INTERFEROMETRIC METHOD FOR MEASURING THE ANGULAR DIAMETERS OF STARS.

A. A. Mikhailov.

The image of a star, or of an ideal luminous point, in a telescope is represented, as a result of diffraction phenomena, in the form of a small bright disk surrounded by several concentric rings of rapidly decreasing brightness. The dimensions of the central disk and of the rings surrounding it, though very small, are nevertheless sufficiently large that they can be distinctly seen in a strong eyepiece. Their diameters, other conditions being equal, are inversely proportional to the diameter of the objective. For the largest refractor in the world—the 40-inch telescope of the Yerkes Observatory near Chicago—the radii of the diffraction rings are as follows:

Further rings, because of their faintness, are in most cases invisible. The quantities given here are theoretical, applying only to an ideal objective; however, modern opticians have attained such perfection in the manufacture of objectives that, at least near the optical axis, no noticeable deviation from theory is observed.

It is obvious that small details in the object under consideration—whether details on the disk of a planet, the duplicity of an observed star, or the existence of a disk in the object observed—if, owing to their smallness, they are projected within the limits of the central disk, can no longer be detected at any magnification, and an instrument with a larger diameter of objective, or mirror, is required for their examination. Hence every instrument has its own limit of “resolving power,” which for a flawless objective of diameter \(d\) cm is approximately equal to \(\frac{11''}{d}\). As regards double stars, a reliable

A. A. MIKHAILOV

their measurement is possible only in the case when the central disks of their images are completely separated from one another, i.e., when the angular distance between them is not less than the diameter of the central disk. For a large 40-inch telescope this is about \(0''.2\). At smaller angular distances, from \(0''.2\) to \(0''.1\), only an elongation of the image is noticeable, while measurement becomes almost impossible.

It has proved possible, by means of an interferometric method of observation, to penetrate within the limits of the central diffraction disk and to reveal, to a certain degree, its internal structure, which depends on the form of the object. In this case a cover with two small apertures near the edges of the objective, situated at the ends of a common diameter, is placed over the objective. The form and size of the apertures are immaterial, but with very small apertures there is too great a loss of light. The diffraction image is then sharply changed. In the first place, it becomes of much larger dimensions owing to the diminution of the effective aperture of the objective; then it ceases to be circular if the apertures are not of circular form; but the main point is that the central disk becomes striped, i.e., consisting of a series of bright and dark bands perpendicular to the diameter at whose ends the apertures are situated.

Fig. 1.

The reason for such striping is not difficult to see. Let, in Fig. 1, \(A\) and \(B\) be the centers of our apertures. Then, if the optical axis of the objective \(OF\) is directed at a luminous point, for example, at a star without a perceptible disk, the rays \(AF\) and \(BF\), being of equal length, arrive at the point \(F\) with the same phase and produce here a bright band perpendicular to the plane of the drawing. If we take another point \(G_1\) in the focal plane, lying inside the central diffraction disk, then two rays \(AG_1\) and \(BG_1\) will arrive at it, already having a certain path difference, and if \(BG_1 - AG_1\) is equal to half a wavelength, \(\frac{\lambda}{2}\), then we shall obviously obtain here a dark band parallel to the preceding bright band. If for the point \(G_2\) the condition \(BG_2 - AG_2 = \lambda\) is satisfied, then a bright band again passes through it, and so on. Elementary

calculation shows that the distance between two adjacent bright fringes is equal to \(\lambda \dfrac{f}{d}\), where \(f\) is the focal length of the objective (or, more precisely, the distance of the plane of the diaphragm from the focus; in the drawing \(f=OF\)), and \(d\) is the distance between the centers of the apertures \(AB\). Taking for visual observations \(\lambda=0.55\,\mu\), as corresponding to the brightest part of the spectrum, we obtain the angular distance between the fringes

\[ \delta=\frac{11^{\prime\prime}.3}{d}, \]

where \(d\) is expressed in centimeters.

