The Deflection of Light in the Gravitational Field of the Sun (Results of the English Expeditions to Observe the Solar Eclipse of 1919).

G. S. Landsberg.

1. The general principle of relativity leads to the consequence that a light ray must undergo curvature in a gravitational field, and Einstein’s calculations make it possible to estimate the magnitude of this curvature as a function of the intensity of the field of gravitation. A total solar eclipse makes it possible to photograph stars situated in the immediate vicinity of the Sun, and a comparison of the visible positions of the stars thus obtained with their usual positions, when the rays they emit do not pass near the massive sphere of the Sun, can serve as an experimental test of the conclusions of Einstein’s theory. Such indeed was the task of two English expeditions, equipped to Sobral, in northern Brazil, and to the island of Principe (in the Gulf of Guinea), for the observation of the total solar eclipse of May 29, 1919, the results of whose work have recently been published1.

According to plans worked out by a commission of outstanding English astronomers and astrophysicists—Dyson, Eddington, Fowler, and Turner—it was necessary to photograph the stars situated near the Sun during the total phase, and the same group of stars in their usual nocturnal positions, in order to verify whether there exists a systematic deflection in the visible positions of the stars during the eclipse. Since the maximum expected displacement on the plate was not to exceed \(1/60\) mm, the comparison of the photographs had to be carried out with the aid of a measuring apparatus used for measuring plates; for this purpose, besides the photographs taken during the eclipse and the comparative night photographs, it was also necessary to have a standard exposure, made on a plate turned toward the objective by its glass side, which made it possible to superpose the negatives being studied upon the standard emulsion to emulsion.

Of the two expeditions, the most complete and definite results were obtained by the Brazilian one with the aid of a 4-inch refractor with a focal length of 6 mt. The other observations of the same expedition are unreliable because of the imperfection of the apparatus, while the results of the second expedition are compromised by cloudy weather, which did not permit sharp images of the stars to be obtained.

2. On a photographic plate an image was obtained of 7 stars, whose coordinates on the negative are determined by the numbers \(x\) and \(y\), the center of the plate serving as the origin of coordinates. Seven such photographs were obtained during the total phase. For comparison, 7 night exposures were used, taken as far as possible under similar conditions. When these negatives are superposed on the standard one, a discrepancy is found between the corresponding stars, caused by the following reasons:

1. Noncoincidence of the origins of coordinates on the two plates (a displacement of the plate being measured relative to the standard one). For all stars on the given plate the coordinate \(x\) changes by a constant amount \(c\), and the coordinate \(y\)—by a constant amount \(t\).

2. Noncoincidence of the coordinate axes (rotation of the plate being measured relative to the standard). The coordinate \(x\) changes by the amount \(b \cdot y\), and the coordinate \(y\)—by the amount \(d \cdot x\).

3. Noncoincidence of scale (Skalenwerte). By scale is meant the distance, expressed in millimeters, on the plate between two stars whose distance on the celestial sphere along a great-circle arc is \(1'\). For ordinary astrographic objectives the scale is approximately 1 mm per \(1'\). However, its value changes slightly for different parts of one and the same plate, these changes increasing in proportion to the distance from the center of the plate. For two different exposures the scales may prove to be different as a consequence, for example, of a change in focusing (temperature effects, inaccuracy of setting, etc.).

Under the influence of this cause the coordinate \(x\) changes by \(a \cdot x\), and the coordinate \(y\)—by \(e \cdot y\).

1. The influence of the deflection of a ray in the gravitational field. This latter, according to Einstein’s theory, is for the coordinate \(x\) equal to \(\alpha \cdot E_x\), and for the coordinate \(y\) equal to \(\alpha \cdot E_y\), where \(E_x\) and \(E_y\) are coefficients supplied by the theory. The deflection caused by this reason depends on the distance of the star from the center of the Sun.

Thus, despite the existence of a whole series of causes producing a noncoincidence of the images on two plates, it is possible to take into account the influence of each of them by making use of the difference of the laws governing the change in the amount of displacement as a function of the position of the star.

Indeed, according to the foregoing, the displacement on the plate is expressed as:

\[ \Delta x = a \cdot x + b \cdot y + c + z \cdot E_x \]

\[ \Delta y = d \cdot x + e \cdot y + t + z \cdot E_y \]

where \(x\), \(y\), \(\Delta x\), and \(\Delta y\) are given by observations, \(E_x\) and \(E_y\)—by Einstein’s theory, while the quantities \(a\), \(b\), \(c\), \(d\), \(e\), \(t\), and \(z\) are to be determined. Thus each plate gives seven pairs of such equations (according to the number of stars), with four unknowns, which are determined by the method of least squares.

