393 
364-366] Long Period Variables 
as different spectral lines tell entirely different stories it seems unlikely that 
their displacements correspond to those of the star’s surface. 
366 . There are cosmogonical reasons for expecting that the long-period 
variables should be in a state of pulsation. A star first born out of a nebula 
has a density so low that the gas-laws must be almost exactly obeyed through 
out its mass. It must consequently be unstable and will contract until the 
gas-laws begin to fail. 
The analysis of § 108 enables us to trace out the star’s motion during this 
period of instability. To a good enough approximation a and /3 may each be 
put equal to zero in the equation of motion (108'1), so that it becomes 
+ G ° d * 
dt 3 
W + v ,n, rh 0 d t A Sr )+ r 
7 Mr 
G V T 0 
^M r G 
A -J- 4 Oq -4- T 0 * 3sA 
+ 
r 0 3 C V T 0 
A + 4 
— n z r + 
A -f- 1 
3 8\ 
A + 1 A + 1 
pC.T, 
a +») 
A + 1 
dt 
(8r) 
(Sr) = 0 (366-1). 
Since 7 M r /r 0 s is large in comparison with G/C V T 0 , it follows that the time 
factors for possible oscillations are of the type 
e ~ Pt ± (p2 - ç)ii , e~ et .(366-2), 
where Q is large in comparison with P and e. 
Suppose first that the gas-laws are so nearly obeyed that s in equation 
(366*1) may be neglected. Then the last coefficient in this equation is 
negative, while the others are positive, so that P and Q are positive while e 
is negative. Thus there are stable pulsations having a period of the order of 
27 t ( 7 M r /r 0 3 ) ~ which is hundreds of days, and also unstable contractions in 
which Sr is doubled in a period of the order of C V T 0 /G 0 , which is thousands 
of years. 
With increasing shrinkage the deviations from the gas-laws increase, so 
that s increases until finally e becomes positive. Both contractional displace 
ments and pulsations are now stable. 
The pulsations retain their identity throughout the whole shrinkage of 
the star, and if they are once excited there is nothing to check them, since 
the time needed for viscosity and other dissipative forces to produce an 
appreciable effect is far greater than the whole time of contraction of the 
star. Thus when the star first reaches a stable configuration, it will be 
affected by the whole of the pulsations which have been set up during its 
birth and contraction from nebular density. Although exact calculations are 
difficult, it seems likely that quite large pulsations must be set up by the 
stars first falling in from the nebular state. If so, in its first era of stable 
existence, the s'tar must be executing pulsations of very large amplitude. 
J 
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