The greatest activity in the area of evolution of the central stars of the planetary nebulae has occurred in the last two decades, although there have been significant contributions by many workers of considerable insight ever since the early studies at the Lick Observatory demonstrated the physical nature of this class of nebula. We shall try to draw together here the many facets of the present picture of this rapidly developing and highly significant phase of stellar evolution. Unfortunately, the picture is not at all as complete as one would like, and there remain several points of scientific contention; however, we should like to present the results of several investigators. A certain amount of redundancy with previous papers given prior will be necessary, but we shall endeavor to treat these topics briefly and in the context of the evolution of the nebulae and their stars.

The suspected rapid rate of evolution of the central stars of planetary nebulae (Shklovsky, 1956) and the discovery by Seaton (1966) of their evolutionary track in the H-R diagram, have led to renewed interest in these stars. According to Seaton the evolutionary track starts at an atmospheric temperature of about 30000°K and a luminosity of 100 L⊙. A maximum brightness is reached of 25000 L⊙at T∗ = 70000°K. Then T∗ increases to 100000°K, at approximately constant L, whereafter, at this temperature, L decreases finally below 100L⊙. This whole evolution is believed to take place in only 5 × 104 years. At such a rapid evolution, short-period fluctuations in brightness are expected and it should even be possible to detect secular variations in a lifetime. Aller and Liller (1957) discovered the variability of the spectrum of IC 4997 and Vorontsov-Velyaminov (1961) found variations in line intensities of NGC 6905. Khromov (1962) tried to explain the variations in both nebulae as a result of temperature changes of the central stars. Variations in the emission spectrum of IC 4997 were explained by Aller and Liller (1966) as a result of nebular expansion.

The observations of nuclei of three planetary nebulae, NGC 6826, NGC 7662 and IC 4593, have been made as part of a more extensive program with the aim of obtaining their absolute spectrophotometric gradients (ϕ) and photographic magnitudes (from the spectrogram). The spectra were obtained with an 8–12′ Schmidt telescope combined with an objective prism (the dispersion is 420 Å/mm at Hγ). The preliminary data which were obtained at different times, particularly the spectrophotometric gradients of the nuclei of these nebulae, did not agree with each other.

During 1960–67 we have studied the variable star FG Sge=377·1943 Sge, which is located exactly in the centre of an image of the planetary nebula Hz 1–5, as shown on Figure 1. The most significant feature of the central star is the continuous increase of its brightness during the past 70 years, as one can see from Figure 2. During the same time, the temperature of the central star has apparently decreased, although direct information to this effect is available only since 1955. In 1955 the star had spectral type B4, while by 1967 it had become A5. The observational data can be explained by the continuous ejection of gas from the central star, which began at the end of the preceding century. The central star at that time must have had T∗≈63 300°K and Mpg≈ + 1·3, on this model.

The occasional appearance of a red giant or long-period variable in planetary nebulae poses a problem for theoretical astrophysics. Such a cool nuclear star would not ordinarily provide a source of ultraviolet radiation necessary for the excitation of the spectrum of a gaseous nebula.

One possible solution of this problem postulates the existence of intense magnetic fields in the star. Second, the star itself has a structure resembling that of a miniature, highly compressed planetary, with a high-temperature nuclear star at the centre and a distended atmospheric shell enveloping chiefly the stellar equator.

The magnetic field induces a sort of pumping action that creates the tire-shaped envelope from matter ejected near the poles. As this shell grows denser, it radiates like a stellar photosphere at low temperature. Eventually the shell becomes unstable and disperses outward to form and maintain the nebula. A quasi-periodic situation occurs, which explains the variation of light. Ultraviolet light absorbed during the minima, when the shell has vanished, adequately accounts for the nebular excitation. A wide variety of such symbiotic stars occurs, including repeating novae as well as the long-period variables.

The calculations that will be discussed in this paper were undertaken to determine if plausible theoretical arguments could be found to indicate that planetary nebulae represent the stage in the evolution of a star immediately before it becomes a white dwarf. The model for the formation of a planetary nebula that will be described may also be applicable for more massive stars than have been studied. However, since it is known that some population-II stars become planetary nebulae and since there is evidence that evolved population-II stars are less massive than 1 M⊙ (Christy, 1966; Hayashi et al., 1962; Faulkner, 1966; Faulkner and Iben, 1966; Schwarzschild and Härm, 1966) only stars whose masses are less than 1 M⊙ have been studied.

