256 
that they do not differ greatly from the stars of 
Classes B and M as regards the degree of their 
similarity to one another in brightness With such 
a probable error of distribution of the absolute magni- 
tudes as has here been derived, the giant and dwarf 
stars would overlap perceptibly in Class G, be just 
separated in Class K, and widely so in Class M, as 
the observational data indicate. 
The questions now arise: What differences in their 
nature or constitution give rise to the differences in 
brightness between the giant and dwarf stars? and 
Why should these differences show such a systematic 
increase with increasirg redness or “advancing” 
spectral type? 
We must evidently attack the first of these ques- 
tions before the second. The absolute magnitude (or 
the actual luminosity) of a star may be expressed as 
a function of three physically independent quantities 
—its mass, its density, and its surface-brightness. 
Great mass, small density, and high surface-bright- 
ness make for high luminosity, and the giant stars 
must possess at least one of these characteristics in 
a marked degree, while the dwarf stars must show 
one or more of the opposite attributes. 
A good deal of information is available concerning 
all these characteristics of the stars. The masses of 
a considerable number of visual and _ spectroscopic 
binaries are known with tolerable accuracy, the densi- 
ties of a larger number of eclipsing variable stars 
have recently been worked out, and the recent in- 
vestigations on stellar temperatures lead directly to 
estimates of the relative surface brightness of the 
different spectral classes (subject, of course, to the 
uncertainty whether the stars really radiate hike black 
bodies, as they are assumed to do). We will take 
these matters up in order. 
First, as regards the masses of the stars, we are 
confined to the study of binary systems, which may 
or may not be similar in mass to the other stars. 
There appears, however, to be no present evidence at 
all that they are different from the other stars, and 
in what follows we will assume them to be typical 
of the stars as a whole. 
The most conspicuous thing about those stellar 
masses which have been determined with any 
approach to accuracy is their remarkable similarity. 
While the range in the known luminosities of the 
stars exceeds a millionfold, and that in the well- 
determined densities is nearly as great, the range in 
the masses so far investigated is only about fiftyfold. 
The greatest known masses are those of the com- 
ponents of the spectroscopic binary and eclipsing 
variable V Puppis, which equal nineteen times that 
of the sun; the smallest masses concerning which we 
have any trustworthy knowledge belong to the faint 
components of ¢ Herculis and Procyon, and are from 
one-third to one-fourth of the sun’s mass. These are 
exceptional values, and the components of most 
binary systems are more nearly similar to the sun in 
mass. 
There appears, from the rather scanty evidence at 
present available, to be some correlation between mass 
and luminosity. Those stars which are known to be 
of small mass (say, less than half the sun’s) are all 
considerably fainter than the sun. On the other 
hand, Ludendorff'* has shown conclusively that the 
average mass of the spectroscopic binaries of spec- 
trum. B (which are all of very great luminosity) is 
three times as great as that of the spectroscopic 
binaries of other spectral types, and may exceed ten 
times that of the sun. Further evidence in favour of 
this view is found in the fact that the components of 
a binary, when equal in brightness, are nearly equal 
18 4. N., 4520, 1¢11. 
NO. 2323, (VOL. 493) 
NATURE 
[May 7, i914 
in mass, while in unequal pairs the brighter star is 
almost (if not quite) always the more massive, but 
the ratio of the masses very rarely exceeds 3: 1, even 
when one component is hundreds of times as bright 
as the other. Very large masses (such as-one hundred 
times the sun’s mass) do not appear, though they 
would certainly be detected among the spectroscopic 
binaries if they existed. It is equally remarkable 
that there is no trustworthy evidence that any visible 
star has a mass as small as one-tenth that of the sun. 
The apparent exceptions which may be found in the 
literature of the subject may be shown to arise from 
faulty determinations of parallax, arbitrary estimates 
of quantities unobtainable by observation (such as the 
ratio of the densities of the two components of Algol), 
and even numerical mistakes. 
It follows from this similarity of mass that we can 
obtain a very fair estimate of the parallax of any 
visual binary (called by Doberck the hypothetical 
parallax) by guessing at its mass, and reversing the 
familiar relation between mass and parallax. If we 
assume that the mass of the system is twice that of 
the sun (about the average value), our hypothetical 
parallaxes, as the existing evidence shows, will 
usually be well within 40 per cent. of the truth, and 
the deduced absolute magnitudes of the components 
will rarely be more than one magnitude in error. 
We may thus extend our study of the relation between 
absolute magnitude and spectrum to all the visual 
binaries for which orbits have been computed. The 
hypothetical absolute magnitudes which we will obtain 
for them will indeed be somewhat in error, owing 
to the differences in their masses; but, for our present 
purpose, the hypothetical values are actually more 
useful than the true values would be. This sounds 
remarkable; but it is easy to show that, if we assume 
that the brighter components of the systems have all 
the same mass (say that of the sun), the resulting 
hypothetical absolute magnitudes will be the actual 
absolute magnitudes of stars identical in density and 
surface-brightness with the real stars, but all of the 
assumed mass. In other words, the effects of differ- 
ences of mass among the stars are eliminated from 
these hypothetical absolute magnitudes, leaving only 
those of differences in density and surface-brightness. 
(This is simply a statement in different form of a 
theorem which has been known for many years.) 
It is therefore desirable to extend our study to as 
many binary stars as possible. The number for 
which binary orbits have been computed is relatively 
small, but by a simple statistical process we may 
include all those pairs which are known to be con- 
nected really physically, however slow their relative 
motion may be.?°® 
Consider any pair of stars, of combined mass m 
times that of the sun, at a distance of v astronomical 
units, and with a relative velocity of v astronomical 
units per annum. By gravitational theory, we have 
v7 =(27)?m(2—17/a)=39-7m(2—1/a), 
where a is the semi-major axis of the orbit Now 
let « be the parallax of the system, s the observed 
distance in seconds of arc, w the observed relative 
motion in seconds of are per annum, and 7, and i, the 
angles which r and v make with the line of sight. 
Then s=rrsini,, w=vrsini,, and our equation 
becomes 
Sw? = 39-775m sin t, sin7t,(2—1/a). 
In the individual case, the last three factors of the 
second member are unknown, and we are no wiser 
19 An outline of this method was given by the speaker at the meeting of 
the Astronomical and Astrophysical Society of America at Ottawa, August 
25, 1911, and published in Sczenxce, N.S., vol. xxxiv., pp- 523-25, October 20, 
tgtt. A similar method was worked out quite indenendently and almost 
simultaneously by Hertzsprung, and published in 4. V., December 19, 1911 
(the date of writing being October r1, 1911). ; 
