dicating that the sun was in a gaseous state with high central condensation when the planets were born. The jets of matter thrown out would also be gaseous, but would rapidly cool in the process of ejection, and might soon liquefy or even solidify. It can be shown that the planets which would be formed out of the middle portion of such a jet ought to be much more massive than those formed near the ends, and this may possibly provide an explanation of the comparatively great masses of Jupiter and Saturn. We imagine that the planets at first described orbits under the combined gravitational action of the sun and the passing star by which the cataclysm was caused, but as this star receded they were left revolving, as at present, around the sun. During their earlier motion they may themselves have been broken up by the tidal action of one or both of the big masses present, and such a process may explain the origin of the satellites of the planets.

Such, in its main outlines, is the tidal theory of the genesis of the solar system. So far as can be seen, a vast amount of further mathematical research is needed before it can be either definitely accepted or finally condemned. For myself, I find it more acceptable than the rotational theory, or any other hypothesis so far offered, of the origin of the solar system. Time does not permit of a discussion of its difficulties, but I may perhaps conclude by stating what seem to me to be its main advantages over the rotational theory.

(1) It escapes the well-known criticism of the rotational theory that the present angular momentum of the solar system is too small to be compatible with a previous rotational break-up, and I do not know of any similar quantitative criticism which can be brought against the tidal theory.

(2) The solar system is arranged with reference to two planes— the invariable plane of the system, which contains the orbits of the outer planets, and a second plane inclined at about 6° to the former plane, which contains the sun's equator and the orbit of Mercury. A system which had broken up by rotation alone ought to be arranged symmetrically about one plane-the original invariable plane of the system. On the tidal theory the two planes of the solar system are readily explained as being the plane in which the tideraising star moved past the sun, and the original plane of the sun's rotation.

(3) Theoretical investigations suggest that there is only one possible end for a rotating system, namely, a binary or multiple star of the type familiar to astronomers, and it is quite certain our system is not of this type. Similar investigations on tidal action suggest that the final end of a system broken up by a tidal cataclysm ought to show many of the features of our present solar system.

THE DIAMETERS OF THE STARS.1

By A. DANJON,

Astronomer at the Strasbourg Observatory.

[With 1 plate.]

Stellar astronomy is now taking its place definitely in the ranks of sciences. For 20 to 30 years the census of the starry heavens grew, star after star, nebula after nebula, without any definite bond, any generalization, any law emerging showing the connections existing among their number. Astronomers seemed yet in the dark as to any general plan of the universe. Similarly before Ptolemy and Hipparchus, except for the existence of the planets, the plan of the solar system was not known. The astronomers of Alexandria made their names forever illustrious in showing that the planets were subject to laws. They gained the first approximation to the plan of the solar system. True, it fell to the lot of others, of Kepler, of Newton, to give a more thorough solution of the problem and a deductive status to planetary astronomy. But without their predecessor's steps, how halting would have been their progress.

The solving of the problems of sidereal astronomy began when physics produced the appropriate tools. The elder Herschel had marked out the way, helped by his great ingenuity but handicapped by paucity of instrumental means. The American school of astronomers has put on foot the work which we of to-day admire both for its rich promise and the germs of truth it contains. Truly hypo thesis plays a preponderant part in the interpretation of the observations. We are yet far from the completion of a magnificent logical structure such as planetary astronomy has gained. A mathematician would be ill at ease before the numerous publications which bring us the echo of the discoveries gleaned from the stellar universe. At present astronomers are garnering facts; the near future may bring the Newton or the Kepler who will place them in their proper

1 At a general assembly of the Société Astronomique de France Professor Michelson described his method of measuring stellar diameters. A survey of some related subjects is useful as an introduction to his work.

Translated by permission from L'Astronomie, November and December, 1921.

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relationships. In a few generations doubtless the time will come to use the rich harvest of observations now patiently being accumulated. Then in its turn sidereal astronomy will become clear, logical, deductive. At least it is a patriotic wish that it may be so, for it is a task especially suited to the French genius.

EQUIVALENT DIAMETERS.

The direct measurement of star diameters fills a whole chapter in astronomy till now rather hypothetical. A long time ago all hope had been lost of detecting the disks of the stars with the ordinary instruments of our observatories. At the focus of our telescopes, diffraction spreads the light received from a luminous point into a circular diffuse spot surrounded by rings. Since such a spot is much larger than the true geometrical image of a star, it is impossible to distinguish between the image of a star and that of a point. If we try to measure with a micrometer the angular diameter of a star, the value obtained will have no relationship with the true dimensions. The measurements will give merely the diameter of the diffraction spot which depends solely on the aperture of the objective of the telescope.2

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It is true, we can measure extraordinarily small relative displacements of stars, of the order of a few hundredths of a second of arc. For instance, the annual parallax has long since been determined. The two problems are really very different in nature and difficulty. We can determine with great precision the center of the diffraction spot and its displacements, although some special device is necessary to force it to deliver up its secret of the true nature of the luminous

source.

