Theoretically it should have been possible to work out all of the foregoing development of the relations between the various components of the physical universe directly from the Fundamental Postulates by mathematical and logical processes without the necessity of checking the results against the actual properties of the existing universe at any stage of the development, and perhaps some one might have had the breadth of vision and the necessary infallibility to have accomplished the task in this manner, but as the work was actually performed each additional point that was established merely set the stage for a limited advance into new territory and a long period of checking against experimental results and reconciling the inevitable discrepancies was almost invariably required before the forward position was sufficiently well consolidated to support a new advance.
As indicated from time to time in the preceding pages there are a number of important physical properties and relationships which had to be omitted from this initial presentation because the detailed analyses of these subjects are still incomplete, and extending onward from the major relations covered in this work there is a never-ending proliferation of subsidiary phenomena. In all of these areas, however, the general nature of the answers is clearly indicated by the principles already developed, and the remaining task is that of working out the details. In another direction we face a different situation. Beyond the frontiers of our present-day knowledge lies an area in which definite correlations with observation and measurement cannot be made because the established facts are too few and their significance is too uncertain. As in the earlier stages of the development of the theories previously outlined, however, we can extend the known principles a reasonable distance into the unknown field with some degree of assurance that the conclusions reached therefrom will be substantially correct in their general aspects, although past experience suggests that accuracy in every detail is unlikely.
We may appropriately begin the theoretical exploration of this field by considering the age-old question as to whether space and time are finite or infinite. The Fundamental Postulates of this work unequivocally support the latter conclusion. There is nothing in these basic assumptions which would establish any kind of a finite limit on either space-time as a whole or space and time individually. Of course, it could be argued that the postulates may be deficient in this respect; that they should perhaps be enlarged to include such limitations. Such an argument, however, is irrelevant. In the preceding discussion it has been shown that a logical and mathematical development of the consequences of the two Fundamental Postulates correctly reproduces the existing universe insofar as it is accessible to observation. We are now attempting to determine what further information these same principles can give us if we make the plausible assumption that they are valid in the unknown regions of the universe as well as in the accessible regions, an assumption which is specifically included in the Postulates as stated. For this purpose it is essential that we maintain the principles in exactly the same form in which they were established as valid in the known region; if we alter them in any way we are no longer examining the effect of extending the range of application of principles of established validity, we are dealing with unsupported hypotheses. It is perfectly in order to make hypotheses and to determine the consequences thereof, but that does not accomplish the objective of this particular investigation.
An important point in connection with this question as to the existence of limitations on space and time is that on the basis of the Fundamental Postulates zero and infinity have equal standing. Zero space is equivalent to infinite time and so on. The concept of zero is much easier for the human mind to accept that that of infinity, but when we postulate space and time as reciprocals the two concepts become one, so far as space-time and its derivatives are concerned, and we can no longer accept one and reject the other.
In addition to defining the physical universe as infinite, the Fundamental Postulates also define it as changeless, when considered as a whole. The myriad of subsidiary phenomena resulting from space and time displacements are, of course, constantly changing but the effect of the reciprocal postulate in combination with the probability postulate precludes any net change in the universe as a whole. There is no mechanism defined by the postulates whereby displacements can be created or extinguished and the total displacement therefore remains constant. Furthermore, the displacements in each direction from the neutral axis must stay in balance, since the two forms of a reciprocal expression are identical from a probability standpoint. It is, in fact, impossible to state which is the original expression and which is the reciprocal.
These conclusions reached from the Fundamental Postulates are in agreement with the so-called “perfect cosmological principle,” which states that the universe has essentially the same aspect from any point in space or any point in time. The validity of this principle so far as space is concerned has been fairly well established by astronomical observations. It is now possible to see far enough into space to eliminate the effect of local irregularities and to confirm the homogeneous nature of the universe from a space standpoint. At the observational limits we are seeing as far into time as into space, but not all observers are convinced that the cosmological principle is applicable in time, because there are so many physical processes that appear to be irreversible. We are accustomed to thinking of an “arrow of time” pointing in a fixed direction and such processes as the observed expansion of the universe and the continual increase in the entropy of the material system seem to confirm the one-way nature of the temporal processes, so that there are formidable obstacles in the way of accepting any conflicting ideas.
In this work we deduce from the Fundamental Postulates that the arrow of time does indeed point in a fixed direction in our part of the universe. The galaxies are actually receding from each other, the general processes of growth and decay are irreversible, and so on. But the Postulates also tell us that we see only half of what is happening. They require the existence of another half of the universe: a non-material sector which is in all respects the inverse of the material sector which we recognize. In that other half of the universe the arrow of time points in the opposite direction and all of the effects of the unidirectional progression of time in our material region are completely nullified in the long run by the oppositely directed progression in the non-material region.
The expansion of the material galaxies carries all of the matter in the universe outward toward infinite space. If this were the only process of its kind the common “explosion” theories of cosmology would have a very strong case, but we find from the Fundamental Postulates that there is a co-existing system of non-material galaxies, equal in all respects to the material system, which is likewise expanding and carrying all of its constituent parts outward toward infinite time. While the material half of the universe moves toward infinite space the non-material half moves toward zero space (infinite time) at the same rate and the net effect on the system as a whole is zero. In order to maintain the constant relationships within the two halves of the system it is, of course, necessary that some conversion process be operating as an interchange between the two. The nature of this process will be examined later.
Because of the permanence of the universe in its general aspects, all major physical processes are necessarily cyclic in character. Where some unidirectional process, such as the increase in entropy required by the Second Law of Thermodynamics, is effective in one area it represents only one phase of the cycle, and in some other area there must be an oppositely directed process which keeps the net balance unchanged. The “heat death” envisioned by the Second Law has no place in the universe defined by the Fundamental Postulates. Instead of a universe that is continually running down and will ultimately reach a dead level of uniformity in which there is no activity at all, the Fundamental Postulates lead to a universe which is forever changing in detail but will always remain the same as a whole. This is a universe of motion, and motion continually alters the relationships of the subsidiary units. It is a universe of mathematical law, and the mathematics of probability lead to a never-ending conflict between individual probability and group probability. The most probable state for the individual is the average. The most probable state for the group is a condition in which there are individual deviations from the average. Each individual tends toward the most probable value, the average, but is continually driven away from that average by the tendency of the group to conform to a probability distribution of individual values.