Deep convective envelopes of the red supergiants with masses smaller than 4 M⊙ are found to be dynamically unstable for the luminosities in the range of 4 × 103–5 × 104L⊙. The total energy of the envelopes is positive when the recombination energy of the ionized hydrogen and helium is taken into account, and is sufficient to expel a typical envelope up to infinite distance at a speed of 30 km/sec. We suggest that planetary nebulae are formed in this way.

For various reasons, we have been studying evolution in the pre-white dwarf phase at Rochester. Our attention to the relevance of this work to planetary nebulae came as a result of calculations of the evolution of a one solar-mass iron star carried out at Rochester by S. Vila. Here the neutrino processes drive the peak luminosity to log L/L⊙=4·26, at an effective temperature log Teff=5·53. Although these models are brighter than 100 L⊙ for 500 000 years, they are brighter than 1000 L⊙ for 4000 years, and exceed 10000 L⊙ for 900 years. We are therefore near the luminosity of the planetary-nebula nuclei, but considerably hotter, for a period of the order of the planetary lifetimes. Except for the temperature discrepancy, these models are in rough agreement with the observationally determined evolutionary sequence found by O'Dell (1963) and by Harman and Seaton (1964) and Seaton (1966).

While the physics of some processes which are going in planetary nebulae grows to be one of the most elaborated parts of theoretical astrophysics, the origin of these objects is enigmatic. Ten years ago Shklovsky (1956) made a supposition that the formation of planetary nebulae occurs at the last evolutionary stage of a red giant as a result of the violation of mechanical equilibrium in the envelope of the star. In connection with this the possibility of the separation of an envelope of a red giant under the influence of a shock wave which moves with the velocity of the order of 100 km/sec was considered by Kaplan and Klimishin in 1959. Later on numerical calculations of the motion of a weak shock wave in the envelope of a red giant, were made which showed the basic possibility of formation of a planetary nebula with a mass of the order of 0·5 M⊙ and an initial velocity about 95 km/sec (Sakashita and Tanaka, 1962). The ballistic character of the mechanism of the formation of the planetary nebulae has been recently discussed also by Abell and Goldreich (1966).

Before discussing observational inputs and actual model calculations, I want to give a very elementary review of the relevant parts of stellar evolution theory. We shall only be dealing with stellar masses M below the Chandrasekhar limiting mass Mch(~1·2–1·45 M⊙, depending on chemical composition). The inequality M<Mch implies that relativistic effects are not of overriding importance and I will not mention them further (however, all quantitative model calculations which I will mention later include all the relativistic corrections to the equation of state, including radiation pressure which is also not very important). Let us try to estimate how the central temperature Tc and (total bolometric) luminosity L varies with central density ρc or radius R(ρc~MR−3) for a star of fixed mass.

We have followed the evolution of stellar models of 1 and 1·2 M⊙ from a non-degenerate configuration to the stages where degeneracy becomes important and eventually causes the star to become a white dwarf. By detailed computer calculation using an evolution program kindly lent to us by Schwarzschild and Härm, we have obtained evolution sequences of stars of the mentioned masses with a core composition of 80% O, 10% Ne and 10% Mg and a thin outer envelope of He (10–6 of total mass) under the assumption of energy loss due to photons alone, and also under the assumption of losses due to photons plus neutrinos.

This contribution makes a few points which are relevant to the question of the fraction of white dwarfs formed via the planetary-nebula phenomenon.

It now seems to be reasonably well-established that the central stars of planetary nebulae evolve directly into white dwarfs. Evidently a knowledge of the chemical composition of the white dwarfs would therefore be of considerable importance in helping to identify the point in the evolution at which the mechanism responsible for expulsion of the nebular shell becomes operative. For this reason I would like to present some evidence which provides a direct suggestion for the internal composition of some of the white dwarfs and to examine briefly the implications of this suggestion for the relation between the planetary nuclei and the white dwarfs.

