Stellar Astronomy

Stellar Astronomy

 

a branch of astronomy that studies the general regularities in the structure, composition, dynamics, and evolution of stellar systems and investigates the applications of these principles in our stellar system— the Milky Way Galaxy. The specific investigations of other galaxies and extragalactic objects branched off, in the middle of the 20th century, from stellar astronomy into a special discipline of astronomy—extragalactic astronomy. In contrast to astrophysics, which studies the nature of individual stars and nebulas, stellar astronomy investigates groups of these objects. It is subdivided into stellar statistics, stellar kinematics, and stellar dynamics.

Every star can be characterized by a series of parameters, some of which depend on the position of the star with respect to the sun. Such visible characteristics are the spherical coordinates of the star (the galactic system of celestial coordinates is usually used in stellar astronomy), the apparent magnitude of the star in various photometric systems, the observed color index, the color excess, the extent of absorption and polarization of light, the distance to the star, the proper motion of the star, the parallax, the tangential and radial velocities, and the apparent rotation velocity. Some of these parameters, namely the absorption and polarization of light and the color excess, depend chiefly on the quantity and properties of the light-absorbing dust material found between the sun and the star. Other parameters are the intrinsic characteristics of the star, which do not depend on the relative positions of the star and the observer. These are the coordinates of the star determining its spatial position in the Milky Way Galaxy, absolute stellar magnitude, luminosity, intrinsic color index, spectral class, temperature, mass, radius, velocity components in our galaxy, and intrinsic rotation velocity.

Stellar astronomy closely interacts with other branches of astronomy—astrometry and astrophysics—in the determination of stellar characteristics.

Stellar statistics. The study of the structure of the Milky Way Galaxy and the elucidation of the characteristics of the stellar population in its various regions can be carried out using the methods of mathematical statistics. Such an approach is used to study the distribution of stars, having one or another characteristic, in different directions or in different regions of the Milky Way Galaxy as well as in various groupings of the galaxy, such as open clusters, globular clusters, and stellar associations. Statistical regularities thus obtained are called distribution functions. For example, the brightness function determines the distribution of stars according to apparent magnitude. Luminosity functions indicate how stars are distributed according to luminosity in various regions of our galaxy. This function has been most reliably determined for the solar neighborhood and for nearby open clusters. The stellar density function expresses the distribution of stars according to their distance in a given solid angle. The light absorption function shows how the absorption of the light of stars (expressed in stellar magnitudes) changes in a given direction as a function of distance. Many distribution functions in stellar statistics are interrelated by equations. For example, the brightness function, stellar density function, luminosity function, and absorption function are related by equations that are termed the fundamental equations of stellar statistics. The equations of stellar statistics always contain, together with the distribution functions of visual properties, distribution functions of true properties. One of the important problems of stellar statistics is the use of these equations to find functions of true properties by means of observationally obtained functions of visually observed properties. For example, solving the equation that relates the distribution function of the apparent surface stellar density in a globular cluster with the true spatial stellar density in this cluster, we find the second function by means of the first function, whichis determined from observation. Investigations of multidimensional distributions of stellar properties are of great importance, since many properties are statistically related to one another. These statistical relations are usually complex and therefore are usually represented with the aid of diagrams. For example, the statistical relation between stellar spectra and absolute magnitudes is represented by a diagram that reveals a series of sequences in the stellar population having evolutionary significance (Hertzsprung-Russell diagram). The following diagrams are also significant in characterizing the stellar population: color-absolute magnitude, color-apparent magnitude, and mass-absolute magnitude diagrams and a two-color diagram (for two colors, each of which characterizes the proportion of radiation energy in two different regions of a star’s spectrum).

Stellar statistics also investigates the distributions of properties of variable stars (shape of the brightness variation curve, period and amplitude of the brightness variation, amplitude of the color index variation), double stars (angular separation between components, difference in apparent magnitudes and spectra of the components, orbital elements), multiple stars and star clusters (diameter, number of stars, principles of the apparent and spatial density distributions, color-apparent magnitude diagram), dark nebulas (dimensions, transparency coefficient), and other objects in our galaxy. Since stars of each spectral class and type (for example, different types of variable stars) are arranged in space in a particular way (the Milky Way Galaxy, as it were, consists of numerous mutually interacting subsystems), many studies in stellar statistics are carried out individually for stars of each spectral class or type.