Let us now examine some of the more specific problems. Since the stars are the most prominent actors on the astronomical stage, where the drama of the universe is enacted, it is appropriate to begin with the question of the path of stellar evolution. We have already deduced from the Fundamental Postulates that all basic natural processes such as this are cyclic in character and we may therefore start our consideration at any phase of the cycle. For convenience we will select a starting point somewhere on the main sequence. Whether the stars move up or down the main sequence in their evolutionary course is not clear from observation since we have only what amounts to an instantaneous picture, and we must therefore resort to theoretical consideration. It has been established both theoretically and from observation that stellar temperature is a function of mass, and since this is a rather obvious result of generating energy by processes which are proportional to the cube of the diameter (the total mass) and dissipating it by processes which are proportional to the square of the diameter (the surface area) no detailed discussion of this point would seem necessary. If the existence of the stars is to be regarded as primarily devoted to expending their substance in producing radiation to be dissipated into the depths of space, there can be no escape from the conclusion that they were originally hot and massive units and are gradually moving down or off the main sequence toward eventual extinction. But in order to meet the cyclic requirement it would then be necessary to find some process whereby cold dwarf stars are reconverted into hot massive stars, and there is no apparent foundation on which any such process could be based.
In recent years astronomers have begun to appreciate that a downward course is not the only possibility, and it is now generally agreed that the stars within dense dust clouds are acquiring enough material by accretion from the surroundings to more than compensate for the loss of matter by radiation and are actually growing hotter and more massive. We thus recognize that the direction of evolution along the main sequence is not necessarily downward as formerly believed; the net movement is the resultant of two opposing factors, the loss of mass or its equivalent by radiation and the gain in mass due to accretion. The conclusions of this present work are that the amount of interstellar matter and potential matter is considerably greater than has heretofore been realized and that there is a substantial accretion even where nothing more than the general interstellar haze is present. Furthermore, the radiation losses are reduced very sharply as the temperature falls, since they vary as the fourth power of the temperature. It therefore appears that even in the regions where the accretion of matter is at a minimum, a star does not cool down indefinitely; it merely moves down the main sequence to an equilibrium point and remains there until it enters a denser zone. In the regions where the accretion is normal or above normal the star moves up the main sequence, becoming hotter and more massive.
The production of energy to take care of radiation losses and to cause the rise in temperature which is an essential feature of this evolutionary course is initially due to certain processes, to be discussed later, which are a direct result of the manner in which the star is formed. As the temperature rise continues, a point is ultimately reached which represents the destructive thermal limit for the heaviest element present One of the magnetic displacement units of this element is then destroyed in the manner previously described and the rotational motion is converted into energy. The amount of energy thus released is very large and this process makes a practically unlimited source of energy available to the main sequence stars. There is a small proportion of heavy elements in the stars as originally constituted, and a similar proportion in the material acquired by accretion from the surroundings. Inasmuch as the entire stellar structure is fluid, the heavy elements necessarily make their way to the center. Here they reach the destructive thermal limits, are converted into energy, and replenish the stellar energy supply which is constantly being depleted by radiation.
As the mass increases and the temperature rises, successively lighter elements are made available as stellar fuel. Since none of the heavy elements is present in more than a relatively minute quantity in a region of minimum accretion, the availability of an additional fuel supply due to the attainment of the destructive limit of one more element is not normally sufficient to cause any significant change in the energy balance of the star. The stars of the upper portion of the main sequence are subject to somewhat higher rates of accretion but they are able to absorb greater heat fluctuations, for reasons which will be developed later, and the main sequence stars are therefore relatively quiet and unspectacular as they gradually increase in mass and temperature and move upward along their evolutionary path.
When the temperature corresponding to the destructive limit of the iron-nickel group of elements is reached, a totally different situation prevails. These elements are not limited to small amounts; they are present in concentrations which represent an appreciable fraction of the total stellar mass. The sudden arrival of this large quantity of material at the destructive limit activates a potential source of far more energy than the star is able to dissipate through the normal radiation mechanism. The initial release of energy from this source therefore blows the whole star apart in a tremendous explosion. Because of the relatively large concentration of the nickel-iron elements in the central core of the star the explosion takes place as soon as the first portions of this material are converted into energy and the remainder is dispersed by the explosion-generated velocities. This carryover of material from one cycle to the next enables the iron group elements to continue building up as the over-all age of the system increases, whereas the heavier elements have to start all over again after each explosion.
This sequence of events is, of course, purely theoretical, but it is the result of a straightforward application of the principles developed from the Fundamental Postulates, and where not actually corroborated by observation it is at least consistent with the observational data. Some observers will no doubt contest the assertion that there is sufficient accretion of mass to cause the upward progression along the main sequence which is required by theory. It is evident, however, that any conclusion on this score based entirely on the results of observation cannot be more than an opinion, in the existing state of knowledge. The existence of some accretion of mass is incontestable; the only open question concerns the quantities. In this connection it is probably significant that within very recent years general astronomical opinion has moved a long way in the direction of recognizing the importance of interstellar dust and gas; from a concept of interstellar space as essentially empty to a realization of the fact that the total amount of interstellar matter is at least comparable to the amount of matter concentrated in the stars.
The chemical composition of the stars and the distribution of elements in the stellar interiors are also debatable subjects, but again the deductions that have been made from the previously established principles do not conflict with the actual observations; they merely conflict with some interpretations of these observations. While the gravitational segregation of the stellar material which puts a relatively high concentration of the heavier elements into the central core is not entirely in agreement with current astronomical thought, it should be emphasized that such a segregation is the normal result in a fluid medium subject to gravitational forces and a theory which requires the existence of normal conditions is never out of order where the true situation is unknown.
