New Observations of the Deflection of Light Rays in the Gravitational Field of the Sun

Yu. M. Kushnir and V. S. Fursov, Moscow

One of the remarkable consequences of the general theory of relativity, as is well known, is the necessity of the deflection of a light ray in the field of an attracting mass. The formula given for the angle of deflection by the general theory of relativity, in the form in which it was developed by Einstein in 1916, is

\[ E=\frac{4kM}{c^2\Delta}; \]

here

\(k\) is the gravitational constant \(=6.67\cdot10^{-8}\ \mathrm{cm^3\ g^{-1}\ sec^{-2}}\);
\(M\) is the attracting mass in grams; \(\Delta\) is the distance along the normal from the ray to the center of mass; \(c\) is the velocity of light \(3\cdot10^{10}\ \mathrm{cm/sec}\). The magnitude \(E\) will be sufficiently noticeable only in the case when the attracting mass \(M\) is very large.

For a light ray passing at the very limb of the Sun, formula (1) gives the following numerical value:

\[ E=1'',75. \]

The first successful observation of such a deflection was accomplished in 1919 by two English expeditions: one in northern Brazil (Sobral), the other on the island of Principe off the coast of Africa. The following results were obtained:

\[ \begin{aligned} \text{exped. at Sobral} \quad & E=1'',98\pm0'',18,\\ \text{exped. on Principe Island} \quad & E=1'',61\pm0'',45, \end{aligned} \]

i.e., values in good agreement with the theoretical one.

Even more striking data were obtained during the eclipse of 1922 in Australia by the American expedition of Campbell and Trumpler.

Expedition in Australia — $E = 1''.78 \pm 0''.16$.

However, the numerous objections and doubts regarding the results of these expeditions made it necessary to repeat the observations with the utmost precautions, so as, insofar as possible, to eliminate all extraneous causes capable of producing a displacement. For this purpose, in 1929 an expedition of German astronomers under the direction of Freundlich set out for North Sumatra (to Takengon). The site of the 1929 expedition was chosen on the grounds that there the number of cloudless days, the humidity of the air, and other meteorological conditions proved to be most favorable. Photographs of the sky were taken with the aid of two horizontally mounted tubes lying in one plane and forming a certain angle with one another. The objectives of the tubes were directed toward the vertex of this angle. A coelostat was placed here. The two tubes served for simultaneous photography of two regions of the sky: one in which the Sun was located, and a second located $25^\circ$ of the celestial sphere away from it. The diameter of the tube objectives was $20\ \mathrm{cm}$, the focal length $8.5\ \mathrm{m}$; the size of the photographic plates was $45 \times 45\ \mathrm{cm}$. The region photographed by each plate covered $3^\circ \times 3^\circ$ of the celestial sphere.

For time control the expedition was provided with a small receiving radio station, which made it possible to receive European and colonial stations. The entire installation was carefully protected against shocks and temperature effects.

With the first tube the following photographs were taken: 1) the Sun during the eclipse and the stars surrounding it—the so-called eclipse field—and 2) the very same stars in the very same position, but without the Sun (at night), about a month and a half later. With the second tube: 1) the stars of a certain region of the sky $25^\circ$ from the Sun and lying at the same zenith distance as the Sun—the so-called control field (photographed simultaneously with the photograph of

on the first tube) and 2) the very same stars in the same position, at night, about half a year later.

Assuming that the observing conditions on the day of the eclipse and four months later were exactly the same, one may obtain:

A—by comparing photographs 1 and 2, taken with the first tube, the magnitudes of the displacements of the stars caused by the fact that during the first photograph the light of the stars was deflected by the Sun, whereas during the second photograph it was not deflected;

