Observation of the Einstein Effect During Solar Eclipses*)

A. A. Mikhailov

Let us imagine a material particle moving with the speed of light in interstellar space. From the standpoint of Newtonian mechanics, when such a particle approaches a cosmic mass—a star—it will describe a hyperbolic orbit with its focus at the center of the star, differing very little from a straight line. This orbit will have only a slight curvature—a bending toward the star near the vertex of the hyperbola, coinciding with the periastron, the point closest to the star. The angle between the asymptotes to the hyperbola is equal to

\[ 2\alpha = 2f \frac{M}{c^2 R}, \tag{1} \]

where \(f\) is the gravitational constant, \(M\) is the mass of the star, \(c\) is the speed of light, and \(R\) is the distance of the periastron from the center of the star. If the star is the Sun, whose surface is touched by the particle’s trajectory, then \(M\) must be taken equal to its mass, and \(R\) equal to the radius of the Sun. After substituting the corresponding quantities into formula (1), we find that \(2\alpha = 0''\!.87\). If a ray of light were a stream of such particles, then by this angle the direction of a ray coming from a star located far behind the Sun and visible to us at the very edge of the solar disk would change; as a result, the star would appear displaced by this angle radially away from the Sun. Light from a star visible at an angular distance \(r\) from the center of the Sun is deflected by an angle

\[ \frac{0''\!.87}{r}, \]

if \(r\) is expressed in units of the angular radius of the Sun. Thus, all the stars projected on the celestial sphere around the Sun would appear to be spread apart and shifted

*) Report read at the meeting of the Einstein Session of the Division of Physical and Mathematical Sciences on November 30, 1955.

from their normal positions by angles \(\frac{0''.87}{r}\) in directions away from the Sun.

In 1915 Einstein showed that the general principle of relativity requires a displacement twice as large, so that the coefficient in the preceding formula must be increased to \(1''.75\). Observation of this “Einstein effect” was possible only during a total solar eclipse, when near the Sun, covered by the Moon, stars can be seen and photographed. An attempt to detect the deflection of light during the eclipse of 1918, undertaken by an expedition of the Lick Observatory, was unsuccessful. The eclipse of May 29, 1919, promised good chances, since at that time the Sun was projected onto a region rich in bright stars in the constellation Taurus.

The importance of having a sufficient number of bright stars around the Sun is explained by the following considerations. First of all, let us note that observations can be made only photographically, with the aid of a long-focus astrograph camera. For example, suppose that the focal length of the objective is \(6\) m. Then \(1''.75\) corresponds on the plate to a linear length of \(0.051\) mm. Such would be the displacement of a star situated at the very edge of the Sun. However, such a star cannot be observed, since its light will be drowned in the bright parts of the Sun’s inner corona. In the best case one can photograph a star of sufficient brightness, for example of magnitude \(7\)—\(8\), at a distance of one solar radius from the edge of the Sun, i.e. for \(r = 2\). If such a star is photographed, this by itself gives nothing, since its normal, undisturbed position will not be known, because a photograph gives not absolute coordinates of stars, but only relative ones. It is therefore necessary that several stars be recorded on the photograph, in the ideal case arranged symmetrically with respect to the center of the Sun. If, say, there were a second star with the same value of \(r\), but on the other side of the Sun, then the distance between the two stars would prove to be increased by the doubled Einstein effect, in our example by \(50\mu\). However, in order to measure this increase, one must know what this distance is for the normal positions of the stars. It cannot be calculated from catalogued positions of the stars: first, because of the insufficient accuracy of the catalogues, which contain errors of several tenths of a second of arc; and second, because the exact scale of the photograph is not known, since it depends on focusing and on temperature. How accurately the scale of the photograph must be known is clear from the following reasoning. Two stars situated symmetrically with respect to the Sun at a distance \(2r\) from its center are separated from one another by \(4r\), or, approximately, by \(4000''\). To guarantee \(0''.05\) in this distance, one must know the scale with relative accuracy to \(1/80000\), which corresponds to \(0.075\) mm in the position of the plate relative to the objective. Thus,

A change in focusing or in the focal length by \(0.1\) mm will already cause a noticeable error in the scale. For stars located farther from the Sun, for which the displacement is smaller and the distance greater, an even more precise knowledge of the scale is required.

