282
IONISATION, DIFFUSION, ROTATION
Rotating Stars.
198. We prove first a very beautiful theorem due to H. von Zeipel*.
If a star, rotating as a rigid body with angular velocity co, is in static
equilibrium, the rate of liberation of energy e at points in the interior is given by
It is assumed that the physical characteristics of the material (opacity,
molecular weight, liberation of subatomic energy, etc.) depend on T and
p only; this would be true of a star of strictly homogeneous composition.
We take axes rotating with the star and include centrifugal along with
gravitational force so that the combined potential is
where </> 0 is the pure gravitational potential. Poisson’s equation is then
so that dP = 0, when d<f> = 0. Hence P is constant over a level surface;
that is to say, P is a function of <f> only.
so that p is a function of </> only.
Since P and p are functions of </> only, T must be a function of f> only;
and all other physical characteristics which depend only on the two
variables T and p defining the statistical state of the material will be
functions of f> only and constant over a level surface. The gradients of any
of these quantities will be normal to the level surface.
The flow of radiation H being along the normal, we have by (7IT)
where dn is along the outward normal to the level surface. We can write
(198-1).
f> = <f >0 + \w 2 (x 2 + y 2 ),
V 2 </> = V 2 </> 0 + 2 a; 2
= — 4:TrGp + 2co 2
The surfaces over which <j> is constant are called level surfaces.
From the usual hydrostatic equations
(198-2).
dP ty dPdj> dP df
dx P dx ’ dy dz ^ dz ’
df>
we have
dP = pdf
(198-3),
c dp R _ c dp R df>
kp dn kp df> dn ’
so that
(198-42).
(198-41),
* Festschrift für H. v. Seeliger, p. 144 (1924).