COMPUTING THE MODIFIED INDEX OF REFRACTION. M 



77 



Equation (20) is a small correction which must be 

 taken into account when t, the average temperature, 

 differs appreciably from zero. Since this term con- 

 tributes only 1 per cent to the refractive index in the 

 extreme case of h = 500 m, t = 40, it is sufficient to 

 replace the exponential by its value at the middle of 

 the height range. This approxiiuation does not lead 

 to an error of more than 0.1 M units at these low 

 altitudes. 



Peessure versus Height 



The differential equation connecting height with 

 pressure may be written 



and, for this linear case 



(21) 



dp _ _ g dh 



IT ~ R Y ' 



where g — acceleration of gravity, 

 R = gas constant for air. 



Since g is not strictly constant (it varies slightly with 

 height and locality) and since R, to a slight extent, is 

 dependent on the percentage of water vapor in the 

 air and, finally, since T may be an arbitrary function 

 of height, this differential equation cannot be inte- 

 grated exactly. However, a careful consideration of the 

 order of magnitude of changes in the pressure brought 

 about by the slight changes of g and R leads to the 

 conclusion that such variations may be neglected, 

 particularly as these changes have practically no effect 

 on the slopes of M curves. Picking the best overall 

 values of g and R {g = 9.80665 and R = 287.05 in 

 the units used in this report) gives a = g/R = 

 0.034163 as the value to be used in equation (8). 



The variation of temperature with height cannot, 

 however, be neglected in the integration of equation 

 (21). The integrated form of equation (21) is equa- 

 tion (8), where the approximation 



/ 



'' dJi 

 T 



h 

 T 



(22) 



has been made. T is the average temperature defined 

 by equation (9) or, more roughly, by equation (10). 

 An estimate of the order of accuracy of this approxi- 

 mation may be obtained from an examination of the 

 case in which T varies linearly with h. Let the refer- 

 ence level temperature be T„ at /i = and the tem- 

 perature at the height /i^ be Tj. Then 



hi 

 f 



2/ii 



/•". dh ^ h j^^ T\ 

 •'0 T Ti — i Tq 



ri-ToLTi 



To 1/Ti-r, 



+ To 3 



(^)'-] 



^0 + Ti 



(23) 



(24) 



by a well-known expansion for the natural logarithm. 

 The first term in the series (24) gives exactly equation 

 (23). Hence, the approximation (22) amounts to 

 dropping the higher-order terms in equation (24). 

 The ratio of the second term to the first is only 

 Vs [(Ti — ro)/(ri + ro)]' or about two parts in 

 10,000 for the first 2,000 m of the standard atmos- 

 phere. This comes out to give an error of about 0.006 

 nib in the pressure at this height. This is certainly 

 negligible. 



For a nonlinear atmosphere the question of the 

 error in equation (22) is chiefly a question of the 

 accuracy in determining T, since any such atmosphere 

 can be broken up into a number of layers in each of 

 which T is linear in h. 



Constants of the Index of Eefraction 

 Formula 



The formula for the ordinary index of refraction, 

 n, which has been used in calculating these tables, is 



(n-l)lO^=^-^-f1?^, (25) 



where A = 79, D = 11, 5 = 3.8 X 10^. (26) 



The formula given in reference 11, in the units 

 adopted here, is the same as (25) but with 



A = 78.7, D = 11.2, B = 3.77 X 10^ (27) 



The formula used by Bell Telephone Laboratories 

 (Monograph B-870, 1935) is also (25) but with 



A = 79.1, D = 10.9, B = 3.81 X 10^ (28) 



The third significant figure in all these constants 

 is questionable. Moreover, the absolute value of n 

 (or M) is not important but only the slopes of M 

 curves. For this purpose it is sufficient if the right 

 form of equation and approximately correct values of 

 the constants are chosen. Hence, in these tables, equa- 

 tions (25) and (26) were adopted. 