Furthermore, even though these conclusions which have been reached as to the amount of iron and heavier elements present in the stellar interiors are beyond the possibility of direct verification, it will be brought out in the subsequent discussion of the solar system that some strong evidence as to the internal constitution of the stars can be obtained from collateral sources. The spectroscopic information from the stars is only of limited value since these data only tell us what conditions prevail in the outer regions. Even from this restricted standpoint the evidence may actually be misleading since it is more than likely that the spectroscopic results are affected to a significant degree by the character of the material currently being picked up throng the accretion process. The observed differences in the stellar spectra that can be attributed to variations in chemical composition are probably more indicative of the environments through which the stars happen to be moving at the moment than of the average composition of the stars themselves. The presence of substantial amounts of elements’ such as technetium, for example, in the outer regions of some stars presents a formidable problem if we are to regard this as an actual indication of the composition of the stars, but it is easily explained on the basis that the technetium has been derived from captured material and is on its way down to the central regions where it will add to the fuel supply. This element is stable wherever the magnetic ionization level is zero, as it usually is in the inter-stellar dust clouds, and relatively heavy concentrations could conceivably be produced in special areas which are left undisturbed for long periods of time.
Growing recognition of the importance of the capture of inter-stellar material has already begun to make an impression on astronomical thought. One of the current theories of the sun’s corona, for instance, is the “infall” theory, which attributes the corona to gas and dust particles being pulled in from the surroundings by the gravitational attraction of the sun. Similarly, the irregular fluctuations of the so-called “nebular variables” are explained as a result of variations in the rate of capture and digestion of material from the relatively dense dust clouds with which these stars are associated. Both of these theories are entirely consistent with the conclusions of this work.
The explosion which theoretically occurs at the destructive limit of the nickel-iron elements is consistent with observation as it can be identified with the observed phenomenon known as a supernova and the theoretical products of the explosion can be correlated with the observed residue of the supernova. So far as can be determined from the information now available the star that becomes a supernova is a hot, massive unit before the explosion, which agrees with the theoretical deduction that such an explosion occurs when a star reaches the upper end of the main sequence. As has been indicated, only a relatively small proportion of the mass of the star needs to be converted into energy in order to produce the explosion and the remainder, constituting the bulk of the original mass, is blown away from the original location at extremely high velocities. We therefore find the site of a relatively recent supernova explosion surrounded by a cloud of material moving rapidly outward. The Crab Nebula, which has been identified as the product of a supernova observed in 1054 A.D., is the typical example.
Inasmuch as this expansion takes place against the force of gravity and against some resistance from the inter-stellar material, it cannot continue indefinitely and at some time in the distant future the expansion of the Crab Nebula will cease. At this stage it will be merely a cloud of cold and very diffuse material occupying a tremendous expanse of space. Gravitational attraction between the particles will be small because of the huge distances involved, but nevertheless it will exist and once the expansion has ceased, a contraction will be initiated by the force of gravity. Another long interval must pass while this minute force does its work but ultimately the constituent particles will be pulled back to where the interior temperature of the mass can rise enough to produce radiation within the visible range and the star will have been reborn.
It is not to be expected that exactly the same mass will necessarily be reassembled into the reconstructed star. The force of the supernova explosion will no doubt give some fragments sufficient velocity to enable them to escape from the gravitational attraction of the remainder, but the cloud will be moving through interstellar matter and accretions from this source will more than offset the losses of the original material. The interstellar matter will also help to minimize the losses, as a part of the translatory velocity of the particles will be dissipated in collisions with this material. In the long interval that elapses between the explosion of the star and the birth of its successor the cloud of matter may also be altered substantially by encounters with other pre-stellar objects, but this does not change the nature of the final result. Eventually the diffuse cloud, whether modified or unmodified, will again condense into a star or be absorbed into existing stars.
At the stage when it first becomes visible a star is still extremely diffuse; in fact it has been said that such a star is nothing more than a red hot vacuum. But the work of gravity is not yet complete. The star still continues to contract, and as it does so it moves toward the main sequence, which we may regard simply as the location at which the gravitational forces are in equilibrium with the forces resisting further contraction. The evolutionary path of any star which has not yet reached the main sequence is thus determined by two separate factors: it is moving toward the main sequence to attain gravitational equilibrium and at the same time it is moving parallel to the main sequence to attain thermal equilibrium.
The contraction process due to the gravitational forces transforms potential energy into kinetic energy and is therefore one of the stellar energy sources during the period in which it is operative. The other major energy source in this early stage of the evolutionary cycle is radioactivity. Inasmuch as a large part of the matter which is assembled into the new star has been obtained by capture from the interstellar material, the magnetic temperature is lower than that of the exploding star and the ’unit magnetic ionization level is not regained until after the condensation into the new star is quite well advanced. We may, in fact, regard the attainment of the unit ionization level as the event which marks the dividing line between dust cloud and star, since the
immediate result of the ionization process is to make the heaviest elements radioactive, thereby activating a source of thermal energy within the cloud and causing the increase in temperature which distinguishes the infant star from a mere cloud of particles.
These newly formed stars, the red giants, are in the upper right section of the Hertzspring-Russell diagram, Figure 42. Let us assume the existence of such a star at the point marked A. This star is more luminous than a main sequence star of the same surface temperature because it is radiating from a very much larger surface. As the gravitational contraction proceeds this extended surface becomes smaller and the emission decreases accordingly, moving the star downward on the H-R diagram. At the same time, however, the contraction and other energy producing processes are increasing the stellar temperature and the star therefore moves toward the left of the diagram as well as downward. If the star is located in a region where the density of the interstellar material is relatively low the downward movement predominates and the evolutionary path is a curve such as AB. On reaching the vicinity of the main sequence in the neighborhood of point B the direction of further movement is determined by the location of the point C or C’ at which the star will reach thermal equilibrium on the main sequence. If the star is formed in a high density region or enters such a region during the early evolutionary stages, the rate of accretion may be rapid enough to relegate the attainment of gravitational equilibrium to a subordinate role and to move the star almost directly toward its ultimate destination at the upper end of the main sequence along a line such as AD. In either event we have now traced a full cycle and we are back to the main sequence where we began our examination of the evolutionary process.
