222 
THE COEFFICIENT OF OPACITY 
Each emission results from a capture, so that each free electron is captured 
4-45.10 10 times per second. The reciprocal of this gives the time of free 
paih* 2 -25. 10 _n secs (15T6). 
The average speed of an electron at the temperature 7-2.10 6 is 
V = 1-67.10 9 cm. per sec (151-7). 
Multiplying by the time of free path, the length of free path is 
A = -0375 cm (151-8). 
Here again an averaging factor will slightly modify the result since the 
slower moving electrons are likely to be the more easily captured. 
The “free path” in the sense of the theory of gases, i.e. from collision 
to collision, is much less than -0375 cm. The electron hits a large number 
of atoms before it meets one in such a way as to be captured—or at least 
it hits what would have been the atom if the atom had not been reduced to 
small dimensions by ionisation. We introduce the idea of a target in the 
atom, i.e. a sphere of size such that the probability of capture is equal to 
the probability of hitting the sphere. This representation is primarily 
intended to be statistical and leaves open the question whether capture 
is actually determined by hitting such a target. If a is the radius of the 
target and N the number of targets per cu. cm. it is shown in the theory 
of gases that ^ 
A = 
ttN a 2 
.(151-91), 
the paths being treated as rectilinear. 
We have N = p/Aii (151-92), 
where A is the atomic weight and h the mass of a hydrogen atom. For 
iron (A = 56) at the density of the centre of Capella, this gives 
N = 5-88.10 20 (151-93), 
so that by (151-91) and (151-8) 
a = 1-20.10 -10 cm (151-94). 
This is a little less than the radius of the K ring in iron, which at the 
centre of Capella is all that is left of the system of satellite electrons. But 
there is no special importance to be attached to this coincidence. In 
(151 • 91) the paths are treated as rectilinear so that the target is the apparent 
target aimed at by the electrons. Owing to the attraction of the nucleus, 
the true target, or target actually hit, may be much smaller. 
* At first sight this makes no allowance for the time spent in the captured state; 
but this is compensated, because if each electron spent one-tenth of its time in a 
captured state the number of electrons concerned in the emission would be ten-ninths 
of the number (151-4) free at a given moment.