VARIABLE STARS 
197 
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density interior to . If o- is the ratio of the mean density of the star to 
the mean density interior to f 0 , (132-3) becomes 
F x = 0-2% - (0-10 + l-75cr) (133-1). 
With the aid of Table 27 the following values are found— 
£o 
*1 
1 pm > i 
3p 0 *°dfo 
Sum 
0 
- -85 
•00 
- -85 
1 
- -87 
- -02 
- -89 
2 
- -96 
- -14 
- MO 
3 
- 1-16 
- 1-05 
- 2-21 
4 
- 1-52 
- 6-28 
- 7-80 
5 
- 2-12 
- 46 
- 48 
Here the unit of amplitude is that of at the centre, which is roughly 0-7 
times that of £ x at the boundary, or 0-7 8R/R. 
If we ignore the variation of e within the star so that e m = e, the last 
column gives — dQ/edt by (132-5). For example, at £ 0 = 3 
~ = 2-21e x 0-7 + const (133-2). 
The adiabatic approximation neglects this periodic gain and loss of 
heat and we can now show that the approximation is justified. For half 
the period, say 2 days, the region at | 0 = 3 is gaining heat at an average 
rate about (taking 8B/E about -05), equivalent to day at the rate e. 
The total heat inside the Cepheid represents about 100,000 years’ supply 
of radiation. Hence the heat gained in the half-period is to the heat 
already present in the ratio of Jy day to 100,000 years. This heat is lost 
in the next half-period. The result is a temperature variation with amplitude 
of the order 0°-01. This is, of course, superposed on the main temperature 
oscillation, due to the adiabatic compression and expansion, which has an 
amplitude of some half-million degrees and differs 90° in phase. Clearly 
we were justified in assuming that in the main part of the star the leakage 
effect is trivial. It appears that the adiabatic approximation is much 
more accurate for a Cepheid than for ordinary sound waves. 
The negative sign of F x shows that the flow of heat is greatest when 
g x is least, i.e. at the moment of greatest compression. The positive sign 
of dQ/dt shows that a region gains most heat at the time of greatest 
expansion, i.e. when it is coolest. It is important to notice that the negative 
sign of F x arises from both terms in (133-1); the increased flow at greatest 
compression is partly due to diminished opacity but there would be some 
increase even if the opacity were constant. 
There will be a region near the boundary of the star where the adiabatic 
approximation ceases to be valid; the heat content there is small and the 
leakage becomes relatively important.