To overcome these difficulties, it is necessary to photograph the same region of the sky a second time, with the instrument in the same setting, at a time when the Sun is far from this region. Such a photograph, giving the normal, undisplaced positions of the stars, must be taken in the night sky, best of all six months after or before the eclipse, when the Sun is in the diametrically opposite position. The influence of the proper motions of the stars over such an interval of time is, as a rule, insignificant and can be taken into account with the aid of catalogs. However, the state of the instrument during the control photography will in general be different, which will cause some change in the scale, and this must be taken into consideration.

Let it be established by measuring two plates, one taken during the eclipse (we shall call it the eclipse photograph) and the other taken at night (we shall call it the control photograph), that some star, situated at a distance \(r\) from the center of the Sun, is displaced in the eclipse plate away from the center of the Sun by an angular amount \(\delta r\). If the Einstein effect were present, this could be represented as

\[ \delta r=\frac{A}{r}, \]

where \(A\), according to the theory of relativity, is equal to \(1''\!.75\), if \(r\) is expressed in units of the apparent radius of the Sun. But a change in scale will cause an additional term proportional to the distance, and therefore the measured displacement must be represented by the following conditional equation:

\[ \delta r=\frac{A}{r}+Br, \tag{2} \]

where \(B\) is the coefficient of the correction for scale.

For a reliable determination of the two unknowns \(A\) and \(B\) from such conditional equations, it is necessary to have a sufficient number of stars with values of \(r\) differing as greatly as possible. But we have already mentioned that stars with \(r<2\) can hardly be observed; therefore, along with stars having such smallest values of \(r\), it is also necessary to have stars with the largest possible values of \(r\), not less than 5 or 6 solar radii. Hence follows the necessity of using comparatively wide-angle astrographs with a field not smaller than \(3\times3^\circ\). Then the distant stars will serve chiefly for determining the scale, i.e., the coefficient \(B\), while the stars close to the Sun will give the value of the Einstein displacement, i.e., the constant \(A\).

Of course, in deriving \(\delta r\) it is necessary to take into account the influence of all factors that distort the positions of stars on the plate. These include dif-

differential refraction and differential aberration. They can be taken into account in the usual way adopted in the reduction of astrophotographs when measuring the rectangular coordinates of stars, for example, by the so-called method of six constants; however, the differential method of measurement is more advantageous, taking refraction and aberration into account by the complete formulae. The nonperpendicularity of the plate to the optical axis of the objective may also affect \(\delta r\). This influence must be eliminated by careful construction of the cassette part of the instrument and by good adjustment of it; otherwise, quadratic terms must be taken into account in the reduction.

The differential method of measurement requires a special procedure for making the control photographs, which are taken “through the glass,” i.e. a photographic plate with the reverse side carefully cleaned is inserted into the cassette with its reverse side, emulsion away from the objective, of course changing the focus depending on the thickness of the plate glass. When the eclipse plate is measured, it is placed with the control plate layer to layer, so that the distance between the images of each star on the two plates does not exceed \(1\) mm. To avoid a possible shift of one plate relative to the other during the process of measurement, the plates are glued at the corners or pasted along the edges with adhesive tape. With the aid of the ocular micrometer of a measuring microscope, the differences \(\Delta x\) and \(\Delta y\) of the rectangular coordinates between the images of each star are measured. These differences depend on how the two plates are placed together: on how far apart the optical centers of the two plates are and on the rotation of one plate with respect to the other. In addition, the Einstein displacement is also present in these differences. The aim of the reduction consists precisely in revealing the latter and in eliminating the first two effects. This is achieved by setting

\[ \left. \begin{aligned} \Delta x &= a + bx + cy,\\ \Delta y &= a' + b'x + c'y, \end{aligned} \right\} \tag{3} \]

where \(x, y\) are the rectangular coordinates of each star, expressed in divisions of the measuring-machine scale, usually in millimeters, and \(a, b, c, a', b', c'\) are unknown constants to be determined from the conditional equations (3). The Einstein displacement is not included in these formulae, since the law of its action is different, and therefore it is retained in the residual terms after equations (3), written for all measured stars, have been solved by the method of least squares.

