224 
THE COEFFICIENT OF OPACITY 
Consider an electron with initial velocity F which would, if undisturbed, 
pass at a distance a from a nucleus of atomic number Z. Under the attrac 
tion of the nucleus it will describe an orbit which will be a hyperbola— 
unless the approach is so close as to make it necessary to allow for change 
of mass with velocity. 
The acceleration will be 
r = /x/r 2 , fj, = Ze 2 /m 
(153-2). 
Let the equation of the hyperbola be 
l , 
- = 1 + e cos 0 
r 
(153*3). 
If h is the constant of areas 
o dO , T/ 
r 2 = h = aV 
dt 
(153-4), 
and by the usual astronomical equation 
ct 2 F 2 = h 2 = ¡¿I 
(153-5). 
Also by the well-known property that a (the perpendicular from the focus 
on the asymptote) is equal to the minor axis of the hyperbola 
e 2 - 1 - l 2 ¡a 2 = cr 2 F 4 //x 2 (153-6), 
and if 2(f) is the angle between the asymptotes 
tanc/> - V(e 2 - 1) = aV 2 //i (153-7). 
By (153-3) and (153-4) 
2 e 2 r , 2 e 2 fi 2 dt 
= 1 iwh (1 + e oos 6)2 de < 153 ' 76 )- 
Hence the total radiation during the encounter is 
Q = 3 cVhj- t „-J 1 + eCOse) * de 
= | {("• - <f>) (sec 2 <j> + 2) + 3 tan ...(153-8), 
where e has been replaced by its value sec </>. 
For not too large values of crV 2 , </> is a small angle; that is to say, the 
orbits are practically parabolas. The result then reduces to 
27re V 277-e 2 /x 4 2nZ i e 10 
^ cH 2 h c s h 5 c 3 m 4 <r 5 F 5 uod-yp 154 
154. As a slight digression we may notice that if Q is greater than 
|mF 2 the electron loses more than its free energy and must be captured. 
Thus there is an apparent target for capture with radius a given by 
, T/2 27rZ 4 e 10 
V 2 = 
c z m i aPV b '