REDUCTION OF STELLAR MASS DUE TO ENERGY RADIATION1

G. Vogt.

Not only the proponents of the theory of relativity, but also those scientists who stand on the ground of classical physics, now believe that matter is only a special form of energy, and that the inertial mass of a body is a measure of the energy contained in it. The relation connecting the energy of a body \(E\) and its mass \(m\), as is known, has the form

\[ m=\frac{E}{C^{2}}, \]

where \(C\) is the speed of light. Proceeding from this conception, one is already inevitably compelled to admit that stars, as a consequence of the radiation of energy, must, in the course of time, decrease in mass. If a star radiates in one second an energy \(dE\), then its mass in the same time must decrease by the amount \(\frac{dE}{C^{2}}\). From the amount of energy which we receive on the earth from the Sun, it can be calculated that in one second the Sun radiates into space \(9.4\cdot 10^{25}\) gram-calories. This corresponds to a loss of mass of 4,000,000 tons per second. The decrease in mass of a giant star whose absolute brightness is equal to \(-5\) mg must be approximately 10,000 times greater.

Do observations provide such data as would testify that such a decrease in stellar mass actually occurs? Here it is necessary first of all to note the following: whether those losses of mass which stars undergo in the course of their development as a result of the radiation of energy can in general attain an appreciable magnitude—this, of course, depends on the duration of the star’s life; for the numbers which we indicated above for the decrease in the mass of the Sun or of stars, in comparison with the entire mass of a star (\(10^{27}\)–\(10^{28}\) tons), are vanishingly small. If it is assumed that the absolute brightness of a star is a function of its mass—as follows from Eddington’s

of the theory of the internal structure of stars,—then it can be shown that the lifetime of a star must be of the order of \(10^{12}\) years, provided only that the decrease in mass connected with radiation plays an appreciable role. This conclusion can be subjected to verification on the basis of statistical investigations. The assumption of so great a lifetime for a star contains nothing at all improbable. In fact, geologists find for the age of certain minerals on the earth numbers of the order of \(10^9\)—\(10^{10}\); meanwhile, the age of these minerals constitutes an insignificantly small part of the age of the Sun, or of a star. Further, the position of stars in space, the distribution of their velocities, and the large eccentricities of the orbits of binary stars make it possible to conclude that a star, in the course of its life, experiences, generally speaking, one or several collisions with other stars. And so—according to the estimate of Schwarzschild and Jeans—a star on average experiences one such collision in the course of \(10^{12}\)—\(10^{13}\) years.

Thus, as far as the scale of time is concerned, it fully permits one to detect, on the basis of statistical observations, a decrease in the mass of stars as a result of radiation,—provided only that this process actually takes place. In what follows we shall see that there exist such observational results which, apparently, support the hypothesis of a decrease in the mass of stars.

1. It is known that the so-called Russell diagram (H. M. Russel), which establishes the relation between the absolute luminosities and the spectral types of stars, is interpreted also as a diagram of the history of a star’s development. According to this conception, every star begins its life as a gas sphere of large volume, of very small density, and with a low temperature at the surface—a red giant star. Then the star runs through the separate spectral classes in the order M, K, G, F, A, B, and at the same time its mean density and its temperature at the surface become ever greater. When the star in the form of an A- or B-star (a white star) reaches the highest point of its development, the course of evolution reverses, and the star runs through the same spectral classes a second time, but in the reverse order. At the same time its mean density continues to increase, while the surface temperature will already decrease until the star, having become a dense and cold body, becomes invisible to us. But Eddington1 has recently shown that, if one moves along the Russell diagram—which, as has already been said, also represents the course of stellar development—then one encounters stars with constantly decreasing mass. If this is correct, then the Russell diagram cannot represent the course of development of stars with constant

mass, as has been assumed up to now. Only two possibilities remain: 1) The mass of a star remains constant during its development; but then the Russell diagram, contrary to modern ideas, would have nothing in common with the course of stellar evolution—it would give only the absolute luminosities and spectral types at which stars, depending on their masses, attain stable states. 2) The Russell diagram—in agreement with modern ideas—gives the course of a star’s development; but then, during its development, a star must decrease in mass—and moreover decrease quite considerably.

