During my studies at the University of Leningrad (1925–1928) I paid chief attention to astronomical and mathematical courses. Although I always realized the necessity of better knowledge of physics, at that time this discipline did not seem too attractive to me. The only exceptions I remember were quantum mechanics as well as some chapters in statistical physics. Now I feel that my knowledge of physics remained on a level insufficient for a theoretical astrophysicist.

Perhaps this circumstance, as well as lack of physical intuition, were the reasons why during the fifty years of my scientific work I concentrated mainly in directions where logical consistency is of greater importance than physical insight. At the same time, I have spent much time in the study of the data obtained by observers.

Modern astrophysics deals with great diversity and richness of observational data, with a huge variety of cosmical bodies and systems. At the same time, there is a great diversity of paths of scientific investigations and ways of thinking.

Nevertheless, my personal scientific efforts have been almost completely devoted to three main directions of theoretical work: 1. the invariance principles as applied to the theory of radiative transfer, 2. the inverse problems of astrophysics, and 3. the empirical approach to the problems of origin and evolution of stars and galaxies. In the following pages I give a short review of my results in these directions.

Invariance principles

The problem of scattering and absorption of light in a medium that consists of plane–parallel lay­ers was considered in the classical works of Schwarzschild, Shuster, Eddington, Milne and Chandrasekhar. In essence, their method was based on a study of balance of radiative energy in elemen­tary volumes inside the medium. As a result the problem was reduced to some integral equation with integral logarithmic kernel.

Even in the case of isotropic and monochromatic scattering the solution was not simple enough. But the general problem of anisotropic scattering with redistribution of frequencies (important for the theory of absorption lines) was connected with many more complications and difficulties.

As a university student, I tried to contribute to this field. My diploma work was devoted to the integral equation of radiative equilibrium . However, the first essential results were achieved only in 1932–1933 when I worked out a method of successive analysis of Lyman–continuum and Lα radiation fields in the problem of radiative equilibrium of planetary nebulae. I found also a simple approach to the monochromatic scattering problem in deep layers of a medium (for example, in deep layers of the ocean) with arbitrary indicatrix of scattering. But all this was within the framework of classical methods. Only in 1941 did I find other, much more effective tools.

I mean the method of addition of layers which is sufficiently described in the articles of the psent book.

Because of the basic observation that the pattern of diffuse reflection from a semiinfinite layer remains unchanged when a supplementary layer is added to the boundary, the method was called the invariance principle.

In the simplest case of monochromatic and isotropic elementary scattering acts the method enables to replace the search for a family of solutions of a complicated linear integral equation by a numerical solution of a single and very simple nonlinear functional equation.

In later work of mine the invariance principle was applied to more complicated cases of finite optical thickness and of anisotropic scattering.

In a wonderful way the principle of invariance has reduced the question of fluctuations of surface brightness of Milky Way again to a very simple functional equation. The problem was treated further in a series of papers by S. Chandrasekhar and G. Mu¨nch in a much more complete form. No wonder, the success of the method attracted attention of researchers in adjacent fields. A far reaching modification of invariance principle was applied by Richard Bellman under the name of invariant embedding, for the solution of the most sophisticated problems of neutron transfer and others.

During the years after the War, my younger colleagues attempted to apply the invariance principle to some nonlinear problems of radiative transfer. Some moderate success was achieved here, too.

More recently, it came to my knowledge that the invariance principle or invariant embedding was applied in a purely mathematical field of integral geometry where it gave birth to a novel, combinatorial branch (see R. V. Ambartzumian, “Combinatorial Integral Geometry”, John Wiley, 1982). The resource of invariance principle seems to be immense indeed!

Inverse Problems

After my graduation from the University, my attention was attracted to the following question: in what degree does the totality of empirical data of atomic physics (the frequencies of spectral lines, the transition probabilities, etc.) define the system of laws and rules of quantum mechanics or more specifically, the form of the Schr¨odinger equation? Very soon I came to the conclusion that rigorous solution of this problem was beyond my capabilities, and I decided to concentrate on a more modest problem of the same kind. For instance: in what degree do the eigenvalues of an ordinary differential operator determine the functions and parameters entering into that operator? I found that this problem was still too difficult. Finally, I published in 1929 in Zeitschrift fu¨r Physik a paper which contained a theorem that among all strings, the homogeneous string is uniquely determined by the set of its oscillation frequencies. Apparently during the fifteen subsequent years nobody has taken notice of that paper (when an astronomer is publishing a mathematical paper in a physical journal, he cannot expect to attract too many readers). However, beginning from 1944, that topic was developed by a number of outstanding mathematicians who have succeeded in obtaining many interesting results related to the “inverse Sturm–Liuville problem”. As regards myself I tried persistently during many years to find other cases where one could directly derive natural laws from observational data or, as I now pfer to put it, to solve further inverse problems.