Let these residual terms be, respectively, \(\delta x\) and \(\delta y\). Then, in order to obtain the Einstein displacement \(\delta r\), these differences must be projected onto the direction passing through the center of the Sun, whence

\[ \delta r = \delta x \sin P + \delta y \cos P, \]

where \(P\) is the position angle of the star relative to the center of the Sun, usually reckoned from the positive direction of the \(y\)-axis (north) to the positive direction of the \(x\)-axis (east).

The \(\delta r\) obtained in this way serve for finding \(A\) from the conditional equations (2).

The first observation of the Einstein effect was made by two English expeditions during the eclipse of May 29, 1919. The path of totality passed through Brazil, the Atlantic Ocean, and Central Africa. One expedition was on the island of Principe off the coast of Africa, the other in northern Brazil at the locality Sobral. The expeditions were equipped with several astrographic telescopes with focal lengths from 3.4 to 5.8 m, lying horizontally and fixed; the rays were directed into the telescopes after reflection from coelostat mirrors. It should be mentioned here that during an exposure, which lasts several tens of seconds, the images of the stars must remain strictly motionless on the plate. But, owing to the diurnal rotation of the celestial sphere, the images move, with an objective focal length of 6 m, at a speed of about 25 mm per minute. Therefore either the astrograph tube must, with great precision, move after the stars, which requires an equatorial mounting, or, with the tube immobile, a coelostat must stand before the astrograph objective, its mirror being turned so that the rays reflected from it enter the astrograph while preserving an unchanged direction. In both cases high demands are made on the precision of motion of the corresponding instrument, since the positions of stars on the photograph are measured with an accuracy down to a micron.

The African expedition of 1919 had only partial success, since part of the plates was spoiled by clouds. On two photographs it proved possible to measure 5 stars on each. The comparison plates were taken in England before the expedition was dispatched, which, of course, is prejudicial, since the instrument was dismantled in the interim and its condition could have changed. Nevertheless, the displacement obtained, i.e. the coefficient \(A\) in formula (2), proved equal to \(1''.65\), sufficiently close to the theoretical value given by the principle of relativity.

The observations of the other expedition, at Sobral, were more successful. On the photographs as many as seven stars appeared, of which two were at a distance of two solar radii from the center of the Sun. The plates obtained with the long-focus astrograph were quite sharp. The comparison plates were taken, without dismantling the instrument, two months after the eclipse. The displacement coefficient proved equal to \(1''.98\), but the residual terms after solving equations (2) had syste-

mathematical character, which was attributed to the cylindrical deformation of the coelostat mirror under the influence of heating by the Sun. Recalculation of the measurements with allowance for second-order terms gave almost the same result. However, in our opinion, the published observed deviations for individual stars are best represented by the coefficient \(A=2'',07\). Be that as it may, the observations established the existence of a displacement, fairly close in magnitude to that predicted on the basis of the theory of relativity and in any case much larger than was required by the corpuscular theory of light. Nevertheless, it could not be considered that the question of the Einstein displacement needed no further investigation. The observations of 1919 must be regarded only as a successful reconnaissance, a test of the observational and reduction method. The chief shortcoming of these observations was the small number of stars and their strongly asymmetric distribution around the Sun. The weak point proved to be the coelostat, whose mirror was subject to thermal deformations. The importance also became clear of a good determination of the scale of the eclipse photographs.

