THE ECLIPSE METHOD FOR DETERMINING THE DIAMETER OF STARS

Knowledge of the sizes of stars is of exceptional interest for astrophysics. However, up to now it has been possible to measure directly only the diameter of the Sun, since the dimensions of all other stars without exception lie beyond the resolving power of modern telescopes. Existing indirect methods for estimating stellar diameters have so far made it possible to obtain information only about very few stars. The most general method is based on the use of the known dependence of a star’s radiating capacity on the temperature of its surface and on the circumstance that the luminosity of a star having a given temperature is proportional to its surface area. Thus, in the case of red giants, such as Betelgeuse and Antares, proceeding from the fact that their luminosity is approximately 3600 times greater than the solar luminosity and that their temperature is close to 3000°, values of their diameters are obtained that exceed the diameter of the Sun by several hundred times. This conclusion is also confirmed by other methods, namely by studying eclipses of binary stellar systems in which one of the components is a red giant, and also with the aid of Michelson’s stellar interferometer. Both of these methods also make it possible to obtain only comparatively rough estimates of the size of a star. Thus, with the aid of the stellar interferometer the diameter of a star is determined from the change in the visibility of the interference pattern as a function of the distance between the mirrors of the interferometer. Here an essential assumption is that of uniform luminosity over the entire visible surface of the star. From this point of view, improvements in the interferometer recently proposed by I. L. Bershtein and G. S. Gorelik1 and, apparently, radically increasing the very limited number of stars now accessible to measurement cannot lead to a substantial increase in the accuracy of the results obtained by this method. Therefore the technique used by the authors of the paper under review2, which, as will be seen below, permits more detailed information on the structure of a star to be obtained, is of undoubted interest. This technique, consisting in determining the changes in the intensity of the light sent by a star during the process of occultation of the star by the lunar disk, is by no means new. It was proposed about half a century ago, but only in the most recent years has it become realizable with sufficient accuracy, thanks to the development of photoelectric methods of measurement.

The principle of the measurements is very simple. As the stellar disk is covered by the limb of the Moon, the amount of light reaching the terrestrial observer will decrease, becoming zero at the moment when the star is completely hidden by the Moon. By measuring the interval of time during which the weakening of the star’s light occurs, and knowing the relative motion of the star and the Moon, it is not difficult to find the diameter of the star. However, the practical implementation of this method encounters serious difficulties. First of all, with present-day measuring techniques reliable data can be obtained only for stars with comparatively large angular sizes, and above all for red giants. The occultation of such stars by the Moon is, however, a rather rare event, especially since measurements are feasible only when the star is occulted by the dark limb of the Moon, since otherwise the light scattered by the lunar surface completely masks the effect being measured.

Furthermore, it is necessary to take into account diffraction phenomena at the limb of the Moon. Finally, a very serious difficulty is posed by the guiding, i.e. ensuring that the image of the star falls at the center of the small entrance diaphragm of the photoelectric apparatus, especially at the moment when the star emerges from behind the lunar disk. If one adds to this the very high requirements imposed on the photoelectric scheme, it will become clear that

measurements of this kind represent by no means a simple task.

The object of observation was the star Antares, whose angular dimensions, according to data obtained by other methods, amount to 0.040 arc seconds (the presumed distance is 6 km). The duration of the process of occultation of the stellar disk by the Moon is about 0.1 sec. Observations were carried out from two points situated in South Africa at a distance of about 1500 km from one another, namely from Pretoria (with a 74-inch reflector) and from Cape Town (with a 24-inch refractor). By the time of the last observations (April 13, 1952) a more advanced photoelectric installation had been constructed in Pretoria, making it possible also to detect changes in brightness connected with the eclipse of the blue companion of Antares; moreover, the results agreed well with those which were to be expected for a point source situated far from the principal star.

In all, the measurements covered 4 dates:

May 4, 1950—the opening of the stellar disk (Cape Town),

June 27–28, 1950—the closing of the stellar disk (Pretoria),

July 15, 1951—the closing of the stellar disk (Cape Town),

April 13, 1952—the opening of the stellar disk (Cape Town—almost central, Pretoria—far from the center).

For the processing of the measurement results the following method was applied. It was assumed, as a first approximation, that the change in the brightness of the star over some interval of time is equal to the brightness of the region of the star uncovered (or covered) by the Moon over the same interval of time. As a result, a certain distribution was obtained

Fig. 1. Brightness distribution over the surface of Antares on April 13, 1952. Solid line—Cape Town; dashed line—Pretoria. The horizontal segment corresponds to 0.01 arc seconds.

Fig. 2. Comparison of brightness distributions over the surface of Antares for 1950 (solid line), 1951 (dashed line), and the weighted mean for Pretoria and Cape Town 1952 (dotted line). The horizontal segment corresponds to 0.01 arc seconds.

of brightness by regions. Proceeding from this distribution, corrections for diffraction were determined, depending comparatively weakly on the form of the distribution because of the small angular dimensions of the star. Then the resulting brightness distribution was corrected for diffraction. The results of the measurements are given in Figs. 1 and 2, in which the abscissa axis gives the angular distance from the center of the star, and the ordinate axis gives the brightness of the region of the stellar disk, uncovered (or covered) by the lunar limb, situated at this distance from the center of the star. As is evident from the figures,

in all cases a sharply expressed “hump” was observed, corresponding to the principal part of the star’s luminosity, as well as “wings” extending rather far in both directions; moreover both the “hump” and the “wings” proved to be sharply asymmetric. Although the observing conditions on April 13, 1952, in Pretoria and Cape Town differed considerably (a different disposition of the regions of the star successively occulted by the Moon, a 2.5-fold difference in the duration of the eclipse process, different parts of the lunar limb, etc.), the results turned out to be similar (Fig. 1), while the differences were of secondary character. On the other hand, the results of the measurements of 1950, 1951, and 1952 (Fig. 2) agree substantially with one another. (The values of the star’s diameter proved to be \(0.''030\), \(0.''033\), and \(0.''043\), respectively.) From this the authors conclude that the star has a nucleus, probably not spherical, and is surrounded by an irregular atmosphere. At the same time this nucleus underwent a noticeable expansion over the two-year period of the measurements. The authors suppose that Antares, like some other red giants, pulsates, which is in agreement with the data of earlier spectroscopic observations that showed variability of the star’s radial velocity. The authors note that Antares is an irregular variable star. Its photographic magnitude in 1950 was 3.12, in 1951—3.05, and in 1952 apparently about 2.90.

Thus the method of measurement described makes it possible not only to determine the value of a star’s diameter, but also to draw certain conclusions about its structure and about the processes taking place on the star.

R. G.

CITED LITERATURE

1. I. L. Bernstein and G. S. Gorelik, DAN SSSR, 86, No. 1, 47 (1952).
2. D. S. Evans, South-African J. Sci., 49, No. 2, 41 (1952).