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No. 7.] 247 75. An Explanation of the Periodic Variable Stars. By Kiyotsugu HIRAYAMA, M.I.A. Astronomical Observatory, Azabu, Tokyo. (Comm. July 13, 1931.) The binary hypotheses, initially proposed to explain the periodic variable stars, were defective. The pulsation theory seemed more promising ; but even this is disappointing in explaining the varieties and the minor details, of the phenomena. The following explanation ,1) although qualitative, seems satisfactory for those. The hypothesis is based on the capture theory of the stars,2) recently proposed by the writer, and is inconsistent with the fission theory. Cepheid Variables. The Cepheid variable is supposed to be a contact system of a giant star and a dwarf star which is almost dark. The relative orbit is supposed to be nearly circular, in accordance with the general tendency of the binary stars. For the rotation of the primary, the following qualifications are made : (1) the axis is nearly perpendicular to the orbital plane ; (2) the sense is the same as that of the revolution ; (3) the period is longer than that of the revolution. No assumption is made for the rotation of the companion, although the libration is very probable. Taking as a model, the stars of spherical form of the masses 10 and 1 solar units, and the period of revolution, 0.02 year, the mean distance comes out as 0.16 astronomical unit, and the velocitiesrelative to the centre of gravity, as 22.3 and 223.5 km sec-1. As a consequence of the relative motion of the companion, a trail, or an abrasion, so to speak, is left on the surface of the primary, which may be recovered as the time goes on. The temperature of the star being supposed to increase rapidly with the depth, the portion affected by the abrasion will emit intense light of an earlier spectral type, and will fade away in some rate. Supposing, for the sake of simplicity, that the inclination of the orbit is nearly 9•‹, i.e., that the sun is nearly in the plane of the 1) This explanation was originally presented by Dr, Hellerich for the Cepheid variables. A.N. 224 (1925), 277. 2) Proc. 7 (1931), 182. 248 K. HIRAYAMA. [Vol. 7, orbit, the companion will make the transit over the disc of the primary. The minimum of the light variation occurs, then, near the 2nd contact, and the maximum, near the 4th contact. The interval of time from minimum to maximum depends on the relative size of the dark companion ; but it is always less than of the period of revolu- tion. The case of ƒÄ Geminorum is an extreme. The size of the companion must be very small in this case. The effect of the absorption of light near the limb, and that of the slope of the bright surface will advance the time of the maximum. If the inclination of the orbit is considerably smaller than 900, and at the same time the size of the companion is not small, then the reflection of light, which is really observed in some eclipsing vari- ables, will become effective, and the second maximum, or j period later than the principal maximum will be observed. The cases of ƒÅ Aquilase, S Sagittai, etc. can be explained in this way. The eccentricity of the orbit, even small, will affect the light curve, considerably. If the stars pass their common priastron near the time of the maximum, a very bright flash-like light may be observed, just as in the case of Z Lacertae. The velocity curve is greatly affected by the rotation. The effect is, qualitatively, 1st contact, positive and small, 2nd " positive and large, 3rd " negative and large, 4th " positive and large. The maximum velocity is retarded and the minimum velocity is ad- vanced by this effect. The former will thus approximately coincide with the minimum brightness, and the latter, with the maximum brightness. The eccentricity, also, will affect these epochs sensibly. The velocity curve of ƒÄ Geminorum, with a rapid decrease soon after the velocity maximum, is a good example showing the effect of rotation. The effect of rotation at the maximum brightness must be greater for the early-type spectral lines than for the later-type lines. This is real, as was shown by the observations of Drs. St. John and Adams. It is known statistically that the spectral type of the Cepheid variables is earlier as the period is shorter. This is probably due to more frequent agitation of the surface. In our hypothesis, the energy of the radiation comes out mainly from the star itself, and only slightly from the resistance at the con No. 7.] An Explanation of the Periodic Variable Stars . 249 tact point. The effect of the latter, although small, may shorten the period of revolution, and will become sensible in a long interval of time. The case of ƒÂ Cephei was thoroughly investigated by Dr. Hertz- sprung, and a small shortening of the period was proved to be real. As the star must contract by the loss of energy due to the radiation, and at the same time, the orbit must contract by the resist ance, the contact of the two stars may continue for a long time. If the star contracts more rapidly, the system will become an eclipsing binary, and if the orbit contracts more rapidly, the companion will drop into the primary. The case when a Cepheid variable falls into a dense nebula may be very interesting. Then the stars must expand by the absorption of energy from the nebula, and, by consequence, the companion must drop violently into the primary. The result will be a nova. The fact that both novae and ordinary Cepheids are strongly concentratednear the galaxy, supports this view. If the Cepheid variable is seen orthogonally to its orbital plane, then the system will be observedas a simple star, with the spectrum exactly alike. The case of (1 Persei or of y Cygni is such. For a given mass of the Cepheid variable, its size and luminosity must be greater as the amount of the stored energy is greater. The well-known relation between the period and luminosity seems to exist on this ground. Slow moving stars near the galactic plane, absorbing more energy from the nebulae, are brighter in general, as is the case in the B-type stars. The Cepheids with the period longer than 1n, possess the similar characters. So, it is natural to assume for them, a larger store of energy and consequently, a larger volume. The period, then, must be longer for them than that of those not possessing the characters, namely, the Cepheids with the period less than 1'. Mira-Type Variables. The M or S spectral type giant is un doubtedly larger than the ordinary giant. Rather, it may be looked as a mass of nebulous matter enclosing a bright star near centre. This central star, as is thermally conserved by the surrounding matter, may be very bright and of the spectral type Oe or Be, possibly. A nebulous star of this type is supposed to form a close binary system with another star almost dark. Then, the densest portion of the nebulous envelope is swept along and the light of the central star is transmitted outward. The case is similar to that of the Cepheid variable, except that the companion moves within the nebulous matter, and that the rotationof the primary is naturally insignificant. 250 K. HIRAYAMA. [Vol. 7, The absorption spectrum seems to emanate from the outer envelope. The relative radial velocity, Vn-Ve, near the maximum, as was measured by Dr. Merrill, justifies this assumption. The fact, moreover, that the maximum brightness occurs nearly one month earlier than the maximum approaching velocity, is in good agreement with our supposition. A conspicuous shortening of the period, as is noticed in the light variations of R Hydrae and R Aquilae, confirms the idea of the nebulous envelope. An apparent objection to the above hypothesis is that the relative velocity, Va - Ve, increases in general, as the period becomes longer. This objection, however, may be removed by the consideration of the eccentricity, which, as is well known in the case of the binary stars, increases with the period. The orbit of the Mira-type variable of longer period may be more eccentric, and the periastron of the dark companion may be directed toward the sun. Such a system, if seen from the opposite direction, may be a simple M or S type star. Lastly, I should like to acknowledge my indebtedness to the re- markable work of Prof. Ludendorff, „"Die veranderlichen Sterne •J (Handbuch der Astrophysik, VI), without which this hypothesis could not be advanced..

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