Considerably more certain results were obtained by the American expedition in Australia during the eclipse of September 21, 1922. A double astrograph on a parallactic mounting, and therefore without a coelostat, had a focal length of \(4.6\) m and covered a large field—the plates were \(43\times43\) cm in size. Four successful photographs were obtained, on which, on the previous evening, another star field \(90^\circ\) from the Sun had been taken for comparison and for obtaining the scale. Six months before the eclipse, on the island of Tahiti, control photographs of the eclipse field together with the same comparison field were made with the same instrument. Images of 118 stars were obtained; the nearest of them was at a distance of two radii from the center of the Sun. The authors—Campbell and Trumpler—consider the most probable result of their investigation to be \(A=1'',72\), although the use of the comparison region for determining the scale increased this value of \(A\) to \(2'',05\).

The reduction and the result of this determination were criticized by several specialists. The chief shortcoming of this investigation of the deflection of light, as indeed of all the others, consists in the fact that the law of deflection expressed by the formula \(A/r\) is assumed in advance, i.e. that the deflection on approaching the Sun increases along a hyperbolic curve, inversely proportional to the distance from the center of the Sun. This law is given by theory, the correctness of which the observations are supposed to test. Instead, the observations give only the value of the constant \(A\), on the assumption of the correctness of the law itself. The reason for this lies in the smallness of the deflection, which for individual stars is smaller than the error with which the position of a given star is measured. The scatter of the results for individual stars is too great for one to

could be used not only to derive the value of the constant \(A\), but also to verify the law of dependence of the displacement on the distance to the center of the Sun. To demonstrate this circumstance let us turn to Fig. 1, borrowed from an article by the director of the Paris Observatory, Esclangon, in l’Astronomie for 1924, and showing the displacements of individual stars obtained from observations of the eclipse of September 21, 1922. In the figure, the angular distances of the stars from the center of the Sun in degrees are plotted along the abscissa axis. On the ordinate axis

Fig. 1.

the observed displacements are plotted on another scale. The black dots representing the displacements of the stars have different diameters depending on the statistical weight of the measurement of the position of the given star. The coordinate axes are not perpendicular to one another, which takes into account a small correction for scale. The hatched strip represents the solar disk. The three dotted dashed lines give the displacement for three values of the constant \(A\), namely \(1'',75\), \(2'',05\), and \(2'',50\). From inspection of the diagram it is difficult to say which of these three lines corresponds more closely to the observation. Only two or three stars nearest the Sun fit better near the lines \(A = 1'',75\) and \(2'',05\); the remaining stars do not make it possible to decide which value of \(A\) is best. Only application of the method of least squares indicates a certain advantage for the value \(A = 1'',72\), but a small change of scale, i.e. of the angle between the coordinate axes, noticeably changes this best value. If one speaks of determining the law of displacement from these observations, then the dashed straight line drawn by eye, whose equation is

\[ \delta r = -0'',1r + 1'',0, \]

A. A. MIKHAILOV

hardly represents the observed displacement any better. But, of course, in view of the large scatter of the points, there is no reason to speak of the reality of such a law of displacement. This formula does show, however, how important an independent determination of the scale is; by changing the scale, though by an inadmissibly large amount, all the deviations can be reduced to a constant value, since the influence of the scale gives precisely a term proportional to the distance \(r\).

Hence the further attempts to observe the Einstein effect in a more refined manner are understandable, with special attention paid to obtaining the scale of the plates independently—not from conditional equations of the type (2), but externally, for example by imprinting on the eclipse plates a certain standard angle. For purity of the procedure, however, such imprinting must be done with the instrument in the same state as during the eclipse, preferably during the photographing of the eclipse itself.

Such an attempt, carefully prepared, was made by an expedition of the Potsdam Astrophysical Observatory, headed by Prof. Freundlich, during the eclipse of May 9, 1929, in northern Sumatra. The main instrument consisted of two identical tubes with a focal length of \(8.5\) m, arranged horizontally at an angle of \(25^\circ\) to one another. Their objectives were directed at the common mirror of a coelostat. One tube observed the reflection of the Sun, the other a comparison field off to the side of it. It was assumed that, being under identical conditions, the scales of plates covering \(3 \times 3^\circ\) would change identically. Several months after the eclipse, control photographs were taken both of the eclipse field and of the comparison region. In addition, to determine the scale, a rectangular grid-scale was imprinted on the plates with the aid of a collimator. The reduction of the plates gave \(A = 2''.2\).