The absorption of light in space is taken into consideration when calculating the distances to stars by the method of comparing their absolute and apparent magnitudes. The amount of this absorption is estimated from the discrepancy between the color of a star and its spectral class, which is caused by the reddening of the star’s color under the influence of light-absorbing material. Because of the inaccuracy in the estimates of light absorption, which is especially great for distant stars in directions close to the Milky Way Galaxy’s plane of symmetry, there is an uncertainty in the determination of the distances to most stars. This is one of the factors complicating the problems of stellar statistics.

The problems are further complicated because most of the stars in our galaxy cannot be observed owing to the galaxy’s large dimensions and the significant absorption of light near the galactic plane. Even in the immediate galactic neighbor-hood of the sun a certain fraction of low-luminosity stars are still not visible. Nevertheless, the total number of stars accessible to observation is so great that the determination of all the characteristics of these stars is an exceedingly large observational problem. Therefore, many astronomical observatories of the world conduct work according to the so-called plan of selected areas proposed in 1906 by the Dutch astronomer J. Kapteyn: According to the plan, the determination of the characteristics of faint stars should basically be done only in 206 individual areas equally distributed throughout the sky and additionally in 46 areas of special interest. Moreover, it is assumed that the regularities derived on the basis of stellar characteristics as determined in Kapteyn’s areas must correspond to those regularities that could have been obtained using the characteristics of all the stars in the sky. The International Astronomical Union has distributed the work on the determination of various stellar characteristics among observatories in different countries. Part of this work is being performed in the USSR.

Stellar kinematics. The methods of kinematics (a branch of mechanics) and mathematical statistics make it possible to study the distributions of apparent kinematic stellar characteristics (proper motion, radial velocity, tangential velocity, space velocity, apparent velocity of rotation), to find the distributions of true kinematic characteristics (residual velocity components, intrinsic rotation velocity), and to draw conclusions about the general principles of the motion of stellar systems as a whole.

Although a stellar system consists of individual bodies —stars—separated by great distances, properties of continuity are observed in its structure and motion along with discrete properties. Let an arbitrary point in space, occupied by a stellar system, be surrounded by a sphere with a volume that is small in comparison with the volume of the entire stellar system but large enough to encompass sufficiently many (for example, 1,000) stars. Then the average value of the velocities of all the stars in the sphere is called the velocity of the centroid of these stars. If the coordinates of the point in the stellar system change, the velocity of the centroid corresponding to it changes slowly and almost smoothly. Therefore, a continuous field of velocities can be observed in the stellar system. It is natural that in the general case the velocity of a star does not coincide with the velocity of its centroid. In our galaxy, in particular, the sun moves with respect to its centroid. This velocity is called the space velocity of the sun and enters into the velocities of stars that are measured from the earth (which moves together with the sun). Methods of determining the space velocity of the sun through the radial velocities and proper motions of stars have been worked out. Although these two methods use observational material obtained in completely different ways (one from astrophysical and the other from astrometrical measurements), they lead to results that agree well with each other. The space velocity of the sun (relative to the aggregate of all stars brighter than the sixth magnitude) is close to 19.5 km/sec and is directed toward a point in the sky situated in right ascension 18 hr and declination about +30° (the standard solar apex). The study of the velocities of centroids shows that they perform circular motion parallel to the galactic plane around the Milky Way Galaxy’s axis of symmetry. The angular velocity of the circular motions of the centroids varies from place to place, that is, our galaxy does not rotate as a rigid body; moreover, it neither expands nor contracts. Only its central region rotates, evidently, as a rigid body, with a period of about 30 million years. At a distance of 5 kiloparsecs from the center, our galaxy’s period of rotation is equal to 130 million years and in the region of the sun, that is, at a distance of about 10 kiloparsecs from the center, to about 250 million years. The linear rotation velocity of the sun’s centroid around the center of the Milky Way Galaxy is approximately 250 km/sec. If the sun’s space velocity is geometrically subtracted from the observed velocity of a star, then the velocity of the star with respect to the sun’s centroid—the peculiar velocity of the star—is obtained. If the velocity of the star’s centroid with respect to the sun’s centroid is subtracted from the star’s peculiar velocity, then the space velocity of the star—its velocity with respect to its own centroid—is obtained. The geometric sum of the velocity of the centroid with respect to the center of inertia of a stellar system and the space velocity of a star is equal to the total velocity of the star with respect to the system’s center of inertia. The study of the distribution of residual stellar velocities shows that at every point in the Milky Way Galaxy a symmetry condition is fulfilled if very large space velocities are disregarded: the number of stars having space velocities in a given direction is equal to the number of stars with space velocities in the opposite direction. But the mean square space velocities vary with direction. The component of space velocity directed toward the center of the Milky Way Galaxy has the greatest mean square value, the component in the direction of the galaxy’s rotation has the next largest, and the component perpendicular to the galaxy’s plane of symmetry has the least. In the solar neighborhood the mean square values of the space velocity components in the three indicated directions are about 41 km/sec, 28 km/sec, and 21 km/sec, respectively, if stars belonging to the different parts of our galaxy are analyzed together.