B—by comparing the first and second photographs taken with the second tube, to be convinced of the identity of both photographs, for the photographed stars lay \(25^\circ\) from the Sun, i.e., in a region where the deflection of their rays by the Sun, because of the range of the action, is practically absent. The assumption of identity of the experimental conditions during the first and second photographs, however, is practically unrealizable, for, as has been said, several months elapse between these photographs, and it would be imprudent to suppose that all the apparatus and all the conditions remained completely unchanged. One of the causes producing this change is a change of scale, connected with variations, even if only slight ones, in the focusing as a consequence of the difference in temperature conditions during the two photographs. The farther a star is situated from the center of the plate, the greater will be the effect upon it of the change of scale connected with the change in focusing. For the displacement of a star due to the deflection of light, however, assuming that the Sun is approximately at the center of the plate, we shall have just the opposite picture: it is the larger, the closer the star is to the Sun. The separation of these two effects—the displacement due to a change of scale and the displacement due to the deflection of a light ray—is very difficult.

In order to be able to take account of the change of scale, one may proceed in two ways:

1. To photograph on one and the same plate, one after another, the eclipse field and then some region distant from the Sun. On a second plate, half a year later, photograph again

one after another both areas of the sky. Then, comparing images of an area far from the Sun, one can obtain an idea of the change in scale. If the scale has not changed, then when the two plates are superposed all the stars should coincide.

This method was chosen by the members of the expedition for the astrograph—a movable tube directed directly at the sky and having a large field of view (the purpose of the astrograph, see below).

1. The second method for finding the change in scale, adopted by the expedition for the horizontal tubes, consisted in the following: on the photographic plate of each tube, before photographing one or another area of the sky on it, an image of a special grid applied to glass was photographed. The stellar areas of the sky photographed on the day of the eclipse—namely, the eclipse field and the control field—were photographed in the very same position of the celestial sphere already about half a year later, in the absence of the Sun—at night. These photographs were taken with the very same apparatus and at the very same place (for this purpose the entire installation in Takengon was left under the supervision of one of the members of the expedition, Kluver). As special tests showed, during this time the installation changed hardly at all.

During the 298 sec. duration of the eclipse, photographs of the following exposures were made with the first tube: 1) a photograph at 40 sec., 2) at 90 sec., 3) at 60 sec., 4) at 40 sec.

With the second tube: 1) a photograph at 40 sec., 2) at 90 sec., 3) at 60 sec.

Since the arrangement of the stars around the Sun was asymmetrical, the photographs were taken so that the Sun was not at the center of the plate, but somewhat displaced, with the intention that as many stars as possible should be in the field of view.

The times of the nighttime exposures for each tube were: 20, 40, 60, 90, and 180 sec. In addition, the Pleiades were also photographed additionally on each plate (see below).

The method of measurement chosen was the following: at night, ...

direct photographs were taken not only for comparing the eclipse field with the control field, but also two additional photographs through plate glass, on which a mirror image of both regions of the sky was obtained. To one of these plates, for example that of the eclipse field, a plate taken on the day of the eclipse was first pressed emulsion to emulsion and compared; then, in the same way, a plate taken at night was compared. From the results of the comparison, the necessary comparison of the day and night photographs was then obtained by calculation. The measurements were made on an apparatus specially constructed for this purpose; in all, 100,000 separate measurements were made. The work was carried out twice by two observers independently of one another. The aim of all the measurements was to determine the displacements of the positions of the stars on the plates taken on the day of the eclipse relative to the corresponding ones taken at night.

The causes that change the positions of stars in the sky when comparing two photographs taken at different times are the following:

1. Different influence of refraction.
2. Different influence of astronomical aberration.
3. Different inclination of the plane of the plate to the optical axis of the tube.
4. Different values of the scales on the plates, caused by different focal lengths of the tube at the times of the two photographs.

And, finally, in this special case of photography in the vicinity of the Sun during a total eclipse:

1. The influence of unknown physical actions on the propagation of light near the Sun.

Items 1 and 2 were accurately taken into account by known methods. Item 3 could not be taken into account in advance, for the reason that in order to allow for the influence of the inclination of the plane of the plate to the optical axis, one must know the coordinates of the foot of the perpendicular dropped from the center of the objective onto the plane of the plate; the latter could not be determined because of the strong blackening of the plates near the center, caused by the image of the solar corona. Therefore item 3

NEW OBSERVATIONS OF THE DEFLECTION OF LIGHT RAYS

was taken into account already directly, during the comparison.