Table I gives the values of \(\alpha\), determined for each of the seven plates.

Table I.

The second and fourth columns should have given for \(\alpha\) a value equal to zero, since on the comparison plates the influence of the gravitational field is absent. The small value obtained should be ascribed to errors in the positions of the stellar images on the standard plate; these we can exclude by subtracting from the value \(\alpha_F\) the value \(\alpha_B\). Thus the influence of the auxiliary standard plate is eliminated, and from both coordinates we obtain the influence of the gravitational field on the deflection

\[ \alpha = 0^r,100 = 0'',625. \]

This value refers to stars whose angular distance from the center of the sun is \(50'\). The deflection for rays coming from the edge of the sun is

\[ = 1'',98 \pm 0'',12, \]

whereas Einstein’s theory gives \(1'',75\).

The second, rather less reliable group of Brazilian observations gives for \(a\), for the edge of the sun, the value \(= 0'',93\).

The observations on the island of Principe also give a less reliable figure,

\[ a = 1'',61 \pm 0'',30. \]

In the appended graph (see Fig. 1), the angular distances of the seven observed stars from the center of the sun are laid off along the abscissa axis, while the ordinates of the plotted points are the radial displacements of the stars.

(i.e., displacements along the line connecting the center of the sun with the star). The thick solid line corresponds to Einstein’s theory.

Fig. 1.

The actually observed points group themselves around the thin solid line. The dashed line corresponds to the deflections which material particles flying with the speed of light would undergo as a result of solar attraction (Newton’s theory of emission).

Thus, photographs taken during the solar eclipse at Sobral and on the island of Principe undoubtedly reveal a systematic displacement of stellar images, as though the light ray is bent in the gravitational field. The magnitude of the displacement agrees well with the predictions of the theory of relativity.

3. Hypotheses which have attempted to explain the observed deflection without the aid of the principle of relativity cannot be considered sound. Such hypotheses were:

a. Anomalous refraction in the earth’s atmosphere, caused by lateral refraction within the cone of shadow.

However, in order to explain the observed effect by this cause, it would be necessary to assume, during the total phase, a fall of temperature of \(20^\circ\) per minute (Eddington). The small lowering of temperature which actually occurred could, owing to lateral refraction, produce an effect of no more than \(0'',1\), i.e. about 5% of the observed magnitude.

b. Refraction of light in the atmosphere surrounding the sun. However, the displacement caused by refraction in an atmosphere surrounding the center of gravitation should not decrease in proportion to the increase of the star’s angular distance from the center of the sun, as is observed on the photographs obtained.

Moreover, an atmosphere of such density that the effect could reach the observed value would have to weaken the light passing through it to the point of complete invisibility of the stars (a diminution of brightness, in round numbers, by 200 stellar magnitudes).

c. The phenomenon of so-called “annual refraction” (jährliche Refraction). According to Courvoisier’s observations, the coordinates of stars, determined for the moment when the star is at a distance of \(90^\circ\) from the sun, show a systematic displacement of the star by \(0'',1\). With a change in the star’s angular distance from the sun the angle of refraction changes, and for a distance of \(20^\circ\) it reaches \(0'',4\). For the limb of the sun Courvoisier extrapolates \(0'',6\). In order to take into account the influence of annual refraction, Courvoisier inserted into the formula for the displacement \(\Delta x\) a fifth term, depending on the ve-

of the magnitude of the annual refraction \(\rho\), and calculated its influence. It turned out that \(\rho = 0\), i.e., the law of variation of the observed displacement with distance is not compatible with the assumption of annual refraction. The supposition, however, that the observed law is distorted by possible irregularities in the positions of the stars at the edge of the plate, and precisely in such a way as to confirm the principle of relativity, is too unlikely. Finally, the cosmic cause of the annual refraction itself is open to doubt. It is possible that we were dealing with a phenomenon of physiological origin or one dependent on the instruments.

1. Philosoph. Transactions of the Royal Society of London, ser. A, vol. 220, pp. 221—333. In view of the fact that in Russia this volume of Phil. Trans. has not yet been received, the material for the present report has been borrowed from the article by E. Freundlich (Die Naturwissenschaften 8, p. 667, 1920). ↩