Let us now imagine that we have two luminous points, for example a double star, with angular separation \(\rho\), equal to \(\dfrac{\delta}{2}\), situated relative to one another in the direction of the diameter \(AB\). Then we shall obtain two systems of fringes; moreover, evidently, the bright fringes of one system will coincide with the dark fringes of the other, and as a result the central diffraction disk will be uniformly illuminated, i.e. the fringes will disappear. The same will occur when the angular distance between the luminous points is equal to \(\dfrac{3}{2}\delta,\dfrac{5}{2}\delta\ldots\), etc. Conversely, at distances \(0,\delta,2\delta\ldots\), the fringes will be distinctly visible. Let now the distance between the luminous points slightly exceed \(\delta\). Then the fringes will not completely disappear, but will be visible, though faintly. We shall rotate our diaphragm about the optical axis of the objective. Then, upon rotation through an angle \(\pm\varphi\), for which

\[ \rho \cos \varphi=\frac{\delta}{2}, \]

the distance between the fringes of both systems will prove equal to half the interval, and the fringes will disappear completely. From this one can find \(\rho\), knowing \(\varphi\) and \(\delta\). Further, if one continues to rotate the diaphragm, the fringes will appear again and will be most distinct at \(\varphi=\pm90^\circ\). In this position the luminous points are situated perpendicular to the diameter \(AB\). From this one can find the position angle of one star with respect to the other. In the case where both luminous points are not of equal brightness, it will not be possible to attain complete disappearance of the fringes, but only a greater or lesser diminution in their distinctness. Thus the measurement of a double star, i.e. the determination of the angular distance between the components and their position angle, is reduced to finding those positions of the diaphragm in which the fringes have the greatest or the least visibility.

If one compares the resolving power of the instrument in the interferential method of observation with the ordinary one, a considerable

growth of it. Thus, with a 40-inch objective one can accurately measure double stars with a separation of only \(0,''06\), and detect the duplicity of a star from the weakening of the fringes in much closer pairs.

If the light source has appreciable angular dimensions, then we have a case to some extent similar to the preceding one. Each point of the stellar disk will give its own system of fringes, owing to which the fringes become blurred and disappear altogether when the disk has sufficient dimensions. Michelson’s theoretical investigations showed that, for a circular uniformly illuminated disk, the fringes disappear at a distance between the apertures satisfying the equality

\[ \rho = 1,22 \times \frac{11,''3}{d}. \]

But already much earlier, at a considerably smaller \(d\), there occurs a noticeable decrease in the sharpness of the fringes, indicating the perceptibility of the star’s disk.

Although the existence of a disk in a star produces the same effect as its duplicity, these two cases are very easy to distinguish: in the first case, obviously, the character of the fringes will not change when the diaphragm is rotated.

The described method of measuring the diameters of stars was first tried by Stephan in 1873 at the Marseille Observatory with an objective \(80\ \text{cm}\) in diameter. The result was negative—the fringes remained clearly visible for all the stars observed, and this indicated that the visible diameters of the stars must be less than \(0,''15\).

In 1891 Michelson, while at the Lick Observatory, measured, with the aid of only a 12-inch refractor, the angular diameters of four satellites of Jupiter, obtaining for them values from \(0,''94\) to \(1,''37\)¹).

In 1919 he renewed experiments with the 40-inch refractor of the Yerkes Observatory, and it became clear that, contrary to expectation, the diffraction fringes are visible even with very poor images, and that the chief enemy of accurate measurements—the unrest of our atmosphere—has only little effect in the diffraction method of observation. After this Michelson tried this method on the 60-inch reflector on Mount Wilson and then on the 100-inch reflector of the same observatory.

In view of the interest presented by this reflector, the largest in the world, it will be appropriate to describe it briefly. The greatest difficulties encountered in constructing a reflector of such dimensions consist in making the parabolic mirror. The glass disk for

¹) These measurements are mentioned in Michelson’s book Light Waves and Their Uses.

A Method for Measuring the Angular Diameters of Stars

the mirror must have a thickness of not less than \(1/8\) of the diameter in order to be sufficiently strong and unchanging. As a consequence, its weight reaches four tons. The disk must be quite homogeneous and free from internal stresses. In view of this, the annealing of such glass presents great difficulties.

The disk for the 100-inch mirror was cast at the Saint-Gobain works in France shortly before the destruction of the factory by the Germans during the war. At first the disk was rejected because of numerous bubbles, but all attempts to obtain another disk were unsuccessful, since the castings cracked during annealing. The grinding of the mirror was carried out on Mount Wilson under the direction of Prof. Ritchey, and for testing the parabolic mirror another—plane—mirror 60 inches in diameter was specially made. The parabolic mirror has a diameter of 256 cm, a focal length of 12.8 m, a thickness at the edges of 32 cm, and in the middle of 29 cm. The perfection of its form is seen from the fact that the greatest difference in focal length for its different parts is only 0.14 mm, i.e. about \(1/100000\) of the whole focal length. Besides the Newtonian type, when the focal length of the telescope is 12.8 m, it is possible, by introducing an additional hyperbolic mirror, to give the reflector a Cassegrain construction and bring the focal length up to 90 m.