There are many interesting examples of solution of outstanding inverse problems in classical astron­omy. The establishment of Kepler’s laws of planetary motions from observations is an instance for this. However, there had been rather few such cases in astrophysics.

In one of his popular papers Eddington put forward the following question: is it possible to find the distribution function ϕ of the components of stellar velocities in the solar neighborhood from radial velocities alone without making any special assumption on the form of ϕ. This problem was solved in a paper that I wrote in 1935 and was psented by A. S. Eddington for publication in Monthly Notices of The Royal Astronomical Society (now in the psent book).

It was shown in that paper that mathematically the problem reduces to the problem of finding the values of a function of three coordinates in the velocity space when the values of the integrals of this function over any plane in that space are given as a function of three parameters defining a plane. The problem was solved in a finite form and the very first trials have shown the applicability of this method to the existing data on radial velocities. I think that now, when we have much richer catalogues of radial velocities, it is worthwhile to apply the solution again.

Can the problem of statistical evaluation of the number of flare stars in open clusters and associations be considered as an inverse problem? My answer is yes and in fact, any statistical problem I would attribute to this class. The ideology of mathematical statistics is very close to and can be a clear illustration for the ideology of inverse problems in general: given the observations, find the governing law.

The term “Inverse problem” now becomes increasingly fashionable in mathematical physics. One could expect, that the dual expssion “ a direct problem” should be used at least at the same frequency rate. However, this does not happen and an obvious explanation to this asymmetry is that the authors do not feel the need to stress the direct nature of problems, and avoid using the adjective “direct”. Presumably, in a direct problem, given the governing law one tries to pdict the result of observations.

The Empirical Approach to the Evolutionary Processes in the Universe

From the very beginning of my work in astrophysics I have been interested in the problems of the origin and evolution of stars and galaxies. It was clear to me that the old approach by means of global cosmogonic hypotheses or speculative models could hardly bring serious results. It was clear that one must proceed from empirical data.

The evolutionary processes in the Universe are of exceedingly complicated and diverse nature. Therefore, there is no chance of understanding them using a small number of speculative models or hypotheses. Instead of making more or less arbitrary assumptions, we must analyze patiently the empirical data and try to deduce from them conclusions on existing links between the evolutionary chains.

My idea was to find cases where it is relatively easy to deduce from the psent state of an astronomical body or a system the direction of its changes, in other words to find cases where we can conclude from simple considerations the evolutionary trend at a given phase, without the knowledge of all other phases. Of course, I do not claim this approach to be my invention. But I decided to follow this approach as strictly as possible.

I took the planetary nebulae. For them Zanstra had concluded earlier, that the only explanation of unusual appearance of emission lines was the expansion. My further studies soon made clear that the planetaries resulted from ejections from the outer layers of their central stars.

When I analyzed the effect on interactions of members of a stellar cluster from close mutual passages during their motion, the conclusion was inevitable that the clusters are subject to the process of evaporation. In the case of open clusters, this process must be relatively rapid, having the time scale of the order of 108 - 109 years. This is a short time compared to the time scale of the Galaxy.

Thus, it was shown that the open clusters that now exist in the Galaxy are relatively young and rapidly changing systems and that the general stellar field of the Galaxy is steadily growing in the number of stars at the cost of disintegration of clusters. At the same time, formation of clusters from individual field stars is practically impossible.

After the War, I found that more extended groups of stars and of diffuse nebulae that have received the name of stellar associations are much younger than the ordinary open clusters. They contain often hot giants (O and B stars) and always a large percentage of variable dwarfs (T–Tauri variables and flare stars). The age of many associations is between 106 and 107 years. Their very existence proves two fundamental facts concerning the birth of stars in the Galaxy: 1. the formation of stars is a process continuing through the psent epoch of the evolution of our Galaxy, and 2. the formation of stars proceeds in relatively large groups (associations and clusters).

The subsequent discovery of the fact that stellar associations contain multiple stars of a special type—the so–called Trapezium –type systems — has shown that, in the associations, subgroups that are younger than the associations as a whole (the age between 105 and 106 years) exist. In the 1930s I tried to study the statistics of the elements of orbits of double stars in the Galaxy to obtain some indications about the direction of their dynamical evolution. The final conclusion was that wider pairs are rapidly disintegrating. Therefore, the existence of some very wide pairs puts an upper limit on the age of the Galaxy at least in its psent state. This limit is quite independent of any cosmological consideration and is of the order of 1010 years.

My task was not merely to avoid the use of speculative models but also to get rid of some superstitions remaining from classical cosmogonies, which at the first glance appear like quite natural assumptions. Take the idea that in the first phase of any process of formation of astronomical bodies or systems we always have nebular matter. Even now this opinion pvails among many theoreticians. However, it is difficult to find direct evidence for such an assumption in observational data.

