94
Gaseous Stars
[ch. Ill
determines the state of actual stars, and finds that atoms of different weights
would be nearly evenly distributed throughout the star, even if there are no
convection currents. I have, however, calculated* that this special solution
assigns a positive charge of about 9 x 10 11 coulombs to the sun, whereas the
limit of charge, fixed by the condition that it shall just restrain negative
electrons from leaving the sun, is 4 x 10 9 coulombs, or only 00045 times that
given by Rosseland’s solution; the factor of 00045 is, moreover, independent
of the special constitution of the sun, and is the same for the starsf. Thus in an
actual star the effect discussed by Rosseland only goes about a two-hundredth
part of the way towards producing a general mixing of atoms, and so may be
neglected for all practical purposes.
v. Zeipel’s Theorem.
In 1924, H. v. ZeipelJ published a remarkable series of investigations
which claimed to shew that the generation of energy per unit mass Gin a
uniformly rotating star is given by
where w> is the angular velocity and c is constant throughout the star. The
theorem is obviously impossible when to has any appreciable value different
from zero, since it makes G assume violently negative values near the surface
of the star (p = 0). In the special case of a very slow rotation, v. Zeipel’s
result takes the form that G must be constant throughout the star. If, for
instance, a star consisted only of atoms of lead and of uranium, there could be
no equilibrium until complete mixing of these atoms was attained.
Milne and Eddington have interpreted v. Zeipel’s condition as one which
the star would evade rather than conform to§, the latter suggesting that an
actual star would evade it by setting up a system of rotatory currents in
meridional planes.
The simple explanation of the puzzle appears, however, to be that v. Zeipel
did not derive his theorem from the exact equations of equilibrium (73T) but
from Eddington’s inexact equations (72'3). The latter, as we have seen, are
only true in the special case of G = 0. Thus to reconcile v. Zeipel’s theorem
with the assumptions from which it is derived, we must assign to the un
determined constant c the special value c = 0. The theorem now takes the
harmless form that when G — 0, then G = 0 . If the exact equations (73T) are
introduced, the theorem is found to fail altogether.
Thus stellar matter appears to be free to arrange itself under the influence
* M.N., R.A.S. lxxxvi. (1926), p. 561.
t l.c. p. 562.
| M.N. , R.A.S. lxxxiv. (1924), p. 665, and subsequent papers.
§ Observatory , xlviii. (1925), p. 73.