local anomalies), assuming the International 

 Ellipsoid represents the figure of the earth. 



The value of R obtained by means of equations (6) and (7) is a fictitious quantity 

 satisfying these two conditions and does not represent the radius vector of either the 

 geoid or the International Ellipsoid of Reference. Effects of the nonspherical figure of 

 the earth, its mean density distribution, and centrifugal acceleration are thus taken into 

 account at mean sea level. Use of the inverse-square law in conjunction with this value 

 of R then yields a satisfactory approximation for the relation of geopotential and geo- 

 metric height even up to heights of several hundred kilometers. 6 The validity of the fore- 

 going technique for relating geopotential and geometric height is uncertain for heights 

 above 600 kilometers. To obtain relationships of greater reliability for great heights it is 

 necessary to have recourse to the more advanced theory given by Helmert. 7 



Note. — The notion of geopotential is derived from purely statical considerations. Therefore the 

 concept of geopotential breaks down absolutely at distances from the earth where the gravitational 

 attraction and the force of centrifugal acceleration are equal and oppositely directed. This occurs at a 

 distance from the earth's center of 6.6 terrestrial radii in the plane of the Equator, about 4.4 terrestrial 

 radii on the earth's axis extended. Within the spheroidal surface at which this absolute break-down 

 occurs, in the statical case, there are high levels where the gravitational attraction and the force of 

 centrifugal acceleration are of the same order of magnitude. Here the vertical component of the force 

 on a moving body due to the Coriolis acceleration which does not have a potential may be significant 

 in relation to the other forces. Where moving bodies are involved at these levels some caution in the 

 use of the concept of geopotential is necessary. Also, the definitions of geographic latitude and 

 elevation become ambiguous at these heights. 



Description of tables. — Table 49 provides values of the quantities R and ( ^ v J as 



functions of latitude, for every whole degree. The last figure in the tabular values of 



R and I -^— J is not significant but is given to permit obtainment of smooth interpolated 



6 Lambert, W. D., Some notes on the calculation of geopotential, unpublished manuscript, 1949. 



In the fifth edition of these tables, use was made of the assumption of a constant vertical gradient 

 of g, as in Helmert's equation, g = g^ — 0.000003086 Z. This yields results which become appreciably 

 erroneous at heights of the order of 10 km., and neglects the latitudinal variation. 



7 Helmert, F. R., Die mathematischen und physikalischen Theorieen der hoheren Geodasie, vol. 2, 

 1886. 



(continued) 

 SMITHSONIAN METEOROLOGICAL TABLES 