Let us now return to a further consideration of the supernova explosion. At the very high temperatures prevailing in the interiors of the stars at the upper end of the main sequence the thermal velocities are approaching the unit level, and when these already high temperatures are still further increased by the processes which lead to the break-up of the star the velocities of many of the interior atoms rise above unity. When the explosion occurs the inner atoms are blown apart in time by these greater-than-unit velocities in the same manner as the outer atoms are blown apart in space by less-than-unit velocities to form the diffuse clouds of matter which eventually coalesce into the giant stars. In both cases the atoms which were in close contact in the hot massive star are widely separated in the product of the explosion, but in this second product the separation is in time rather than in space.
This does not change either the mass or the volumetric characteristics of the atoms of matter. But when we measure the density, m/V, of the giant stars we include in V, because of our method of measurement, not only the actual equilibrium volume of the atoms but also the empty three-dimensional space between them, and the density of the star calculated on this basis is something of a totally different order from the actual density of the matter of which it is composed. Similarly, where the atoms are separated by time rather than by space the volume obtained by our methods of measurement includes the effect of the empty three-dimensional time between the atoms, which reduces the equivalent space (the apparent volume), and again the density calculated in the usual manner has no resemblance to the actual density of the stellar material. In the giant stars the empty space between the atoms decreases the measured density by a factor which may be as high as 105, or 106. The time separation produces a similar effect in the opposite direction and the second product of the explosion is therefore an object of small apparent volume but extremely high apparent density: a white dwarf star.
When judged by terrestrial standards the calculated densities of these white dwarfs are nothing less than fantastic and the calculations were originally accepted with considerable reluctance and only after all conceivable alternatives had been ruled out for one reason or another. The indicated density of Sirius B, for instance, is in the neighborhood of 68,000 g/cm3 and other stars of the same type apparently have still greater densities. In the light of the relationships developed in this work, however, it is clear that this very high density is no more out of line than the very low density of the giant stars; each of these phenomena is simply the reverse of the other.
Gravitational forces in the white dwarf stars tend to draw the constituent atoms closer together in time just as the same forces in the giants tend to draw the atoms closer together in space. The white dwarfs therefore decrease in apparent density as they contract and they become more and more like the giants which are approaching the normal from the other side. This means that on the H-R diagram the white dwarfs are also moving toward the main sequence and once having attained that location, the volumetric normal, the giants and the white dwarfs are indistinguishable from the standpoint of the variables portrayed in the diagram. There is a very marked difference in composition since the white dwarfs were formed from the material in the center regions of the exploding star, whereas the giants were formed from the lighter material in the outer regions. We will consider this point in some detail shortly.
In Figure 42 the zone of formation of the white dwarfs is at the lower left of the diagram, directly opposite the zone of formation of the red giants. In this area the luminosity is low because the equivalent surface area is small, but the temperature is high because the thermal energy is concentrated in a relatively small equivalent volume. The normal evolutionary path is XY, the inverse of the normal path of the giant stars. As the star contracts in time, increasing the equivalent volume, the temperature drops accordingly but the luminosity increases because of the greater surface area available for radiation. In a region where the accretion rate is high the drop in temperature is minimized and the movement on the H-R diagram is nearly vertical, along a line such as XZ.
The general features of the binary and multiple star systems are readily explained by this dual evolutionary cycle. The seemingly incongruous associations of stars of very different types are seen to be perfectly normal developments. Combinations of giant and dwarf stars are not freaks or accidents; they are the natural initial products of the star formation process. As will find later when we examine the quantitative relations, the vast distances which we observe between the star systems are a permanent feature of the stellar distribution and there is no interaction between systems other than the escape of some diffuse material from one region to another. Every system that has been through the explosion process therefore contains two components: an A component on or above the main sequence and a B component on or below the main sequence. Since the evolutionary path for both components is first toward the main sequence and then up along that line there are no associations of dissimilar stars in the upper (more advanced) portions of the main sequence. Many of these stars are binaries, but they are pairs of stars of the same or closely related types.
In the earlier stages the pairing varies with the evolutionary age of the system. Immediately after the explosion the A component is merely a cloud of dust and gas which appears as a nebulosity surrounding the white dwarf B component. Later the cloud develops into a pre-stellar aggregate and then into a giant infra-red star, and since these aggregates are invisible the white dwarf appears to be alone during this phase. When the giant gets into the high luminosity range this situation is likely to be reversed as this bright star then overpowers its relatively faint companion. Further progress finally brings the giant down to the main sequence. The development of the white dwarf is usually slower and there is normally a stage in which a main sequence star is paired with a white dwarf, as in Sirius and Procyon, before the mature status as a pair of main sequence stars is attained by the system’ It is true that some of the double stars which have been reported by observers do not fit into the evolutionary picture. For example, Capella is said to be a pair of giants. Neither of these stars can qualify as the B component of a binary, hence on the basis of the theory that has been developed herein we must conclude that Capella is actually a multiple system rather than a double star and that it has two unseen white dwarf or faint main sequence components. The Algol type stars in which a main sequence star is accompanied by a sub-giant are similarly indicated as multiple systems, and in Algol itself at least one and possibly both of the theoretical B components have been identified. Further consideration will be given to the multiple systems when we take up a consideration of the different stellar populations.