Another instrument was a wide-angle astrograph with a focal length of \(3.4\) m and a field of \(7.5 \times 7.5^\circ\), mounted parallactically. To determine the scale, during the total phase of the eclipse the tube was shifted to another region, distant from the Sun, which was photographed on the same plate. These photographs gave an unsatisfactory result.

A detailed discussion of the results obtained with the main instrument, in which specialists in adjustment computations and in applications of the method of least squares from the Potsdam Geodetic Institute participated, showed with complete definiteness that, for a reliable determination of the scale and for obtaining not only the constant \(A\), but also the law of the deviation of light, a very precise and independent determination of the scale of the eclipse plates is necessary, and that the methods tried up to that time for this purpose were unsatisfactory.

The eclipse of June 19, 1936, was of exceptional interest to Soviet astronomers. The path of totality passed

throughout the entire Soviet Union from the Black Sea to the Far East. The duration of the total eclipse was short—about \(2\frac{1}{4}\) minutes—but the time of year and the considerable altitude of the Sun favored observations. The study of the Einstein effect was included in the program of work of the expedition of the Sternberg State Astronomical Institute, headed by A. A. Mikhailov.

Fig. 2. Lower part of the instrument for observing the Einstein effect during the 1954 eclipse. At upper right the clock mechanism is visible.

For the photography a two-lens objective was used, 15 cm in diameter and with a focal length of 6 m, specially made at the State Optical Institute under the direction of D. D. Maksutov. Plates measuring \(35 \times 35\) cm covered an area of \(3\frac{1}{3} \times 3\frac{1}{3}\) degrees. They were made by special order on polished mirror glass 6 mm thick and weighed almost 1.5 kg each. The manufacture of cassettes of such size, ensuring focusing accuracy to 0.05 mm and allowing rapid replacement so that several photographs could be taken during the total phase, was practically infeasible. This difficulty was overcome by the fact that the plates without

of each cassette were placed directly on the rounded ends of three screws, which also served for centering the plate and for fine focusing. But for this purpose the pavilion itself was light-tight and represented a photographic camera, inside which was the observer with his assistant. The observer

Fig. 3. Exterior view of the pavilion for observing the Einstein effect during the 1954 eclipse in Pyatigorsk.

placed the plate on the screws and operated the shutter; the assistant removed the plate after the exposure was completed. In this way it was possible to change plates within four seconds, and during the entire duration of the total phase to take four plates with exposures of 25 and 35 seconds. The objective was connected with the frame, into which the aforementioned screws were screwed, by a strong truss welded from iron tubes. The upper part of the truss with the objective was slightly projected out of the pavilion and was suspended on a cardan mounting, allowing it to turn in any direction. The lower part of the truss rested by means of runners on a steel cylinder along which it could slide, being carried along by a long screw rotating ча-

with a clock mechanism at the required speed to paralyze the diurnal rotation of the celestial sphere. Most of the weight of the truss was relieved by counterweights. The driving cylinder was adjusted parallel to the tangent to the Sun’s diurnal parallel at the mean moment of totality.

For an independent determination of the scale, it was intended to use the following method. In front of the objective, at an angle of about \(45^\circ\) to the optical axis, a plane-parallel glass plate of high accuracy, also made by D. D. Maksutov, was mounted. Its two surfaces reflected, in all, about \(9\%\) of the light coming from a region of the sky situated almost \(90^\circ\) from the Sun, so that the stars located in this region were photographed simultaneously with the stars in the eclipse region, but weakened by \(2\frac{1}{2}\) stellar magnitudes. As the comparison region, a star cluster in the constellation Coma Berenices was chosen, where, over the area covered by the photographic plate, there are about ten stars of magnitude \(5\)—\(6\). Weakened by reflection from the glass, they should have appeared on the plate as stars of magnitude \(7\)—\(9\), the same as most of the stars of the eclipse field, and would have served for determining the scale of the plates. The advantage of this method is that the comparison field is photographed simultaneously with the eclipse field, with the instrument in one and the same state and with one and the same exposure. Control photographs of the same regions, taken under night conditions, would give the stars of the eclipse field in their normal, undisplaced positions, and the stars of the comparison field would make it possible to tie the eclipse plates very accurately to the control plates, i.e., would make it possible to take account reliably of the scale differences, as well as of refraction.