For large space velocities, exceeding 70 km/sec in the solar neighborhood, the symmetry condition ceases to hold. Large space velocities are absent that have directions forming acute angles with the direction of rotation of the centroid around the center of the Milky Way Galaxy. But at the same time velocities of such magnitude directed opposite to the galaxy’s rotation are observed. This phenomenon, called the asymmetry of the space velocities, is explained by the fact that the total velocity of a star, which is equal to the geometric sum of the centroid velocity and the star’s space velocity, is larger the smaller the angle between these velocities and, in the case of a small angle, the larger the space velocity. For a space velocity greater than 70 km/sec and directed in the direction of the Milky Way galaxy’s rotation, the total velocity of the star would surpass the critical escape velocity in the solar neighborhood, and the star would leave our galaxy. The critical escape velocity in ,the region of the sun is about 320 km/sec.

The radial velocities and proper motions of stars are the fundamental observational material of stellar kinematics. Contours of spectral lines with wavelength λ = 21 cm emitted by neutral hydrogen, which is located primarily near the galaxy’s plane of symmetry, have also been widely used since 1946 for the investigation of the kinematics of our galaxy. The radiation is not absorbed by the galaxy’s dust matter. In addition, because of the different angular velocities of the centroids in our galaxy, the radial velocities of the masses of hydrogen located in the line of sight are different, and the masses of hydrogen situated nearby do not absorb the radiation emitted by distant masses. Owing to this, the 21-cm radiation from the most distant regions of our galaxy reaches terrestrial radio telescopes and is registered by them. Statistical methods of studying the contours of the line λ = 21 cm make it possible to clarify the law of rotation of the Milky Way Galaxy, to investigate the density distribution of neutral hydrogen, and to mark the position of the spiral arms of our galaxy.

The various objects that make up the population of stellar systems are divided into two populations; moreover, each of these occupies definite regions of the stellar systems. Stellar Population I is located near the planes of symmetry of spiral galaxies, concentrating, moreover, in the spiral arms and avoiding the region of the nucleus. Stellar Population II pre-dominates in the regions of spiral galaxies that are remote from their plane of symmetry; it forms the nuclei of spiral galaxies, and elliptical galaxies and lenticular galaxies of type SO are formed from it. Population I stars include blue-white giant and supergiant stars, long-period cepheids, novas, and supernovas, open clusters, hydrogen clouds, and dust nebulas. Population II stars include red subdwarfs, red giants, short-period cepheids, and stars of globular clusters.

The idea of the division of galactic populations is more thoroughly worked out in the concept of subsystems of stellar systems. Stellar subsystems, which include all objects of one or another spectral class or type, are distinguished by sepa-rate values of the characteristics of spatial distribution (gradients of stellar density along the radius of the Milky Way Galaxy and perpendicular to its plane of symmetry) and by the features of the distribution of the objects’ velocities. Sub-systems of different objects are interpenetrable, and a stellar system is thus an aggregate of subsystems. Each subsystem is approximately an oblate ellipsoid of revolution, with the ob-lateness being different for different subsystems. Reflecting this fact, the subsystems are classified according to three component types of our galaxy: flat, spherical, and intermediate.