In the earlier discussion of the stellar energy generation process it was pointed out that the increase in energy output resulting from the attainment of the destructive limit of one additional element is not normally sufficient to disturb the energy equilibrium within a star which is located in a region of minimum accretion. Detailed calculation of the various factors involved in this energy balance is outside the scope of this work, but it is evident without calculation that at some point there is a minimum below which the thermal equilibrium will not be affected enough to cause any noticeable irregularity. Since the stars which follow the gravitational path AB are observed to be stable it can be deduced that the variations in energy release in these stars are below this minimum. It is also apparent, without the necessity of numerical calculation, that complete stability and a violent explosion are not the only alternatives when a new destructive limit is reached. An intermediate possibility is that the sudden release of additional energy from this source may be sufficient to produce a substantial change in the physical condition of the star without being adequate to blow it apart. We can determine the qualitative effects of such an energy release and when we find that these effects are actually recognizable in certain classes of stars which should theoretically be subject to greater rates of energy release than the stars on the gravitational path AB, it is in order to conclude that the observed effects are due to this cause.
The stars which can be expected to show effects of this kind are those whose normal supply of fuel in the form of heavy elements is being augmented by a relatively heavy inflow from the surroundings: the stars which are following the evolutionary path AD. Let us examine the result of reaching a new thermal destructive limit in one of these stars. Since the star is unable to dissipate the additional output of energy by the normal heat transmission processes, the suddenly released excess heat will cause a rapid expansion. After the expansion has accomplished its purpose inertia carries it beyond the equilibrium point and this cools the interior of the star, which in turn drops the temperature in the central regions below the recently attained destructive limit and shuts off the extra supply of energy, accentuating the cooling effect. Ultimately the cooling causes a contraction of the star, whereupon the temperature again rises, the destructive limit is once more reached, and the whole process is repeated. The evolution of a star along the path AD, where it experiences a substantial accretion of mass from the surroundings, is therefore likely to be characterized by a pulsation. Such a star is classified as an intrinsic variable.
The length of the cycle or period of the variable star depends on the time required to restore the original conditions after the expansion takes place. Since the initial excess production of thermal energy which causes the expansion varies much less than the stellar temperature, the initial conditions are restored more rapidly in the hotter stars, and the period is therefore an inverse function of the temperature. The relatively new stars just entering the pulsation zone are long period variables, with periods ranging from 100 days to several years. More advanced stars with shorter periods that extend down to minutes are classified as Cepheids. Various subdivisions of both the Cepheid and long period classes are recognized, and there are also some other less common and less distinctive types of variables in the remaining sectors of the high density region.
Within a group of stars of the same temperature the period depends on the stellar volume, since the reaction of a more extended volume to any specific force of compression or expansion proceeds more slowly. Inasmuch as luminosity is a function of surface temperature and surface area, this means that the more luminous stars have the longer periods: the celebrated period-luminosity relation. The results of this present investigation suggest that this relation does not have the degree of accuracy in application to the entire Cepheid population that is usually assumed, since it is affected by both the stellar temperature and the rate of accretion, but it is approximately correct over a wide range of temperature and has therefore been a very valuable astronomical tool. Its deficiencies show up conspicuously at the two extremes; it is not applicable to the long period variables, and it has to be modified in application to the very short period cluster variables.
The region of the H-R diagram occupied by the variables is the triangular area between the gravitational path AB and the main sequence. The great majority of the stars in this zone are intrinsic variables; some observers even say that they are all variables. On the right of the variable region the irregularities in the rate of release of energy are too small to produce pulsation; along the main sequence the response of the system is too rapid and the period is negligible. As would be expected from the nature of the process which is responsible for the variability, the most prominent classes of variable stars are found in certain definite locations within this zone of instability. Each of these locations undoubtedly represents a stage at which the interior temperatures of the stars reach the destructive limit of an element or group of elements which is present in a higher concentration than the average heavy element. In Figure 43 we see that the region of the “classical” Cepheids, the best-known of the intrinsic variable stars, is a relatively narrow band running diagonally upward from left to right in the low temperature zone of the region of variability. The RR Lyrae stars, or cluster variables, the principal class of variable stars in Population II, are located on a downward extension of this band into the region of less luminosity and shorter period.
Inasmuch as the central temperature of a larger and more luminous star is higher than that of a smaller and less luminous star of the same surface temperature, it is apparent that the diagonal Cepheid band represents a zone of approximately equal central temperatures. The particular elements whose destructive limits are reached at this temperature cannot be positively identified without further investigation, but since the lead-mercury group is not only the first group of moderately abundant elements in the descending order of atomic mass but also the only such group in the upper half of the atomic series, we may at least tentatively correlate the destructive thermal limits of these elements 80 to 82 with the central temperature corresponding to the Cepheid band. It should be noted in this connection that lead is the heaviest element that is stable against radioactivity in a region of unit magnetic ionization and it therefore occupies a preferred position somewhat similar to that of iron.
The long period variables can be correlated with the elements above lead in the atomic series. Here the quantities of excess energy are smaller since these elements are relatively scarce, but each increment of energy has a greater effect on the stellar equilibrium because of the smaller heat storage capacity of these low temperature stars. This situation accentuates the effect of minor variations in the incoming flow of matter from the environment and as a result these long period variables are less regular than the Cepheids. On the other side of the Cepheid zone these relations are reversed. Because of the higher temperature and greater mass the heat storage capacity of each star is much greater and any variations, either in the rate of accretion of matter or in the abundance of the elements whose destructive limit is reached, are to a large extent smoothed out. In general, therefore, these stars are not separable into easily recognized groups on the order of the Cepheids.
Let us now turn to the opposite side of the main sequence. When we examine the stability situation in this area we find some important differences. The gravitational forces in the white dwarf stars are inverse; that is, they operate to move the atoms closer together in time rather than in space. At the location where these gravitational forces are the strongest, the center of the star, the compression in time is the greatest, and since compression in time is equivalent to expansion in space the center of a white dwarf star is the region of lowest density. The expansion due to the generation of thermal energy within these stars does not oppose the effect of gravitational compression as in the giant stars; it merely adds to the gravitational effects. The conflict of forces which is responsible for the pulsation effects in the giants is therefore absent in the white dwarfs.