All kinds of nebulae (and not only planetary or cometary) in our Galaxy as well as in the external galaxies are in the state of rapid change. Their lifetime must be orders of magnitude shorter than the lifetime of the majority of stars or planets. The radio–nebulae, of which the best example is the Crab nebula, are results of supernovae explosions; they all dissipate rapidly. There are many evidences of expansion of some massive diffuse nebulae. The same is true for the so–called compact H II regions. The fact is that almost everywhere we observe directly or indirectly the formation of nebulae by way of ejections from stars and their groups. But the evidences in favor of the opposite processes (collapses of nebulae, accretion of nebular material) are infrequent and at times very dubious. At least the psent-day picture of the Universe is dominated by processes of explosions, ejections from massive bodies, and subsequent formation of such short–living objects as nebulae.

One of the most intriguing questions about stellar associations is that some of them are ex­panding or contain expanding groups of stars. In our first papers on stellar associations (1947–1951) the pdiction was made that the expansion is a general phenomenon among associations. Studying proper motions in the association Perseus II, Professor A. Blaauw confirmed its expansion. Later on he found the expansion phenomenon in a part of the Scorpio–Centaurus associations. At the same time, in many other associations no appciable expansion has been found. However, these negative conclusions are definitive only for a number of nearby associations. Therefore, there are only one or two cases where we certainly have no simple expansion phenomenon. At the same time, the existence of at least some expanding groups is the evidence of some kind of explosion processes connected with the birth or with the early stage of evolution of young stellar groups. Here again, the empirical data do not favour the theories of condensation of diffuse matter into the stars.

In the years 1955–1965 I turned my attention to the phenomena in and around the nuclei of galaxies. In the past, astronomers and particularly theoreticians showed little interest in the properties of the nuclei of the galaxies. In a report delivered to the Solvay Conference of 1958 I showed that these nuclei are often centers of large scale activity which proceeds in different forms. I suggested that the ratio galaxies are not the products of collisions of galaxies, as it was accepted at that time, but are systems in which ejections of tremendous scale from the nuclei take place. As a consequence of such ejections, clouds of high energy particles are formed.

The subsequent discovery of quasars added one more form of nuclear activity by which a con­siderable part of liberated energy is emitted as the nonstellar optical radiation of the nucleus. In such cases, the luminosity of the nucleus often exceeds 1011 or 1012 times (sometimes even more) the luminosity of our Sun.

In another important development, the astronomers B. Markarjan, E. Khatchikjan, and others who worked with me at Byurakan Observatory initiated a more systematic observational study of the optical manifestations of activity in galaxies such as the ultra–violet excess and strong emission lines. One of the results of this work was a tenfold increase of the number of known Seyfert galaxies.

At the symposia organized in 1966 at Byurakan Observatory and in 1970 at the Vatican Academy of Sciences, different forms of activity of nuclei including the phenomena in QSO’s and in Seyfert galaxies were thoroughly discussed. Since then, a huge volume of observational work has been carried out. However, the theoretical interptation has made little progress as yet.

While the observed forms of the activity of nuclei speak directly in favor of the fundamental nature of explosion and expansion processes taking place in central parts of galaxies, many the­oreticians are still constructing models of nuclear phenomena in which the ejection processes are pceded by some form of collapse of great amounts of diffuse matter. According to such models, the ejections are only the secondary consequences of more fundamental processes of collapse. It is hardly necessary to say that I am very skeptical about such a speculative mode of thinking. There is no evidence even for the possibility of such a course of events. It seems that such an approach is the remnant of the old notion that the evolutionary processes in the Universe are always going in the direction of contraction and condensation.

Almost all the new interesting discoveries, which were extremely abundant during the last three decades, proved to be great surprises for existing formal theoretical models. Let me mention two cases of complete failure of the speculative approach.

1. The existing theories have completely failed to pdict such an important phenomenon as flare stars. There is no doubt now that the majority of stars after the period of their formation (T– Tauri stage) go through this phase of evolution. Therefore, one of the first tasks of every evolutionary theory must be the explanation of peculiarities of the flare processes.
2. The situation is even worse with fuors (this term I use for the stars of FU Orionis type). The fuor stage is important in the life of at least some categories of stars, and this was rather fatal for many speculative theories. But the things are more serious than it may seem from first glance. It appears now that there is a whole sequence of types, which in their photometric behavior are more or less similar to fuors. The P Cygni star, which brightened almost four centuries ago, is an example. It is well known that in every spiral or irregular galaxy we have many supergiants of P Cygni type. Doubtless, brightenings of p–P Cygni type stars are significant in the evolution of supergiants.

It is natural to try to uncover the secrets of nature by observing the key points where they are hidden. We can hardly reach this aim only by theorizing. We conclude with the statement that observations produce almost innumerable testimonies in favour of ejectious and explosions. At the same time the observations are rather scanty as regards the processes of condensation and collapse. It is not our intention to pronounce an indictment on the ideas connected with condensation process or deny their existence. But in the psent epoch of the life of Observable Universe the opposite phenomena, i.e. expansion and diffusion are responsible for the majority of changes now taking place.

Yerevan, 1993.

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