276 
IONISATION, DIFFUSION, ROTATION 
lighter elements—not quite so abruptly-—drop from the surface to the 
centre. Hydrogen lags behind the others, the reversal occurring when 
¿i 0 /3 0 = 0-5, i.e. in extremely massive stars with 1 — )S 0 > 0-75. Helium 
leaves the surface in stars of more than 10 times the mass of the sun— 
just the stars which show its spectrum most conspicuously. The other light 
elements reverse at smaller masses. 
It would be difficult to reconcile these results with the observed spectra 
at the surfaces of the stars where light and heavy elements appear to 
gether. For some elements the exponent of T is so large as to leave not a 
single atom anywhere near the surface. If we believed these results we 
might have to consider important modifications in the theory of the stellar 
interior—for example, its constitution of heavy elements in small stars 
and light elements in great stars. We must now point out that these 
formulae give the ultimate steady state of the material ; and the question 
arises whether the approach to this steady state is sufficiently rapid to 
effect appreciable separation in the life-time of a star or to overcome the 
mixing tendencies which may be retarding it. 
Thermal Diffusion, 
194. It is desirable to give a warning against possible misuse of (191-3) 
and similar formulae. In a star <f> is proportional to T by (58-3) so that 
gimi-mj) <t>/RT p as the same value at all parts. We might be tempted to 
infer that $j/s 2 is constant so that no separation of the different sorts of 
atoms occurs. This is very far from true. The formulae (191-1) refer to 
thermodynamical equilibrium, and there is no justification for employing 
them even as an approximation in material at non-uniform temperature. 
Our procedure is to differentiate these formulae keeping T constant 
because the formulae apply only to uniform temperature. Thus (191-2) 
gives 
i <‘°g *) - 
m x d<f> 
RT dz 
.(194-1). 
This gives us the density gradients of the different kinds of atoms required 
to keep the composition steady when the material is under a pressure 
gradient but no temperature gradient; that is to say, it gives us the density 
gradients set up to prevent further pressure diffusion. If now we superpose 
a temperature gradient this will set up diffusion on its own account known 
as thermal diffusion. Hence, in general, the required density gradient 
consists of two parts, the one set up by pressure diffusion, and the other 
by thermal diffusion. 
If thermal diffusion can be shown to be negligible the differential 
formula (194-1), but not the integral formula, is valid for non-uniform 
temperature. We then re integrate it allowing T to vary.