Ultimately, however, the continued expansion in the interior of the white dwarf star eliminates the empty time between the atoms in this region and the thermal forces begin to build up a gas pressure. When this pressure is high enough the compressed gas breaks through the overlying material in the manner of a gas bubble forcing its way through a liquid, and the hot material makes its appearance at the surface of the star, increasing the luminosity by a factor which may be as high as 50,000. Within a short time the relatively small amount of ejected material cools by radiation and the star gradually returns to its original status. In this condition it is rather inconspicuous and the first observed events of this kind were thought to involve the formation of entirely new stars, as a result of which the inappropriate term nova has been applied to this phenomenon.
From the foregoing description it is apparent that the nova explosion is another periodic event. As soon as one gas bubble is ejected, the compressive and thermal forces in the interior begin working toward development of a successor. Since the gravitational forces within the star are gradually expanding it toward the gravitational normal represented by the main sequence (that is, they are drawing the constituent atoms closer together in time), the additional expansion required to cause the nova explosion is correspondingly reduced as the star grows older and this reduces the time interval between explosions. The first event of this kind may not occur for millions of years after the original formation of the white dwarf star, but as the star approaches closer to the main sequence the time interval decreases, and some novae have repeated in less than 100 years. Furthermore, there is a special kind of variable star which has all of the earmarks of a small scale nova. This stellar class, of which U Geminorum is the type star, follows the nova pattern in miniature with a very much shorter period, ranging from about a year downward. The U Geminorum stars are reported to be slightly under-luminous for their spectral type; that is, they are somewhat below the main sequence on the H-R diagram, which is just where they belong if they are nearing the end of the white dwarf stage. The long period novae lie still farther down on the H-R diagram and are reported to have densities in the neighborhood of 100 times the solar density. From this it would appear that such stars as Sirius B are still in the early white dwarf stage and have a long way to go before they reach the nova phase.
It is neither feasible nor appropriate to discuss all of the variations in stellar behavior in a general work of this kind, but some comments on the stars with extended atmospheres are in order since these stars furnish some additional information regarding the white dwarf branch of the evolutionary cycle. On the giant side of the main sequence the succession of events from supernova to red giant star is simple: there is first an expansion due to the translational velocity imparted to the stellar material by the explosion, and then a contraction due to the force of gravity. On the white dwarf side a similar process takes place, but since the expansion in this case is in time the entire action takes place in one small region of space and there are collateral effects in the surrounding space that have no parallel on the giant side of the main sequence.
When the explosion first occurs the density of the material expelled from the star is great enough to carry everything in the vicinity along with it, and we see only a rapidly expanding cloud of material such as that which constitutes the Crab Nebula. At this stage the inward-moving component is almost invisible as the radiation which it emits is mostly at extremely short wavelengths, and while the total amount is large because of the very high temperature the emission within the visible range is small. As the expansion progresses the density of the expanding cloud decreases and eventually the point is reached where it passes through the interstellar material rather than carrying that material with it. The interstellar gas and dust then resumes the gravitational flow toward the central star that was interrupted by the supernova explosion. The first material of this kind arriving at the surface of the star finds that surface at an extremely high temperature (calculations indicate temperatures on the order of 500,000° K) and the incoming material is heated to such a degree that it is ejected back into the surroundings. Since both the incoming and outgoing material are at a very low density one flow does not interfere with the other to any serious extent and the cold material continues to flow inward through the outward moving hot material.
The result of this process is a planetary nebula in which a central star of the white dwarf type is surrounded by a large expanding shell of very diffuse matter. As time goes on the surface of the central star gradually cools due to radiation and transfer of heat to the ejected material. Ultimately a point is reached at which the star is able to retain the incoming material and output to the nebula ceases. The shell then continues to expand and cool until it finally merges with the general interstellar medium, while the central star assumes the status of an ordinary white dwarf. From this description it can be seen that the planetaries are short-lived objects, in the astronomical sense, and the only reason why several hundred of them can be observed in our galaxy is that they occupy a definite place in the stellar evolutionary cycle and are therefore produced at a steady rate. It does not necessarily follow, however, that every white dwarf passes through the planetary stage. If the rate of expansion of the explosion products is slower, or if the rate of cooling of the outer surface of the white dwarf is faster, or if the density of the interstellar medium in the vicinity is less, the conditions which lead to the formation of the nebular shell either may not develop at all or may only result in the production of a light and transient nebulosity.
Similar ejection of material on a smaller scale is quite common in various classes of hot stars, and there are a great variety of stars with extended atmospheres which have apparently been produced by a process of this kind. Whether or not the ejection process in these stars is exclusively thermal is not yet certain but the high temperature is at least a major factor and practically all of these stars are in the very hot spectral classes O and B. An interesting group of this kind is the Wolf-Rayet class of stars. The outer regions of these stars are in a state of violent agitation and it is difficult to make accurate observations, as a result of which there is considerable difference of opinion as to the actual conditions, but the most general conclusion is that they are hot massive stars which are continuously ejecting matter. On this basis they are assigned to the spectral class W, which is above class O or at least on a level with the upper portion of class O.
The possibility has been suggested that the continuous ejection of mass by these stars may be an alternate and more peaceful method of eliminating excess mass when any kind of a stellar limit is reached. Such an explanation, however, is open to the objection that a process of this kind could not reduce the mass appreciably below the stability limit and any further accretion from the environment would promptly put the star back into the unstable condition. On this basis the Wolf-Rayet status once attained would be essentially permanent and the number of these stars in the older structures should be very large, which does not agree with observation. The general explanation of the ejection of material from hot stellar surfaces as developed in the foregoing discussion indicates that the Wolf-Rayet stars are simply those stars at the upper end of the main sequence which are near the maximum with respect to both of the variables which determine the amount of material ejected: the surface temperature and the rate of accretion from the surroundings. In other words, this class of star is a special type of incipient supernova. Some of the central stars of planetary nebulae are currently being classed as Wolf-Rayets but this is not a logical grouping as it combines stars of different evolutionary stages and widely different characteristics. The two types are quite similar in their ejection phenomena but the resemblance stops at this point. In almost all other respects the properties of these stars are widely divergent.