The observing site—Eastern Kuibyshevka, not far from Blagoveshchensk—was chosen with the calculation that the comparison region—the group of stars in Coma Berenices—would, during the eclipse, be at almost the same height above the horizon as the Sun, which was advantageous for taking refraction into account. During the eclipse two photographs suitable for measurement were obtained. The instrument was left in place, and in March 1937 control plates were made at exactly the same hour angles. Unfortunately, owing to insufficient adjustment of the instrument, the stars of the comparison field could not be used in the reduction, which had to be carried out in the usual way, determining the scale from conditional equations of the form (2). Another unfavorable circumstance was that the temperature conditions during the photographing of the eclipse differed greatly from those during the taking of the control plates—in the first case the temperature was \(+23^\circ\), and in the second \(-21^\circ\). 29 stars were measured and used in the reduction; for the nearest of them \(r = 1.97\) solar radii. The constant \(A\) was obtained equal to \(2''.74 \pm 0''.31\) (probable error), or one and a half times greater than the theoretical value. A detailed consideration of the result shows,

indicates that this value of \(A\) is based chiefly on the measured displacement of the three stars nearest to the Sun and therefore cannot be considered especially reliable. But the reality of the displacement, though perhaps of smaller magnitude, is proved by the following. The sum of the squares of the measured differences in the positions of the stars, taken in directions from the center of the Sun, i.e., the sum of the squares \(\hat{\xi} r\), turned out to be equal to 29 square seconds; if, however, these differences are freed from the displacement, then this sum decreases to \(15''\). If, for comparison, one takes the sum of the squares of the measured differences of the coordinates of the stars in the direction perpendicular to the radii of the Sun, in which the Einstein displacement does not take part, then this sum turned out to be equal to \(17''\), i.e., practically coinciding with the sum of the squares of the radial displacements freed from the effect of the displacement.

Let us now note wherein lay the defect in the adjustment of the instrument’s motion which, in our case, did not allow the region of comparison stars to be used for determining the scale. Usually this adjustment of the instrument is carried out so that the image of that stellar field—in the present case, the near-solar one—toward which the instrument is directed remains motionless on the photographic plate throughout the entire exposure. But in order to obtain good images of the comparison field, situated at a distance of about \(90^\circ\) from the Sun, an additional adjustment according to this latter field is also required, since good motion of the instrument following the stars of the first field does not yet ensure equally good motion following the stars of the comparison field. Let us explain this circumstance.

Suppose there is a certain inaccuracy in the setting of the instrument, consisting in the fact that its axis of rotation is not strictly parallel to the Earth’s axis of rotation. With good adjustment of the motion following an imaginary star located at the center of the eclipse field, this star will remain motionless on the photographic plate, but the entire region of the sky will slowly rotate about this center. During an exposure of about one minute such rotation will still remain quite imperceptible, since the most distant stars of this region are no farther than \(2^\circ\) from the center. But a star in the comparison region, \(90^\circ\) away, will have a radius of rotation 30 times greater and, at the same angular speed of rotation, will describe on the plate an elongated trail, unsuitable for precise measurement, or will even not be recorded at all.

Since then we have repeatedly tried to observe the Einstein effect by our method with somewhat improved apparatus during a number of subsequent eclipses. Observations of the eclipse of September 21, 1941, were organized in the vicinity of Alma-Ata. The weather on the day of the eclipse was cloudless. However, because of wartime conditions, it was necessary to bring from Moscow an instrument that was not quite ready and to adjust the rebuilt and unfinished clock mechanism already on site, in the absence of a good workshop.