To exclude the influence of differences in scale, special comparisons were made of the images of the grid on the corresponding pairs of plates, so that the entire measuring work consisted of two parts: comparison of the sizes of the grids and comparison of the positions of the stars.

On the photographs of the eclipse field, 17 and, respectively, 18 stars were obtained whose positions could be reliably determined. With the plate format used, one arc minute corresponded to a linear distance of about \(2.5\) mm; therefore the displacements on the plate due to the supposed Einstein effect should be of the order of hundredths of a millimeter, while the measuring apparatus gave readings with an accuracy of up to half a micron. The exact determination of the absolute value of the divisions of the measuring apparatus in angular units was carried out by measuring the distance between two stars of the Pleiades group, photographed on all the night plates, and the angular distance between which is known very accurately.

The first comparison of the positions of the stars was made without any assumption whatever about the deflection of light, namely: the obtained differences of coordinates, freed from the effects of refraction, aberration, and the different value of the scales, were assumed to be due only to the different orientation of the coordinate axes (shift and rotation) and to the influence of an unaccounted-for term of the 3rd order, and therefore were represented in the following form:

\[ \begin{aligned} \Delta x_i &= C + By_i + px_i^2 + qx_iy_i;\\ \Delta y_i &= D - Bx_i + px_iy_i + qy_i^2. \end{aligned} \]

Here \(C\) and \(D\) take into account the displacement of the origin, \(B\) the rotation of the axes, and \(p\) and \(q\) the different inclination of the plates to the optical axis.

Similar equations were written for all the stars, and from them, by the method of least squares, the values of \(B\), \(C\), \(D\), \(p\), and \(q\) were found. These values were then substituted back into the equations, and the residual term of the displacement of each star was then found. In the absence of light deflec-

...the residual term should have the form of random errors, but already at first glance it is evident (see Fig. 1) that the residual term has a clearly expressed regularity—a systematic displacement of the stars away from the center of the Sun, which quite definitely speaks in favor of the presence of a deflection of the light ray. Therefore a second comparison was carried out, now taking account of the light deflection by adding to the expressions \(\Delta x_i\) and \(\Delta y_i\), respectively, the terms \(\dfrac{E}{r_i}\dfrac{x_i}{r_i}\) and \(\dfrac{E}{r_i}\dfrac{y_i}{r_i}\), where \(E\) is the value of the light deflection in the immediate vicinity of the Sun, and \(x_i\), \(y_i\), and \(r_i\) are the coordinates and radius vector of the star, with the center of the Sun chosen as the origin. After this second comparison, which among the other unknowns also gives \(E\), the residual term indeed assumes the form of random errors (see Fig. 2). And the sum of squares

Fig. 1. Vector diagram of the displacements of stars in the eclipse field.

errors relative to the first comparison is reduced approximately sevenfold.

Thus comparisons were made of all four pairs of plates, as a result of which the following values were obtained for \(E\):

\[ \begin{aligned} &\text{From the plate with exposure }40\text{ sec.} &&—\ E = 2'',25 \pm 0'',19\\ &\text{” ” ” ” }90\text{ ”} &&—\ E = 2'',17 \pm 0'',20\\ &\text{” ” ” ” }60\text{ ”} &&—\ E = 2'',61 \pm 0'',26\\ &\text{” ” ” ” }40\text{ ”} &&—\ E = 1'',81 \pm 0'',19\\ & &&\overline{\text{Mean value } —\ E = 2'',24 \pm 0'',10} \end{aligned} \]

Fig. 2. Errors in the field of view.