According to the foregoing theory the local star system, the group of stars in the immediate vicinity of the sun, should be composed principally of binary stars, if most of these stars are in the same age bracket, as the available evidence would indicate. A large number have actually been identified as binaries. Most of these recognized systems have main sequence stars in both positions but there are a few main sequence-white dwarf combinations. No giant-white dwarf systems are visible but this is probably due to the effect of the time factor on the number of stars in each part of the cycle, as the giant stage of stellar evolution is of short duration compared to the time spent in the pre-stellar and main sequence phases. It should be noted in this connection that this local system is representative only of a particular evolutionary stage; not of stellar systems in general, and the proportions in which the various types of stars occur in this local system are not indicative of the composition of the stellar population as a whole. The white dwarf, for instance, is an explosion product, a star of the second or later generation, and such stars are totally absent from the stellar systems which are composed of first generation stars: those which have not yet passed through the explosion phase of the cycle. It should not be assumed, therefore, that the high proportion of white dwarfs in the local system indicates a similar high proportion throughout space or even throughout the Galaxy.
In addition to the binaries we also observe a considerable number of stars in the local system which appear to be single. Some of these may actually be single stars which have drifted in from younger systems, but we have already noted that the A component of a double star is invisible during a portion of the early evolutionary stage and all we see under these conditions is a lone white dwarf. The white dwarfs are not dispersed in space and they do not participate in this retreat into obscurity, but they may become invisible for another reason: they may be too small to maintain the temperature required for radiation in the visible range. Inasmuch as velocities less than unity are normal in the material sector of the universe a greater proportion of the mass of the parent star is normally dispersed in space (by velocities below unity) than in time (by velocities above unity). If substantially the same amount of material is reassembled into a binary star system the giant member will have the greater mass. In Sirius, for example, the main sequence star, originally the giant, has more than twice the mass of the dwarf. A less violent explosion would result in a still smaller dwarf mass and it is not improbable that in many instances the mass of the dwarf component is below the minimum requirement for a star, in which case the final product is a single star with one or more relatively small and cool attendants: a planetary system.
Since this question of the origin of a planetary system is of considerable interest to the inhabitants of a planet, it will be desirable to examine the theoretical processes leading to the formation of such a system in more detail. When the supernova explosion occurs the material near the center of the star is obviously the part which acquires greater-than-unit velocity and disperses in time. The remainder of the stellar material is dispersed outward into space. In view of the segregation of heavy and light components which necessarily takes place in a fluid aggregate under the influence of gravitational forces the chemical composition of the two components must differ widely. Most of the lighter elements will have been concentrated in the outer portions of the star before the explosion, those heavier than the nickel-iron group will have been converted to energy, except for the stray atoms mixed in with other material, and the central portions of the star will contain a high concentration of the iron group elements. When the explosion occurs the outward moving material, which we may call Substance A, consists mainly of light elements with only a relatively small proportion of high density matter. Substance B, the inward-moving component, consists primarily of the iron group elements with some admixed lighter material.
In each of the two products of the stellar explosion the primary gravitational forces are directed radially toward the center of mass of the dispersed material. Secondary forces can be expected to develop by reason of local aggregation, but each aggregation as a whole is subject to the radial forces. Unless outside agencies intervene it is to be expected that any capture of one subsidiary aggregate by another will result in consolidation, the formation of a binary system being ruled out by the absence of non-radial motions. Ultimately the greater part of the matter in each of the two components will be collected into one unit. The two separate components then acquire orbital motion around each other, consolidation being unlikely in this case as neither unit will be moving directly toward the other unless by pure chance. The ultimate result is a system in which a mass composed principally of Substance B is moving in an orbit around a central star of Substance A. If the B component is of stellar size the system is a binary star; if it is smaller the product is a planetary system. Where interaction occurs before the consolidation process is complete some of the unconsolidated fragments may take up independent orbital positions in the final system, constituting additional planets or planetary satellites.
On this basis we may conclude that at the beginning of the formative period of the solar system a large mass of Substance A with some small subsidiary aggregates and considerable dispersed matter was approaching a smaller and less consolidated mass of Substance B, in which the subsidiary aggregates were relatively more numerous and much larger in proportion to the central mass than in the A component. When the combination of the two systems took place under the influence of the mutual gravitational attraction the major aggregates of the B component acquired orbital motion around the large central mass of the A component. In the process of assuming their positions these newly constituted planets encountered local aggregates of Substance A which had not yet been drawn into the central star and under appropriate conditions these aggregates were captured, becoming satellites of the planets. At the end of this phase all major units of both components had been incorporated into a stable system in which planets composed of Substance B were rotating around a star composed of Substance A, and smaller aggregates of Substance A were similarly in orbits as planetary satellites.
Smaller fragments are more subject to being pulled out of their normal paths by the gravitational forces of the larger masses which they may approach, and while orbital motion of these fragments is entirely possible the chances of being drawn into one of the larger masses increase as the size decreases. We may therefore deduce that during the latter part of the formative period all of the larger members of the system increased their masses substantially by accretion of fragments of Substance A in various sizes from planetesimals down to atoms and sub-material particles, with some smaller amounts of Substance B, also in assorted sizes. After the situation had stabilized we could expect to find a central star consisting primarily of Substance A, with a small inner core of Substance B derived from the heavy portions of the original Substance A mix and the accretions of Substance B. We could expect each planet to consist of a relatively large core of Substance B and an outer zone of Substance A, the surface layer of which would contain some minor amounts of Substance B acquired by capture of small fragments. The satellites, which have comparatively little opportunity to capture material from the surroundings because of their small masses and the proximity of their larger neighbors, should be composed of Substance A with only a small dilution of Substance B. It can also be deduced that after the formative period was complete further accretion took place at a slower rate from the remains of the original dispersed matter, from newly produced matter, and from matter entering the system out of interstellar space, but the general effect of such subsequent additions of material would not differ greatly from that of the accretions during the formative period and would not change the general nature of the result.