Stellar dynamics. Stellar dynamics studies the principles of stellar motion in the force field of a stellar system and the evolution of stellar systems as a result of the motion of the stars. Stellar systems are self-gravitating, that is, the aggregate of stars in a system itself produces the gravitational field that governs the motion of each star. The gravitational field of a stellar system has a complex structure. Because the gravitational force of a point mass diminishes proportionately to the square of the distance, that is, not very rapidly, at every point over most of the volume of the stellar system the total gravitational force of all the objects making up the stellar system significantly exceeds the gravitational force of the object nearest to this point. On the other hand, in the immediate vicinity of stars, dense star clusters, or other compact objects, the force of attraction of such an object equals the total gravitational force of all the remaining objects or may even exceed it. Thus in studying the structure of the force field of a stellar system, it must be regarded as the sum of (1) the regular field of the system, that is, the field produced by the system as a whole, which reflects the stellar system’s properties of continuity, and (2) the irregular field, produced by forces arising during the approach of stars, which reflects the properties of discreteness in the structure of the stellar system. The irregular forces have the character of random forces. The more bodies there are in a stellar system, the greater the role in its dynamics of the regular forces and the lesser the role of the irregular forces.

During its formation a stellar system, as a rule, is in a state of natural equilibrium. Under the influence of the system’s regular and irregular force fields, the distribution of stars and stellar velocities changes within it. The stellar system gradually approaches equilibrium. Since, in a system containing a large number of stars, the regular field acts more quickly than the irregular, equilibrium is first attained in the regular field. In this state the regular field no longer changes the distribution of the stars and their velocities. The time necessary for the transition to a state of equilibrium of the regular field is inversely proportional to the square root of the density of the matter in the system. For stellar systems this time amounts to tens or hundreds of millions of years. In a state in which only the regular field is in equilibrium, the irregular field continues to change the distribution of the stars and their velocities, bringing the system closer to a state of equilibrium of the irregular field. A stellar system cannot attain complete equilibrium since under the influence of the irregular forces certain stars acquire a velocity greater than the critical velocity and thus leave the system. This process goes on continuously. The state in which all changes in the distributions of the stars and their velocities are the result only of the continuous slow departure of stars from the system is called the state of quasi-equilibrium of the irregular field. The time for the attainment of quasi-equilibrium is termed the relaxation time. The relaxation time amounts to a quantity on the order of tens or hundreds of millions of years for open clusters, billions of years for globular clusters, and thousands or tens of thousands of billions of years for galaxies. The time for the complete disintegration of a nonrotating stellar system under the influence of its irregular field is approximately 40 times greater than the relaxation time. The faster a stellar system rotates, the slower the disintegration process.

The age of observable open clusters as a rule exceeds their relaxation time. The majority of observable open clusters have achieved quasi-equilibrium, and many of them have become greatly depleted as a result of the departure of stars. There is reason to believe that most stars of the Milky Way Galaxy once belonged to open clusters and are the result of their disintegration. The number of completely disintegrated open clusters must exceed by many times the number of open clusters presently existing in our galaxy. The age of globular clusters is comparable to their relaxation time. Evidently, the central regions of the globular clusters, where the relaxation time is less, have attained quasi-equilibrium, while the outlying regions are in a state of equilibrium of the regular field. The age of galaxies does not exceed tens of billions of years, whereas the relaxation time for them is hundreds or thousands of times greater. Therefore, galaxies are far from attaining quasi-equilibrium. Some of them, namely the irregular galaxies, are not even in a state of equilibrium either because they are very young systems or because of deformations caused by mutual interactions during the close approach of galaxies.

A stellar system that has achieved a state of equilibrium of the regular field has a plane of symmetry and an axis of symmetry perpendicular to it. A stellar system with a zero value for the principal angular momentum in a state of equilibrium of the regular field may be spherically symmetric. In a quasi-equilibrium state it must be spherically symmetric. The trajectories of stars in a spherically symmetric system are flat. In the general case they are not closed, and the convolutions of a single trajectory fill a ring. In a system with a plane of symmetry and an axis of symmetry the trajectories are not plane curves. The convolutions of a single trajectory fill a three-dimensional region—a torus.

The fundamental problem of stellar dynamics is the investigation of the regularities in the structure and evolution of stellar systems based on the study of the forces acting within them. One of the methods for such investigations is the construction of theoretical models of stellar systems at various stages of their evolution, which correspond to specific, observed stellar systems, including our galaxy, other galaxies, clusters of galaxies, and open and globular clusters. In a theoretical model, complete consistency must be obtained in the mutually interacting distribution and motions of the stars. Empirical models of the Milky Way Galaxy and of other galaxies are also constructed based on observable data on the distribution of the density of matter in them. The distribution of stars and their motions are not self-consistent in empirical models.

