198 
VARIABLE STARS 
Towards the boundary the second term in (132-5) becomes predominant 
We have tabulated F x up to £ 0 = 5. It is difficult to trace it beyond; but 
remembering that there has to be a node at the boundary, we need not 
fear any abnormal increase. Probably the maximum value of — dF 1 /d £ 0 
is about 1 in terms of the central value of | l5 or about -05 in absolute 
amplitude. 
Hence the gain of heat per unit volume per second may amount to 
the factor R'/R being inserted in order to change the unit of length to 
1 cm. Since R/cR' is about 10 seconds 
The heat accumulated per cu. cm. in the half-period (2 days) is then about 
200 uT e 4 . The normal heat content of a cu. cm. is somewhat larger than 
aT i , say 2 aT i . The amplitude T x is about -08. It follows that the tem 
perature oscillation due to heat leakage is about equal to the oscillation 
due to adiabatic pulsation when T = 5 T e . In the region of the star for 
which T < 5T e the adiabatic approximation fails utterly. 
134. The small leakage of heat found in the last section will gradually 
dissipate the energy of pulsation if there is no countervailing agency. We 
shall estimate roughly the rate of decay. 
Suppose that the pulsation of the region is kept steady by supplying 
mechanical work W, so that by the conservation of energy 
for any number of complete cycles. If we substitute in this our expressions 
for dQ/dt such as (133-2) we merely obtain W = 0 to the first order of small 
quantities. We must obtain an expression which will enable us to calculate 
W to the second order. Since the state is steady, the change of entropy of 
the material must be zero for complete cycles, so that 
so that 
dQ _ F 0 dF-y _ H 0 dFy 
dt 4vp 0 g 0 2 dg 0 p 0 dg o ‘ 
Po dt —zV-^o cos 
— ^ acT 6 4 ( R'/R) cos nt, 
Po ^ ~ C0S n ^‘ 
(134-1) 
or since 
Hence 
1 1 _ 8T __ 1 _T X 
T~T 0 TJ ~T 0 ~ T 0 ’ 
[dQ (1 - T,) - 0. 
W + j T x dQ = 0 .. 
(134-2).