This is the theoretical picture as it can be drawn from the principles developed in the earlier pages. Now let us look at the physical evidence to see whether such a theory is tenable. The crucial issue is, of course, the existence of distinct substances A and B. Both the deduction as to the method of formation of the planetary systems and the underlying deduction as to the termination of the dense phase of the stellar cycle at a destructive limit would be seriously weakened if no evidence of a segregation of this kind could be found. Actually, however, there is no doubt on this score. Many of the fragments of matter currently being captured by the earth reach the surface in such a condition that they can be observed and analyzed. These meteorites definitely fall into two distinct classes, the irons and the stones, together with mixtures, the stony-irons. The approximate average chemical composition is as follows:
Chemical Composition of Meteorites
The composition of the iron meteorites is in full agreement with the hypothesis that these are fragments of pure Substance B. The stony meteorites have obviously been unable to retain any volatile constituents and when due allowance is made for this fact their composition is entirely consistent with the deduction that they represent Substance A. The existence of mixed structures, the stony-irons, is easily explained. The evidence from the meteorites therefore gives very strong support to those aspects of the theory which require the existence of two distinct substances A and B. There is no proof that the meteorites actually originated contemporaneously with the planets in the manner described, but this is immaterial so far as the present issue is concerned. The theoretical process that has been outlined is not peculiar to the solar system; it is applicable to any system reconstituted after a supernova explosion and the existence of distinct stony and iron meteorites is just as valid proof of the existence of distinct substances A and B whether the fragments have originated within the solar system or have drifted in from some other system which according to theory has originated in the same manner. The support given to the theory by the composition of the meteorites is all the more impressive because of the fact that the segregation of the fragmentary material into two distinct types on such a major scale has been very difficult to explain on the basis of previous theories.
Additional corroboration of the theoretical deductions is provided by the spectra of novae. Since these are stars of the white dwarf class they are composed of Substance B as originally formed. Within a relatively short time, however, the original star is covered by a layer of light material captured from the environment. This material is essentially the same as that in the outer regions of stars of other types and the composition of the stellar interior therefore is not revealed by the spectra obtained during the pre-nova and post-nova stages. When the nova explosion occurs, however, some of the Substance B from the interior of the star forces its way out as previously described and the radiation from this material can be observed along with the exterior spectrum. As would be expected from theoretical considerations the explosion spectra often show strong lines of highly ionized iron and nickel.
Another theoretical deduction that can be compared with the evidence from observation is the nature of the distribution of Substance A and Substance B in the planetary system. The sun has a relatively low density and we can undoubtedly say that it consists principally of Substance A as required by theory. Whether or not it actually contains the predicted small core of Substance B cannot be determined on the basis of the information now available. The planet that is most accessible to observation, the earth, definitely conforms to the theoretical requirement that it should consist of a relatively large core of Substance B with an overlying mantle of Substance A. The observed densities of the other inner planets, together with such other pertinent information as is available, likewise make it practically certain that they are similarly constituted.
The situation with respect to the major planets is less clearly defined. The densities of these planets are much lower than those of the earth and its neighbors, but we find that their outer portions are composed largely of very light elements, and this leaves the internal composition a wide open question. It seems, however, that there must have been some kind of a stable gravitational nucleus in each case to initiate the build-up of the light material and it is entirely possible that this original mass, which is now the core of the planet, is composed of Substance B. Jupiter has a total mass 317 times that of the earth and even if the core only represents a small fraction of the total it could still be many times as large as the earth’s core. This viewpoint as to the nature of the cores of these planets is further strengthened by observations which indicate that the outermost planet, Pluto, has a relatively high density and may actually have a metallic surface, which would classify it as pure Substance B. We may conclude that, although the observational data on the outer planets do not definitely confirm the theory that they have inner cores of Substance B, the observed properties are not inconsistent with this theory. Since it is highly probable that all of the planets have the same basic structure this lack of any definite conflict between theory and observation is very significant.
The satellites present a similar picture. The verdict with respect to the distant satellites, like that applicable to the distant planets, is inconclusive. The available observational information is consistent with the theory that the inner cores of these objects are composed of Substance A, but it does not exclude other possibilities. The satellite we know the best, like the planet we know the best, gives us an unequivocal answer. The moon is definitely composed of materials similar to the stony meteorites and the earth’s crust; that is, it is practically pure Substance A, as it theoretically should be.
It is appropriate to point out that this theory of planetary origin derived by extension of the principles developed from the Fundamental Postulates is independent of the temperature limitations which have constituted such formidable obstacles to many of the previous efforts to account for the existing distribution of material. The fact that the primary segregation of Substance A from Substance B antedated the formation of the solar system explains the existence of distinct core and mantle compositions without the necessity of postulating either a liquid condition during the formative period or any highly speculative mechanism whereby solid iron can sink through solid rock.
This explanation of the process of formation of the system also accounts for the fact that nearly all of the constituent units have the same direction of rotation. The reason for the near-coincidence of the orbital planes of the planets is not as obvious. The original distribution of the masses of Substance B which are now the cores of the planets should have been roughly spherical and the separation of the planets perpendicular to the orbital plane of Jupiter should have been comparable to that in the plane of the orbit. It is probable that the shift of the orbits to their present locations has been due to the inter-planetary gravitational forces. Jupiter exerts a small but significant force component tending to rotate the orbits of the other planets into its own orbital plane and in the long period of time that has elapsed since the formation of the system even a small force could be quite effective.