1. If the assumption is correct that, in the course of a star’s development, its mass decreases as a result of the radiation of energy, then further conclusions can be drawn from this. First, the decrease in the mass of any star must be proportional to its absolute luminosity, or to the energy radiated by the star per unit time. But, on the other hand, the absolute luminosity of a star is the greater the greater its mass; and moreover—as follows from Eddington’s theory of the internal constitution of stars—the absolute luminosity increases faster than proportionally to the first power of the mass. Consequently, the decrease of mass, expressed in fractions of the total mass for a given star, must be the greater the greater its total mass. But then the ratio of the mass of the brighter component of a binary star to that of the less bright one, in the course of further development, must become ever smaller, gradually approaching unity. At the same time, naturally, the same course of the mass ratio of binary stars must reveal itself if they are placed on the Russell diagram, i.e. distributed according to their spectral types in the order of progressive development. From a special investigation carried out by the author1 it follows that, apparently, on the average the ratio of the mass of the brighter component to that of the less bright one for binary stars becomes the smaller and the closer to unity the farther this star is situated on the Russell diagram.

2. If spectroscopic binary stars are arranged according to their spectral types, the following tendency is found: the mean sizes of the orbits of such binary stars increase with increasing spectral type, so that very close systems are encountered ever more rarely. Further, together with the sizes of the orbits, their eccentricity also increases on average. It can be shown that, if the mass of a binary star undergoes a very slow change, then the product of the mass and the major axis of the orbit of the binary star must remain constant. Consequently, if the mass of a binary star decreases as a result of the radiation of energy, then its orbit must increase; if, for example, the mass is reduced by half, then the major axis of the orbit

must increase twofold. In agreement with the observations, this tendency should be revealed by the mean sizes of the orbits of spectroscopic binary stars, if these stars are arranged according to their spectral types in the order of progressive development. As regards the distribution of the eccentricities of the orbits of binary stars, it cannot be explained by a decrease in the mass of the stars—at least, it cannot be explained in a direct way. For it can be shown that, if a binary star undergoes very slow changes of mass, then the eccentricity of its orbit must remain unchanged. But it may be said that the hypothesis of a decrease in the mass of stars—indirectly—does provide an explanation for the distribution of orbital eccentricities. Indeed, this hypothesis requires such great intervals of time for the duration of the life of stars that each star, in the course of its development, may in general experience one or several close encounters with other stars, as a result of which considerable increases in eccentricities may be produced. The older a binary star is, and the farther apart its components are, the greater the probability that it has already experienced an “effective” collision with another star, and that it therefore possesses an orbit with such a large eccentricity.

However, not only this good agreement of the individual results of observation with theory, but also general considerations speak in favor of a decrease in the mass of stars in the course of their development. Any scientific theory of the Cosmos must proceed from the fact that the universe is in a stationary state; that at any moment of time, on the average, just as many stars flare up anew as become invisible as a consequence of cooling. Otherwise, in cosmic space there would long ago have been only dark bodies, whereas a superficial glance testifies to the fact that the universe is still very far from such a state. And since the number of dark stars—as can be concluded from a whole series of astronomical facts—does not greatly exceed the number of luminous ones, it is not enough merely to assume that, by some method unknown to us, new stars are constantly arising, which shine for more or less considerable intervals of time and then reach their final state in the form of dark bodies. One is forced to conclude that stars not only go out with the passage of time, but also disappear as accumulations of matter; and this is precisely what the hypothesis developed here assumes. This hypothesis must be supplemented insofar as the process of the transformation of matter into energy must also continue in dark bodies, in the form of a general disintegration of atoms, until these dark stars cease to exist as material bodies. Further, we must consistently conclude that in separate regions of space, as a consequence of local regroupings, energy conversely

is transformed into material atoms—such is Nernst’s idea1. These atoms may then gather again into nebulae, from which new stars will subsequently develop. This process of the emergence of new atoms must take place very rarely—at least on our human scale of time—for matter possesses a very long duration of life.

According to this theory—whose correctness, although not proven, is supported by a great deal—the universe undergoes a cosmic cycle connected with the constant emergence and disappearance of matter.

It should also be mentioned that, if the views developed here are correct, then the difficulties that until now have been associated with finding the sources of solar energy disappear of themselves. As is known, for a long time it was assumed that the radiant energy emitted by the Sun is maintained by the mechanical work performed by gravitational forces during the contraction of the solar mass. At present it may be considered proven that the constant output of energy by the Sun over those enormous intervals of time that are obtained for the age of the Earth—and, consequently, for the minimum age of the Sun—from geological and paleontological investigations cannot be explained by the contraction hypothesis. Hence the inevitable conclusion is that, besides contraction, there must exist in the Sun another, far more powerful source of energy. But this source of energy, according to our conceptions, is the Sun itself, whose substance is identical with an accumulation of energy, the latter being transformed into another form of energy—radiant energy—at the expense of the solar mass.

1. W. Nernst, The Universe in the Light of New Research—Advances in the Physical Sciences, III, 1923. Separate edition. Moscow, 1923. ↩↩↩↩