History. The foundation of stellar astronomy was laid at the end of the 18th century by the British astronomer W. Herschel, who carried out statistical investigations (“surveys”) of the stellar sky. Calculating the number of stars visible in the visual field of a telescope in different parts of the sky, he discovered the phenomenon of galactic concentration, that is, the increase in the number of stars as one approaches the galactic equator. This indicated the flatness of our stellar system. Herschel constructed the first model of our stellar system—the Milky Way Galaxy—and determined the direction of the sun’s motion with respect to nearby stars. He discovered a large number of double stars and found the orbital motion of a number of them and thus showed the physical nature of their duality; he also showed that Newton’s universal law of gravitation was valid even beyond the solar system. In 1847 the Russian astronomer V. la. Struve, studying the structure of our galaxy, asserted the existence of light absorption in interstellar space and the increase in (spatial) stellar density in approaching the galaxy’s plane of symmetry. In the middle of the 19th century the Russian astronomer M. A. Koval’skii and the British astronomer G. Airy worked out analytical methods for determining the velocity of the sun from the proper motions of the stars. At the end of the 19th century H. von Seeliger and K. Schwarzschild in Germany developed methods for investigating the spatial distribution of stars according to their surveys. At the beginning of the 20th century the Dutch astronomer J. Kapteyn discovered a preferred direction in the motions of stars and proposed a hypothesis of the existence of two star streams moving toward each other. Later Schwarzschild advanced a proposition concerning the ellipsoidal law of (space) stellar velocity distribution, which more naturally explains the observed regularities in the motions of stars. This period (before 1922) includes the completion by Kapteyn of the investigation of the structure of the Milky Way Galaxy on the basis of the results of star counts and his analysis of the proper motions of stars. Despite the fact that Struve had already in the middle of the 19th century arrived at the conclusion that light absorption existed in our galaxy, at the beginning of the 20th century the conviction prevailed that interstellar space was completely transparent. Therefore, the apparent thinning out of stars with increasing distance from the sun in all directions, which is primarily caused by light absorption in interstellar space, was taken to be an actual decrease in stellar density in all directions from the sun. In Kapteyn’s model the sun is located at the center of our galaxy.

In the first quarter of the 20th century astronomers of the Harvard Observatory (USA) completed a survey of the spectra of hundreds of thousands of stars, and the Dutch astronomer E. Hertzsprung and the American astronomer H. Russell at this time discovered the division of stars of the late spectral classes into giants and dwarfs and constructed a spectrum-luminosity diagram reflecting the statistical relationship between the spectrum of a star and its luminosity. In 1918 the American astronomer H. Shapley found that the center of the system of globular clusters was situated far from the sun. Evidently, the center of the huge system of globular clusters (and not an ordinary star—the sun) must precisely coincide with the center of our galaxy. Shapley determined the direction to the center of our galaxy and estimated its distance from the sun. In 1917 the American astronomers G. Ritchey and H. Curtis discovered faint stars that suddenly appeared and then disappeared in spiral-shaped nebulas, and they determined that these novas were similar to those that are observed from time to time in our own galaxy. It became clear that the spiral nebulas are located at enormous distances outside our galaxy and are of comparable size. From 1924 to 1926 the American astronomer E. Hubble, using a 2.5-m (100-inch) telescope, resolved into stars the outer regions of three spiral nebulas, including the Andromeda Nebula and the Triangulum Nebula; and in 1944 the American astronomer W. Baade, using a 5-m (200-inch) telescope, resolved into stars several elliptical nebulas as well as the nuclei of the spiral nebulas mentioned above. By this it was finally demonstrated that other stellar systems existed in addition to our galaxy; they were also called galaxies.