In the first experiments made with this instrument, the diaphragm was placed not on the mirror, but at a distance of only 1 m from the focus, which, obviously, did not affect the accuracy and sensitivity of the method, but greatly facilitated handling the instrument. At the beginning of 1920 Capella was measured; it had long been known from spectroscopic observations that it is a binary star with nearly equal brightness of both components and an almost circular orbit, with a period of revolution of 104 days. All attempts to resolve this star by the ordinary method had remained without result.

The interferometer detected the disappearance of fringes corresponding to a separation between the components of \(0.''042\), and subsequent measurements made it possible to trace also the orbital motion of the stars, manifested chiefly in the rapid change of the position angle. The measurements made in the interval from December 30, 1919, to April 23, 1920, agree with the calculation on the basis of the spectroscopic orbit to an accuracy of \(0.''0001\) in separation and up to \(1^\circ\) in position angle. This shows the astonishing accuracy of the method.

For Capella, the dimensions of its orbit (more precisely, of the projection of the orbit onto the line of sight) are known from spectroscopic observations in linear measure. The interference measurements give the same dimensions in angular measure. Comparison of these quantities, obviously, makes it possible to determine the distance of the star from the Sun, which turned out to be 55 light-years (parallax \(= 0.''06\)). The distance between the two components is equal

131,000,000 km, and their masses respectively 4.62 and 3.65 times the mass of our Sun.

With these measurements the very slight sensitivity of the method to atmospheric disturbance was once again revealed; it turned out that even with very poor images, when micrometric measurements became already quite impossible, the diffraction method continued to give good results. A much more serious obstacle proved to be atmospheric dispersion, which changed the wavelength \(\lambda\) in different parts of the image and thereby impaired the clarity and regularity of the bands. To eliminate it, the use of special prisms with a small refracting angle, or of inclined plane-parallel plates, was planned.

Fig. 2.

The success of the diffraction method with respect to close double stars made it possible to hope to detect noticeable disks in some stars. According to the modern theory of stellar evolution, stars with low effective temperature, of spectral classes \(G\), \(K\), and \(M\), fall into two sharply separated groups—the so-called giants and dwarfs. The former possess a very low density and a large volume, their linear dimensions sometimes exceeding the dimensions of the Sun by many tens of times. With a not too small parallax, such giant stars should have angular sizes on the order of several hundredths of a second of arc. The limit of the 100-inch reflector with the interference method is \(0.''06\). Therefore, to increase the accuracy, Michelson adapted to the reflector a special interferometer, whose arrangement is shown in Fig. 2. \(S\) is a parabolic mirror; \(A\), \(B\), \(C\), \(D\) are four plane mirrors mounted on a special transverse beam. The path of the rays is shown by dotted lines. The need for a diaphragm with apertures here disappears, since the outer mirrors \(A\) and \(B\) play the role of such apertures. It is evident that the entire device is equivalent to increasing the diameter of the parabolic mirror to the value \(AB\)—the distance between the outer mirrors.

Despite the simplicity of the idea, the construction of such an interferometer presents enormous technical difficulties, since the distances between all the mirrors must remain unchanged

with an accuracy to a small fraction of the wavelength of light, and with the same accuracy it was necessary to eliminate any bending and deformation at different positions of the instrument. The apparatus has such dimensions that the distance \(AB\) could be brought up to 20 feet, or 6 m. On December 13, 1920, the instrument was directed at the star \(\alpha\) Orionis (Betelgeuse), which belongs to the typical giants. At a mirror separation \(AB\) of 3 m, the interference fringes disappeared. Thus, for the first time, the angular diameter of a star was measured, and was found to be \(0''.045\). Since the parallax of this star does not exceed \(0''.03\), the linear diameter of \(\alpha\) Orionis must exceed the diameter of the orbit of Mars.

Among other measured stars, let us note Arcturus, with a diameter of \(0''.022\), and Antares, with a diameter of \(0''.040\). The parallax of Antares is approximately \(0''.01\), and thus the equator of this star is four times larger than the Earth’s orbit, while its diameter is about 400 times greater than the diameter of the Sun.

These results exhaust the incomplete information at our disposal at present. But there is no doubt that, in the very near future, the interferometric method will become a new and powerful means of investigating the structure of stars.