The comparatively large values of the deviation \(E\) from the mean, obtained on certain plates (plate 60 and 40a — \(\pm 0'',4\)), are explained not by inaccuracy of measurements of such an order, but by the fact that most of the stars were located, on average, at a distance of about four radii from the center of the Sun, and therefore \(E\) had to be found by extrapolation to the edge of the Sun, so that the deviations

from the mean order of \(\pm 0''.4\) corresponded to deviations of the actually measured quantities by approximately \(\pm 0''.1\).

In the table given below are summarized the results of measuring the displacements of individual stars on each plate, their mean values, and comparisons with the quantities required by the theory of relativity.

In Fig. 3 the mean values of the deviations of individual stars from the four plates are presented graphically. As

Fig. 3. Displacements of stars as a function of distance. The dotted hyperbola is that required by the theory of relativity; the dash-dotted one is that obtained from the observations.

is evident, the points clearly show (especially star 13, which is only half a radius from the surface of the Sun) an increase in the deviation on approaching the Sun, and are fairly well approximated by the dash-dotted hyperbola. From this it may be concluded that here there is no superposition on the effect, which follows the law of inverse proportionality to the distance from the Sun, of another effect of the same order of magnitude but functionally substantially different.

The vector displacements of the stars also have tangential com-

Table of radial displacements of all stars on all plates of the eclipse field

components, but, as is seen from Fig. 1, they have the character of random errors.

A similar processing of the plates of the control field did not reveal any systematic displacement of the positions of the stars. The displacements of the positions of 27 stars that could be reliably fixed on the plates of the control field, even after the first comparison, appeared in the form of random errors.

The value obtained for the deflection of a light ray passing near the surface of the Sun, \(2''.24\), exceeds that required by the general theory of relativity and at the same time, as Freundlich proves, is in complete agreement with the true results of previous observations published earlier. The agreement of these latter with the value given by the theory of relativity is erroneous and occurred as a consequence of inaccuracies allowed by the authors in comparing the plates, which by chance led precisely to a diminution.

Of the previous observations for detecting the deflection of light in the gravitational field of the Sun, the Lick expedition of Campbell and Trumpler in 1922 to Australia deserves the greatest attention, as having provided the most thorough and extensive material, and the results of which agreed strikingly with those required by the theory of relativity.

Campbell and Trumpler did not take special photographs to determine the difference in the values of the scales; therefore this difference was computed in the following manner. Since, with increasing distance from the Sun, the light deflection rapidly decreases, while the displacement due to the difference of scales, on the contrary, increases, it was assumed for the outermost stars on the plate, situated on average at a distance \(r_0\) equal to approximately ten solar radii, that the light deflection is entirely absent, and that all their displacement occurs as a consequence of the difference of scales; from measurements of this displacement the latter was then determined.

It is clear that by such a determination of the scale there was introduced, though small, a systematic error.

When the necessary corrections were introduced and, in particular, allowance was also made for the change in the apparatus that occurred as a result of transportation from Tahiti to Australia (which Campbell and Trumpler themselves do by means of the reduction of the control-field plates, but mention only in the afterword), the value obtained for \(E\) was \(2''.21\), in good agreement with the latest results—\(2''.24\).

In the reduction of the observations of the English expedition of 1919, an inaccuracy also crept in, consisting in the fact that the determination of the difference of the scale values was made from the raw values of the displacements of the stars, not freed from the effects of refraction and aberration.

Below is a table of the results of the latest expedition and of the new reduction of the earlier ones:

\[ E = 2''.2 \pm 0.10; \]
horizontal camera, \(8.5\) m focal length, Potsdam expedition, 1929.

\[ E = 2''.2; \]
15-foot camera, Lick expedition, 1922.

\[ E = 2''.1; \]
5-foot camera, Lick expedition, 1922.

\[ E = 2''.0 \text{ and } 2''.2; \]
19-foot camera, Greenwich expedition, 1919.

As we see, the results of all the observations quite definitely indicate that the deflection of a light ray in passing near the Sun is greater than that given by the general theory of relativity in the mathematical form in which the theory was developed by Einstein in 1916.