Ionisation 
155 
141-143] 
relevant parts of these curves by a curve such as that drawn thick in the 
diagram, we get a graph of the actual values of (1 — x) p. 
142. We have seen that (1 — x) p will not pass through a minimum 
resulting from the ionisation of the (r + 1) quantum ring, unless 
152 x 10 5 (1421). 
<r (r + 1)" 
As a star of given mass contracts and the various rings of electrons are 
ionised in turn, X and N 3 remain approximately the same, but er and r +1 
decrease with each successive ionisation. Thus a time may come, according to 
the values of X and N, when a new ionisation does not produce a minimum 
of (1 — x) p at all, but instead (1 — x) p increases continuously and without 
limit. When this ring is reached, the graph of (1 — x) p rushes upwards 
without limit until deviations from the gas-laws occur. As before, the effect 
of these is to check the upward rush of (1 - x) p, and ultimately the graph 
of (1 — x) p must become asymptotic to a curve giving values of (1 — x) p at 
which the atoms are jammed so close together that no further compression 
is possible. 
Precise calculations given later (p. 162) will shew that the atomic numbers 
of actual stellar atoms are in the neighbourhood of 95. With this value for 
N, equation (142T) shews that the ionisation of the T-quantum ring will give 
a maximum and a minimum to (1 — x) p only if 
X < 0*444cr (t + l) s . 
For the Xf-ring, t + 1 = 3 and a = 18 so that ilf-ring ionisation will give a 
minimum value to (1 - x) p only if X< 216, a condition satisfied by all stars 
whose mass is greater than about three-quarters of the sun’s mass. 
For the Z-ring, t +1 = 2 and a — 8, so that A-ring ionisation gives a 
definite minimum only if X < 2*9, a condition which is satisfied only by stars 
more massive than Sirius. 
For the XT-ring, r +1 = 1 and a = 2, so that XT-ring ionisation gives a 
definite minimum only if X < 0*09, a condition which is not satisfied by even 
the most massive of known stars. Thus actual stars can shew no minimum 
for XT-ring ionisation, at any rate until the density is so great that our analysis 
has failed through the deviations from the gas-laws becoming excessive. 
143. To examine the state of ionisation in a star in which the deviations 
from the gas-laws are great, we must replace equation (140T) which we have 
so far had under discussion by the more general equation (139’4), namely 
ÿ 2 
3 
y~ 2 
107W . lwyrT 4XA» 
3X dTJ 
T^d\ 
+ 3X dT. 
...(143-1). 
O- (r + l) 3 
The two terms in d\/dT in this equation represent the effect of deviations 
from the gas-laws. It is readily seen* that the term in 0X/02 1 on the left- 
* M.N. iiXxxvii. (1927), p. 731.