As a result, we did not succeed in properly adjusting the clock mechanism, and the stars appeared not as points but as little streaks, which could not be measured. We intended to observe the eclipse of July 9, 1945, in Rybinsk, where everything had been prepared for this purpose; however, during the eclipse there was continuous cloud cover. The eclipse of May 20, 1947, seemed especially favorable; a Soviet expedition was sent to Brazil to observe it. At the observing site—the town of Araxá in the state of Belo Horizonte—the duration of the total phase was almost four minutes, and the month of May is one of the clearest of the year. Unfortunately, on the evening before the eclipse a cyclone moved in from Argentina, and continuous cloud cover made it impossible to carry out the observations. Matters were no better on June 30, 1954, when we intended to observe the eclipse in Pyatigorsk. For this eclipse our instrument had been somewhat rebuilt: instead of the former one, a new objective was installed, with an aperture of 20 cm and the same focal length of 6 m. The inclined plane-parallel glass plate in front of the objective was replaced by a small elliptic diagonal mirror of fused quartz, whose projection was a circle 8 cm in diameter; this was done in order to eliminate the possibility of temperature deformations of the reflecting surface. The circumstance that in this case the eclipse field was photographed by rays passing through the edge of the objective, while the comparison field was photographed through its middle part, was of no significance, since the control photographs had to be taken under the same conditions. Finally, according to V. P. Linnik’s design, a special device was made for impressing, through the same objective, several systems of interference fringes with a strictly constant angular distance between them. We were also preparing to observe the eclipse of June 20, 1955, of very long duration, in Ceylon, but the Ceylonese authorities did not grant the Soviet expedition permission to enter. There is, however, no need to regret this, since almost all the foreign expeditions that were there failed because of bad weather.

The Einstein effect was observed on two more occasions by the American astronomer van Biesbroeck. According to him, independently of us he developed and applied the same method for determining the scale. He also placed a plane-parallel plate in front of the objective, but it was not transparent, as ours was, but lightly aluminized, so that it transmitted about 50% of the light and reflected approximately the same amount. The disadvantage of this was that the eclipse field was greatly weakened, which made it necessary at least to double the exposure time of each photograph and thereby reduced the number of photographs that could be taken during the total phase of the eclipse. The gain in brightness of the comparison field is not of essential importance, since this field can always be chosen ...

with sufficiently bright stars. As the objective, a specially constructed triplet with an aperture of 15 cm and a focal length of 6.1 m was used. The tube of the instrument was mounted on a simplified parallactic mounting. Plates measuring \(43 \times 43\) cm covered an area of \(4 \times 4\) degrees.

The eclipse of May 20, 1947, was also observed by van Biesbroeck in Brazil, at the locality of Bocaiuva, situated almost 500 km northeast of Araxá, and the cyclone which interfered with our observations had not yet reached Bocaiuva on the day of the eclipse. He made only one exposure, with an enormous exposure time of 3 minutes 5 seconds, as a result of which the plate turned out to be very strongly fogged. The images of the stars of the comparison field were distorted by strong astigmatism, which the observer attributed to deformation of the plane-parallel plate from heating by the Sun, but we are inclined to think that this distortion of the images, at least in part, was due to insufficient accuracy in guiding the instrument—especially since such a long exposure imposed particularly high demands on the accuracy of guiding. Be that as it may, it proved impossible to use the comparison field in reducing the photograph, and the scale had to be determined in the usual way, with the aid of the conditional equations (2). The control photographs were taken at the same place in August 1947. In all, 51 stars were measured; of these, the nearest to the Sun was at a distance of 3.33 solar radii, so that the maximum Einstein displacement that could be measured amounted to only \(0''.53\), which corresponded, in linear measure on the photograph, to 15 μ. The measurements were made not by the differential method, twice. As a result, a displacement was obtained
\[ A = 2''.01 \pm 0''.27 \]
(mean error). As the author himself notes, this value is based almost entirely on five stars whose distances from the center of the Sun lie within the range from 3.3 to 3.8 solar radii.