A most puzzling feature of the evolution of the central stars of planetary nebulae is their rapid contraction (O'Dell, 1963; Seaton, 1966); assuming homologous contraction, an enormous amount of gravitational energy is released therefore. For reasons of hydrostatic equilibrium a large part of this energy must be radiated away. As this does not show up in the luminosity of the central stars it must either leave them by some other means – e.g. neutrinos (Vila, 1966) – or it does not originate from the outset; in other words, the actual contraction must deviate strongly from the homologous way. I should like to discuss briefly the second possibility.

Evolutionary sequences of stellar models have been obtained, using the Henyey method, for stars of mass 0·8 M⊙, 0·9 M⊙ and 1·0 M⊙. The starting-point of each sequence is a star with a core of carbon and oxygen, comprising 25% of the star's mass, surrounded by helium-rich material. It is assumed that any remaining hydrogen-rich material has been ejected in the form of the planetary shell. The star contracts gravitationally until helium ignites in a shell at the composition discontinuity, and the subsequent evolution of the shell burning has been followed. No neutrino-loss processes have been considered in this investigation.

When Dr. Seaton asked me to chair the epilogue session of our conference I could not help but wonder if it were to be considered an example of type-casting. He told you that when an Englishman is asked to give a prologue, he immediately begins thinking of quotations from Shakespeare, so I must tell you that when an American is asked to deliver an epilogue, his thoughts turn naturally to Ernest Hemingway. I therefore suggest to you as the theme for our symposium the closing words of the main character of To Have and Have Not, which may for publication here be bowdlerized to read “One man alone ain't no —, — good at all.” Over and over again in the last week we have seen that discussions between astronomers have provided new insights, have opened up new fields for investigation, have revealed unsuspected connections between apparently unrelated problems. I believe this has been a very fruitful conference because there have been so many fruitful discussions among the participants. I shall therefore try to summarize briefly my impressions of the things we have learned and the things we may reasonably hope to learn in the next few years about physical processes in planetary nebulae, and then five of our colleagues will summarize the other sessions of our conference. We invite discussion from all of you after each of these summaries, and also during the general discussion at the close.

The main papers at this symposium dealing with the galactic dynamics and distribution of the planetary nebulae were those of Perek and Cahn, and I shall not attempt to summarize their work again. Rather, I want to draw attention to what I believe are some of the main problems we have to solve in order to make further progress in this field.

I understand that my task is to summarize the present state of radio observation of planetary nebulae, and to give us radio observers our marching orders for the next few years. Where, then, do we stand at the end of this Symposium? Well over 100 planetary nebulae have been looked at by one or more groups at one or more frequencies, and nearly that number have been detected. Most nebulae which have been observed at several frequencies show optically thin spectra, about eight to a wavelength of 40 cm, and three to 70 cm. The fluxes of these nebulae agree well with the values predicted from the observed Hβ radiation, after correction for interstellar extinction. Some are clearly optically thick at long wavelengths, but none have been shown to remain so at the shortest wavelengths. NGC 6853 becomes optically thin near 75 cm, and NGC 6857 may become thin near 5 cm. Eight further nebulae become thin at intermediate wavelengths.

I would probably not be mistaken in saying that for the first time in the practice of solid scientific meetings the problem of the dynamics of planetary nebulae has become a subject of broad and sufficiently many-sided discussion. At the same time, I am inclined to estimate the successes achieved at the present symposium as more than modest. However, there have been no sensations. To the contrary of diminishing the importance and value of the reports made here, I only want to underline the fact that despite its attractiveness, the problem of the dynamics of planetary nebulae remains as before one of the difficult and complicated fields of theoretical and practical astrophysics. It cannot be solved without the cooperation of contiguous branches of science, particularly gas dynamics, magnetohydrodynamics and others. These are new components without which the physics of planetary nebulae has, nevertheless, managed. At the same time, almost all of the physics of planetary nebulae is comprised in the problem of the dynamics. Finally, the problems of the dynamics of planetary nebulae cannot be solved in an approach detached from the problem of the origin of planetary nebulae themselves and also from the origin and development of their nuclei. In this sense, the dynamics of planetary nebulae is becoming to a certain degree a cosmogonic science.