The measurement of stellar diameters is therefore a difficult task to approach since we are unable to produce true images of stars by any direct means. Consequently, for some time we have exercised our ingenuity in avoiding this difficulty by evaluations more or less hypothetical. Sometimes the evaluations have rested on solid foundations. We know, for example, the fruitful researches relative to eclipsing variables, but the information thus gathered does not suffice for an exhaustive study. Let us leave aside this aspect of our subject, very interesting though it be in itself, in order to consider a general method based upon the comparison of the relative brightness of stars and the sun.

2 The diameter of the diffraction spot is inversely proportional to that of the objective. An aperture of 12 centimeters (5 inches) gives a spot about two seconds of arc (2′′) in diameter. An aperture of at least 5 meters (16 feet) would be necessary before the disk of the greatest star (0."05) would begin to be appreciable.

The annual parallax of a star is the angle which the radius of the earth's orbit about the sun occupies as seen from the star.

The apparent intensity of a luminous source depends upon two factors; its intrinsic brilliance and its surface. If two stars have the same intrinsic brilliance, their apparent brightnesses will be proportional to their surfaces.

Let us take as an example Sirius and our sun, of which the "magnitudes" are -1.6 and -26.7. The difference is approximately 25 magnitudes, corresponding to a ratio of apparent brightness

(Sun)/(Sirius)=10,000,000,000.

If we admit for the moment that the two stars have the same brightness per unit surface, we must then also admit that their apparent surfaces are in the same ratio, 10,000,000,000. The apparent diameters are in the ratio of the square root of this, or 100,000. Since the diameter of the sun is 1,800", that of Sirius is 0.018.

Let us consider another example: Betelgeuse is 2.5 magnitudes fainter than Sirius. Its brightness is therefore 10 times smaller and its diameter the square root of 10 times smaller or 0.006.

These diameters correspond to their true diameters only provided that the intrinsic brightnesses of these stars and of the sun are all the same. A priori, nothing could be less certain. However, these values, taken in their proper significance, are precise. They are the diameters which we must assume for fictitious stars of the same surface brightness as the sun, in order that if substituted for the real ones, their brightnesses would be of the same magnitude as the real ones. E. C. Pickering has designated such values as equivalent diameters.

Having reached this far, to go farther we need to know more of the surface brightnesses of the stars. This becomes possible through the advance of spectroscopy and the physics of radiation. From the laws of radiation we glean two essential properties of radiating bodies:

First, the energy of radiation per unit surface increases with the temperature; it is proportional to the fourth power of the absolute temperature when the body is a "black body." (Law of Stefan.)

Second, the spectrum composition of the emitted light also varies with the temperature; in the spectrum of a black body, the wavelength corresponding to the maximum intensity of energy is inversely proportional to the absolute temperature. (Law of Wien.) In simpler words, as a body becomes hotter, its emitted light changes from red to white to blue.

We therefore have reason to think the yellow and red stars are colder than the white or blue ones. Consequently the former radiate much less light than the latter for equal surfaces. The equivalent diameters cannot in general be equal to the true diameters. The stars of superior intrinsic brightness to the sun are smaller than our

calculations indicate, and inversely. That our determinations may be more precise, it will be next necessary to summarize briefly the principal basic facts of stellar spectra.

SPECTRUM CLASSIFICATIONS AND STELLAR TEMPERATURES.

The classification now universally adopted is that of Harvard. We will pass in review the principal classes. The classes have been arranged in the order of decreasing temperatures. We can assure ourselves of this in verifying that, from one class to the next, the maximum intensity retreats toward the red (at the right) and that the blue end progressively disappears.

The first spectrum shows the type of class B, or helium stars. Their temperatures are very high. Their light is very white or blue. Their absorption spectrum contains helium lines. Hydrogen lines present are usually dark but at times bright. The metallic lines are absent. We pass next to the classes A and F characterized by the intensity of the absorption spectrum of hydrogen. The lines of this element belonging to the Balmer series are very noticeable (types A and F and intermediate type F.) huddling together in the violet. The metallic lines make their debut in class A, but they do not become very abundant until in the spectra of type F. The latter class is especially noteworthy for the extraordinary development of the H and K violet lines of calcium which are faint in the preceding classes.

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The temperature becomes still lower in passing to class G where our sun belongs. The metallic lines become definitely preponderant; those of hydrogen are yet more intense (C and F Fraunhofer lines) but less so than those of calcium. The Balmer series ceases to give the spectrum its characteristic aspect. The absorption lines of helium have disappeared. We know that they cannot be observed in the sun's ordinary spectrum. Helium was discovered in the sun because of its bright line spectrum in the chromosphere.

The stars of class K are distinctly yellow or reddish. The metallic lines are so numerous and strong that the continuous spectrum is reduced to a few bright rays upon a sombre background. The scale of the figure is perhaps not large enough to show this structure but what can be surely seen is the weakening of the blue part of the spectrum. This indicates a comparatively low temperature.

The letters designating the classes are not now arranged in alphabetical order. It would take too long to recount the history and evolution of the Harvard classification. Experience has arranged the classes in the order indicated.

We are leaving out of consideration the planetary nebulae which make up the class P, the nebular stars, placed in class Q, and also the so-called Wolf-Rayet stars constituting class O. The spectra of the last class contain a number of bright lines upon a feeble ground. They seem related to novae or "new stars" and are apparently yet hotter than the stars of type B. The study of them is not so far advanced as for the stars described in the body of the text.

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