More successful was van Biesbroeck’s observation of the eclipse of February 25, 1952, in Khartoum (Sudan). The same instrument was used, but with a yellow light filter in front of the objective. Two photographs of the eclipse were obtained, with exposures of 60 and 90 seconds, and two control photographs taken six months later. This time it was possible to determine the scale from the stars of the comparison field. The author gives the following final value of the displacement obtained:
\[ A = 1''.70 \pm 0''.10 \]
(mean error). In all, only 9 stars were measured on one plate and 11 stars on the other. Of these, only one star was at a distance of \(2.12r\) from the center of the Sun, while the next two nearest stars were at distances of \(4.34r\) and \(4.37r\). Therefore the result depends to a considerable degree on these few stars. The residual terms of the conditional equations reveal a noticeable systematic character, which disappears if the observed

The deflection is given by the linear formula

\[ \delta r=-0'',066r+0'',67, \]

which is quite close to the one we derived for the 1922 eclipse. The observed deflections for individual stars, together with the displacement curve and the straight line, are shown in Fig. 4.

How much better the linear formula represents the observations of the 1952 eclipse is seen from the fact that it gives a sum of squares of the residual terms, in microns, equal to 140, whereas the formula \(A=1'',70/r\) gives a total of 272, i.e. twice as much. Of course, the linear formula is only an interpolation formula and loses its meaning for large values of \(r\), but it shows how far the observational results still are from confirming the law of displacement inversely proportional to the distance from the center of the Sun.

Fig. 4. Deflection of the apparent positions of stars according to van Biesbroeck’s observations during the eclipse of February 25, 1952.

In this connection, in 1924 Eddington wrote: “the observations neither confirm nor refute Einstein’s law of deflection. They only indicate, if all assumptions about systematic errors are discarded, the existence of deflections near the Sun, but without establishing a law and without an exact value of the deflection at the solar limb.” These words, perhaps in a softened form, remain valid even now, despite a number of observations carried out in subsequent years. There still remains the difficult but interesting task of a more complete and precise experimental investigation of the Einstein effect. On the basis of the work performed it is fairly clear what instrument is required for this purpose.

commentary and what method of observation should be applied. But, taking into account the great dependence of the observations on the weather, it is necessary that these investigations be carried out more systematically and by a larger number of scientists, without missing a single favorable eclipse.

There remains the question whether there are other possibilities for verifying the deflection of light rays under the influence of gravitating masses, apart from observations of solar eclipses. It may be possible, by applying infrared technique and an instrument such as an extra-eclipse coronagraph, to make observations of stars near the Sun even without an eclipse; but the development of this method will require much preliminary work. As for the use of other deflecting masses besides the Sun, if we confine ourselves to the field of astronomy, such a possibility is almost excluded. Indeed, formula (1), which is also valid for Einstein’s law when the numerical coefficient is doubled, shows that the deflection of a ray is proportional to the gravitational potential

\[ \frac{fM}{R} \]

at the surface of a spherical mass. After the Sun, the greatest gravitational potential is possessed by the planet Jupiter. However, the potential at its surface is 106 times smaller than the potential at the surface of the Sun. Moreover, the probability of observing a star sufficiently close to the limb of Jupiter is very small because of the small apparent dimensions of the planet. Thus observation of the Einstein effect at the limb of Jupiter is practically impossible. Only some chance may be offered by an interferometric measurement of a double star near the very limb of Jupiter, as V. P. Linnik recently pointed out. There remains also the possibility of detecting the deflection of the rays of a distant star projected on the celestial sphere in very close proximity to another star nearer to us, to which G. A. Tikhov recently drew attention. But this interesting question still belongs to a considerable extent to the realm of fantasy. Thus, for the time being, in the investigation of this phenomenon we are limited to observations of solar eclipses; but this does not mean that work should not be done on realizing other, more difficult possibilities as well.