TROPOSPHERIC PROPAGATION AND RADIO METEOROLOGY 139 
slightly different form. Instead of using angles re- 
ferred to the plane surfaces it is now necessary to 
refer the angles to horizontal planes tangent to 
spheres about the center of the earth (see Figure 9). 
The new form, as given on page 165, is 
Mr COS A=ANp COS ao , (4) 
where r and a are values of the radius vector from the 
center of the earth to a point in the atmosphere and 
oh 
Ficure 9. Refraction through a curved layer. 
to the earth’s surface, respectively. a now stands for 
the angle formed by the ray with a plane normal to 
the radius vector. ao and 7 are the values of a and n 
at the ground surface. 
If h is the height above the ground surface, so that 
r= h-+a, the above equation may also be written 
in the form 
. n (1+ 2) cos a = np £08 a1 (5) 
h/aisa very small quantity, and n differs from unity 
by only a few parts in 10,000. Under these conditions 
n(1 + h/a) may be replaced by n + h/a with neg- 
ligible error. The quantity n + h/a is called the 
modified refractive index, or the modified index for 
short Equation (5) then assumes the form 
(» + *) COS @ = No COS an. (6) 
As a result of general agreement it is customary to 
“use, instead of n + h/a, the symbol M defined as 
follows: 
M = (n4+2—1) 108. (7) 
At the surface of the ground M reduces to 
My= (nmo—1) 10°. (8) 
Hence M is the excess of the modified refractive in- 
dex above unity, measured in units of one millionth. 
This unit is called an M unit [MU]. Values of M for 
the atmosphere lie in the range of 200 to 500. Cus- 
tomarily M is referred to simply as the modified 
index of refraction. 
Using the numerical value for the radius of the 
earth, 6.37 X 10°m (21 X 10 ft), the rate of increase 
of M with height, owing to the term h/a, is (1/a) 10, 
which is equal to 0.157 MU per meter (0.048 MU per 
foot). As the result of a large number of experiments, 
carried out chiefly in the northern temperate lati- 
tudes, the rate of decrease with height of the re- 
fractive index has been found, on the average, to be 
an 6 — 11 igs 
adh 105 = ae 10 
= —0.039 MU per meter . (9) 
This is the rate of decrease assumed for the standard 
atmosphere. 
It will be noticed that the average rate of decrease 
of n with height is one quarter of the rate of increase 
of the term h/a which results from the curvature of 
the earth. The fact that these quantities are of com- 
parable magnitude is of great importance, as will be 
seen later. 
Consequently the vertical gradient of M for the 
standard atmosphere is 
dM _ dn il 5 
a= (Gta) 
= Gj — 1 6 
= ¢:) 10° = Tay (10) 
(3) 
which has the value 0.118 MU per meter (0.036 MU 
per ft). The value of M at any height, relative to the 
surface value Mo, for the standard atmosphere, is 
equal to 
M—M,=0.118 h; h in meters, 
M—M,=0.036 h;_ h in feet. (11) 
Equivalent Earth Radius— 
Flat Earth Diagram 
An important conclusion may be drawn from 
equation (11). As will be shown on pages 14'6-147, 
dn/dh is the negative of the curvature of a ray in the 
atmosphere, and 1/a is the. curvature of the earth. 
The algebraic sum of these two quantities (their 
numerical difference) is the curvature of the ray 
relative to that of the earth. The net result is this: if 
the earth is replaced by an equivalent earth with an 
enlarged radius equal to 4a/3 the rays may be drawn 
as straight lines. To state the result in another way: 
using the equivalent earth with radius equal to 4a/3 
corresponds to replacing the actual atmosphere, in 
which the index n decreases with height, by a homo- 
geneous atmosphere with an equivalent index n’ 
which is independent of height (see Figures 10, 11, 13, 
14, and 15). This transformation of coordinates great- 
ly facilitates the calculation and interpretation of cov- 
erage diagrams for the standard atmosphere. 
More generally, if the rate of change of n with 
height differs from the value—(1/4) (1/a) 10° MU 
per meter given above, which may be true in certain 
parts of the world, the equivalent earth radius de- 
parts from the value 4a/3. In general the equivalent 
earth radius is designated by ka. For a steeper drop 
of refractive index with height, & increases and be- 
comes infinite when the curvature of the ray is just 
equal to the curvature of the earth. 
