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(Read April 20, 1912.)

To the student of the stars, who attempts to arrange our existing knowledge in such a manner that some light may be thrown upon the problems connected with stellar evolution, the spectral classification developed at Harvard is of vital importance.

In such investigations, we must deal, if possible, not with single instances, but with representative averages for groups of stars. But really representative averages are often much harder to obtain than might be supposed. Consider, for example, the actual brightness of the stars. We can find this only when we know the distance of the star—and out of the hundreds of thousands of stars which have been catalogued, we know the distance of barely five hundred. But even if we knew the exact distances of the 6,000 or more stars which are visible to the naked eye, we would not have a fair sample of the general run of stars. To explain how this may happen, let us suppose that there were only two kinds of stars, one equal to the sun in brightness, and the other 100 times as bright as the sun, and that these were distributed uniformly through space, in the proportion of 100 stars of the fainter kind for every one of the brighter. To be visible to the naked eye, a star of the fainter sort must lie within about 55 light-years from the sun; but all the stars of the brighter kind which lay within 550 light-years would be visible. We would therefore be searching for these stars throughout a region of space whose volume was 1,000 times greater than that to which our method of selection limited us in picking out the fainter ones, and our list of naked-eye stars would consequently contain ten stars of the brighter kind to every one of the fainter—though if we could select instead the stars

contained in a given region of space, we would find the disparity to be 100 to I the other way.

It is therefore a fortunate circumstance that the stars whose distances have been measured have for the most part been chosen, not on account of apparent brightness, but because of relatively rapid proper-motion—which is found by experience to be a fairly good indication of actual nearness to our system. These stars, therefore, represent mainly the sun's nearer neighbors, without such an egregious discrimination in favor of stars of great actual brightness as we have seen must occur if we choose our stars by apparent brightness alone. Some traces of this discrimination will still be unavoidable, for our knowledge of the proper-motions of the fainter stars is still imperfect, and stops short at a little below the ninth magnitude.

In addition to the stars whose parallax has been directly observed, we have data for many more, which belong to clusters whose distances have been found by combining data regarding their proper-motions and radial velocities. In this case too the absence of proper-motion data (which decide whether or not a star really belongs to the cluster) prevents us from obtaining information about stars fainter than a certain limit; but otherwise our knowledge is probably fairly complete.

In the present discussion of the relation between the spectral type and the real brightness of the stars, those directly measured parallaxes have been employed which are confirmed by the work of two or more observers, and also a few results obtained by single observers whose work is known to be of high accuracy, and free from sensible systematic errors. To these have been added the members of the Hyades, the Ursa Major group, the “61 Cygni group” and the moving cluster in Scorpius discovered independently by Kapteyn, Eddington, and Benjamin Boss. The spectra of a very large number of these stars have been determined at Harvard especially for this investigation, and the writer takes pleasure in expressing his most hearty thanks to Professor Pickering and Miss Cannon for this generous and invaluable aid.

The actual brightness of the stars may best be expressed by

means of their "absolute magnitudes”-i. e., the stellar magnitudes which they would appear to have if each star was brought to the standard distance of 32 light-years (corresponding to a parallax of o".10). The absolute magnitude of the sun on this scale is

about 4.7.

On plotting these absolute magnitudes against the spectral types it becomes immediately evident that most of the stars belong to a series in which the fainter members are redder than the brighter, while a few outstanding stars of each spectral class greatly exceed in brightness those belonging to this series (except for class B, all of whose stars are very bright). The existence of these two series was first pointed out by Hertzsprung, who has called them by the very convenient names of “giant” and “dwarf” stars—the former being of course the brighter.

With the large amount of material now available, especially for the dwarf stars, the results derived from the stars with directly measured parallaxes and from those in the clusters are in striking agreement, as is shown in Table I.

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In the above table, the quantity given under the heading "Abso- . lute Magnitude" is the mean of the individual values derived from the observed magnitude and parallax of each star in the corre

1 Zeitschrift für wissenschaftliche Photographie, Bd. V., p. 86, 1907.

sponding group (giving half weight to a few stars of relatively uncertain parallax or spectrum)-except for the stars of spectrum B with directly measured parallaxes. In this case the parallaxes are so small that a reliable value could be obtained only by taking the mean of the observed magnitudes and parallaxes for the whole group. These stars are of much greater apparent brightness than most of those of class B, and their actual brightness may be greater than the average for the class. No similar error of sampling need be suspected in other cases, except for the faintest stars in the clusters, where it is obvious in going over the lists that only a few of the brightest stars of class K5 are above the limit of magnitude at which our catalogues of stars belonging to the clusters stop, and probable that some of the fainter stars of class K are also excluded.

With the exceptions just explained, the results of the two independent determinations from the measured parallaxes and the clusters are in remarkably good arrangement, considering the small numbers of stars in many of the groups. The absolute magnitudes of stars of the same spectral class in different clusters are in equally good agreement. The relation between absolute magnitude and spectral type appears therefore to be independent of the origin of the particular star or group of stars under consideration.

This relation seems to be very nearly linear, as is shown by the last column of Table I., which gives for each spectral type an absolute magnitude computed by the formula

Abs. Mag. =0.5 + 2.2 (Sp.- A), in which spectrum B is to be counted as o, A as 1, F as 2, etc. It is of interest in this connection to remember that the difference of the visual and photographic magnitudes of the stars is also nearly a linear function of the spectral type.

The individual stars of each spectral class are remarkably similar in real brightness. Excluding those for which the parallax or spectrum is considerably uncertain, there remain in all 218 stars. Of these only 11, or 5 per cent. of the whole, differ more than two magnitudes in absolute brightness from the value given by the

formula for the corresponding spectral class, while 150, or 69 per cent., have absolute magnitudes within one magnitude of the computed value.

The series of stars so far discussed does not however comprise all those in the heavens. Most of the stars of the first magnitude have small parallaxes, and are of great absolute brightness; and a study of proper-motions shows the same to be true of the nakedeye stars in general. It follows that there exists another series of stars, of great brightness, differing relatively little from one spectral class to another. These “giant” stars can be seen at enormous distances, and consequently form a wholly disproportionate part of the stars visible to the naked eye, as has been explained above. The illustration there given greatly understates the actual situation for the redder stars. The dwarf stars of class M, for example, are so faint that not one of them is visible to the naked eye (though one of them is the second nearest star in the heavens), and so the naked-eye stars of this class are all “ giants.”

Relatively few of these giant stars are near enough for reliable measures of parallax, and even for these it is safer to take the mean observed parallaxes and magnitudes of groups of stars, to diminish the effect of errors of observation. Confining ourselves as before to parallaxes determined by two or more observers, or by observers of high accuracy, the existing data may be summarized as follows.

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The stars of class B are repeated here, since they may be regarded as belonging to either series.

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