In 1927 the Dutch astronomer J. Oort worked out a method for studying the rotation of our galaxy, and on the basis of the data on the proper motions and radial velocities of stars, he discovered the phenomenon of rotation and determined its fundamental characteristics. The direction toward the center of rotation coincided with the direction toward the center of the system of globular clusters. In 1932 the Soviet astronomer K. F. Ogorodnikov developed a theory of kinematics for stellar systems, in particular for our galaxy, in which a stellar system is viewed not simply as a collection of separately moving stars but as a single system, in whose motion the entire volume of space occupied by it participates. Between 1915 and 1920, J. Jeans and A. Eddington (Great Britain), and later V. A. Ambartsumian (USSR) and S. Chandrasekhar (USA), worked out the foundations of stellar dynamics. B. Lindblad (Sweden) derived the basic dynamic relationships for our galaxy. In 1930 the American astronomer R. Trumpler, studying a large number of open clusters, determined that their distances were distorted by the presence of light absorption in interstellar space and estimated the absorption for directions close to the Milky Way Galaxy’s plane of symmetry. Hubble investigated the distribution of galaxies over the entire sky. It was found that as one approaches the galactic equator the number of observable galaxies quickly diminishes, and there are almost no galaxies close to the galactic equator (approximately between latitudes -10° and +10°). This showed that the light-absorbing matter was concentrated in a comparatively thin layer near the Milky Way Galaxy’s plane of symmetry. Between 1938 and 1947, Ambartsumian established that the light-absorbing matter in our galaxy had a flakelike structure.

The 1940’s were characterized by investigations that determined characteristics of the distribution and kinematics of various-type stars. It was ascertained that distribution and kinematics are closely linked to the problems of the origin and evolution of stars of a given type, of star clusters, and of interstellar gas and dust. Ambartsumian discovered that hot giants (spectral classes O and B0-B2) form groups called stellar associations. Stellar associations are unstable, and consequently the stars constituting them are young. Their age has proved to equal from 105 to 107 years, that is, less by far than the age of the earth, sun, most stars of the galaxy, the galaxy itself, and of other galaxies, which is estimated in billions of years (up to tens of billions of years). Thus the existence of stellar associations indicates that star formation in the galaxy is continuing.

The Soviet astronomers P. P. Parenago and B. V. Kukarkin and their colleagues have studied the distribution and kinematics of various-type stars, including variable stars, and they have established that our galaxy is an aggregate of sub-systems, each of which has its own distinctive features. Baade has indicated the existence of two types of stellar populations. The development of methods of radio astronomical observation has had great significance for stellar astronomy. Radio observations have permitted the study of the structure of the Milky Way Galaxy’s nucleus as well as a more accurate determination of the location of its plane of symmetry. The investigation of the contours of the line λ = 21 cm emitted by neutral hydrogen (the first work was published by H. C. van de Hulst, S. Muller, and J. Oort in 1954) has made it possible to determine the law of the galaxy’s rotation for a significant range of distances and to obtain information about the disposition of the spiral arms in our galaxy. The beginning of the second half of the 20th century has been characterized by the intensified development of research in stellar dynamics—by the study of the role of regular and irregular forces in stellar systems, the obtaining of estimates of the age of various systems, the study of stellar velocity distributions, the construction of models for spherical and rotating systems, the determination of orbital characteristics of stars in stellar systems, and the investigation of different types of instability in stellar systems. Of important significance are methods for the direct solution of the problems of stellar dynamics with the help of numerical solution by computer of the equations of motion of n bodies.

In the 20th century investigations in stellar astronomy are being conducted at most astronomical observatories in many countries, including the USSR—in Moscow, Leningrad, Abastumani, Biurakan, Tartu, and other cities.

REFERENCES

Chandrasekhar, S. Printsipy zvezdnoi dinamiki. Moscow, 1948. (Translated from English.)
Kukarkin, B. V. Issledovanie stroeniia i razvitiia zvezdnykh sis tern na osnove izucheniia peremennykh zvezd. Moscow-Leningrad, 1949.
Parenago, P. P. Kurs zvezdnoi astronomii, 3rd ed. Moscow, 1954.
Ogorodnikov, K. F. Dinamika zvezdnykh sistem. Moscow, 1958.
Zonn, W., and K. Rudmcki.Zvezdnaia astronomiia. Moscow, 1959. (Translated from Polish.)
Kurs astrofiziki i zvezdnoi astronomii, vol. 2. Moscow, 1962. Chapters 2, 1&;-21.
Stroenie zvezdnykh sistem. Moscow, 1962. (Translated from German.)
Kinematika i dinamika zvezdnykh sistem. Moscow, 1968.
Kurth, R. Vvedenie v zvezdnuiu statistiku. Moscow, 1969. (Translated from English.)
Pahlen, E. \on.Lehrbuch der Stellarstatistik. Leipzig, 1937.
Smart, W. M. Stellar Dynamics. Cambridge, 1938.
Trumpler, R., and H. Weaver. Statistical Astronomy. Berkeley-Los Angeles, 1953.

T. A. AGEKIAN

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