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May 14, 1914] 
more massive body will be the greater, though to a 
less degree than the central temperature. A large 
mass of gas will therefore arrive at a higher maxi- 
mum temperature, upon reaching its critical density, 
than a small one. The highest temperatures will be 
attained only by the most massive bodies, and _ all 
through their career these will reach any given 
temperature at a lower density, on the ascent, and 
return to it at a higher density, on the descending 
scale, than a less massive body. They will there- 
fore be of much greater luminosity, for the same 
temperature, than bodies of small mass if both are 
rising toward their maximum temperatures. On the 
descending side the difference will be less con- 
spicuous. Bodies of very small mass will reach only 
a low temperature at maximum, which may not be 
sufficient to enable them to shine at all. 
All this, again, is in excellent agreement with the 
observed facts. The hottest stars—those of Class B 
are, on the average, decidedly more massive than 
those of any other spectral type. On the present 
theory, this is no mere chance, but the large masses 
are the necessary condition—one might almost say 
the cause—of the attainment of unusually high 
temperature. Only these stars would pass through 
the whole series of the spectral classes, from M to B 
and back again, in the course of their evolution. 
Less massive bodies would not reach a_ higher 
temperature than that corresponding to a spectrum 
of Class A; those still less massive would not get 
above Class F, and so on. This steady addition of 
stars of smaller and smaller mass, as we proceed 
down the spectral series, would lower the average 
mass of all the stars of a given spectral class with 
“‘advancing’’ type, in the case of the giants as well 
as that of the dwarfs. This change is conspicuously 
shown among the dwarf stars in Table VII., and 
faintly indicated among the giant stars. The average 
masses of the giant and dwarf stars appear, how- 
ever, to be conspicuously different, which at first 
sight seems inconsistent with the theory that they 
represent different stages in the evolution of the same 
masses. But the giant stars which appear in these 
lists have been picked out in a way that greatly 
favours those of high luminosity, and hence, as we 
have seen, those of large mass, while this is not the 
case among the dwarf stars. The observed differ- 
ences between them are therefore in agreement with 
our theory, and form an additional confirmation of it. 
It is now easy, too, to understand why there is no 
evidence of the existence of luminous stars of mass 
less than one-tenth that of the sun. Smaller bodies 
presumably do not rise, even at maximum, to a 
temperature high enough to enable them to shine 
perceptibly (from the stellar point of view), and hence 
we do not see them. The fact that Jupiter and 
Saturn are dark, though of a density comparable with 
that-of many of the dwarf stars, confirms this view.?° 
25 In the foregoing presentation of the theory, to avoid interference with 
the progress of the main arcument, no mention has been made of certain 
considerations which should be discussed here. 
(1) It is probable that at stellar temperatures the gaseous matter is very 
considerably dissociat d and ionised. But this will not affect its gaseous 
nature. For our present purpose it amounts to little more than a diminution 
of the mean molecular weight. This will lower the temperature correspond- 
ing to a given density and pressure, and so tend to lower the maximum 
attainable temperature; but as the deeree of dissociation is likely to vary 
gradually with the temperature, it should not affect the orderly sequence of 
changes which form the basis of the previous arguments. 
(2) It is also probable that the available potential energy of a star is not 
entirely gravitational, but partly, if not mainly, of radio-active or similar 
atomic origin. If, as in the relatively very small range accessible to experi- 
mental investigation, the rate of liberation of this energy is independent of 
the temperature and pressure, it would simnly supply a constant annual 
addition to the energy derived from gravitational contraction, and the only 
difference in the course of events would be that a star, on cooling, would 
approach, not complete extinction, but a steady state, of very long duration, 
in which as much energy was annually radiated away as was supplied by 
atomic disintegration. !f the rate of disintegration is increased under the 
extremely violent molecular collisions which must occur in the interior of a 
Star, a great liberation of energy may occur when the interior his got hot 
NO. 2224, VoL. 92] 
NATURE 
285 
We may once more follow the lead of our hypo- 
thesis into a region which, so far as I know, has been 
previously practically untrodden by theory. It is well 
known that the great majority of the stars in any 
given region of space are fainter than the sun, and 
that there is a steady and rapid decrease in the number 
of stars per unit volume, with increasing luminosity. 
The dwarf stars, especially the fainter and redder 
ones, really greatly outnumber the giants, the pre- 
ponderance of which in our catalogues arises entirely 
from the egregious preference given them by the 
inevitable method of selection by apparent brightness. 
What should we expect to find theoretically? To 
get an answer, we must make one reasonable assump- 
tion, namely, that the number of stars, in any suffici- 
ently large region of space, which is, at the present 
time, in any given stage of evolution, will be (roughly 
at least) proportional to the lengths of time which it 
takes a star to pass through the respective stages.*° 
While a star is growing hotter it is large and bright, 
is radiating energy rapidly, and is also storing up 
heat in its interior; while, on account of its low 
density, contraction by a given percentage of its 
radius liberates a relatively small amount of gravita- 
tional energy. It will therefore pass through these 
stages with relative rapidity. Its passage through its 
maximum temperature will obviously be somewhat 
slower. During the cooling stages its surface is 
relatively small and its rate of radiation slow; it is 
dense, and a given percentage of contraction liberates 
a large amount of energy, and the great store of heat 
earlier accumulated in its interior is coming out again. 
It must therefore remain in these stages for ‘very 
much longer intervals of time, especially in the later 
ones, when the rate of radiation is very small. 
This reproduces, in its general outlines, just what 
is observed—the relative rarity of giant stars, the 
somewhat greater abundance of those of Class A near 
the maximum of temperature, and the rapidly in- 
creasing numbers of dwarf stars of smaller and 
smaller brightness. The well-known scarcity of stars 
of Class B, per unit of volume, is further accounted 
for if we believe, as has been already explained, that 
only the most massive stars reach this stage. 
In this connection we will very probably be asked, 
What precedes or follows Class M in the proposed 
evolutionary series, and why do we not see stars in 
still earlier or later stages? With regard to the 
latter, it is obvious that dwarf stars still fainter than 
the faintest so far observed (which are of Class M) 
would, even if among our very nearest neighbours, be 
apparently fainter than the tenth magnitude. We 
cannot hope to find such stars until a systematic 
search has been made for very large proper-motions 
among very faint stars. The extreme redness of such 
stars would, unfortunately, render such a search by 
photographic methods: less profitable than in most 
cases. 
But a giant star of Class M, a hundred times as 
bright as the sun, certainly cannot spring into exist- 
ence out of darkness. In its earlier stages it must 
have radiated a large amount of energy, though 
perhaps less than at present. But as the tempera- 
ture of a radiating body falls below 3000° C., the 
energy-maximum in its spectrum moves far into the 
infra-red, leaving but a beggarly fraction of the whole 
radiation in the visible region. Stars in such stages 
enough, thus increasing the maxima temperature and prolonging its duration. 
But, even on this hypothesis, the number of the violent collisions which 
liberate the atomic energy would increase gradually as the temperature of 
the interior rose, and the general character of the evolutionary changes, 
including the relation of the mass and density of the body to the time of 
their occurrence, would not be radically altered. ) 
It seems, therefore, probable that the previous reasoning would require no 
essential modification on account of either of these factors in the problem. | 
26 Hertzsprung, Zeitschrift fiir Wissenschaftliche Photographie, vol. iii, 
|p. 442, 1905. 
