The white dwarf situation is a good example of the way in which an erroneous basic concept can cause almost endless confusion in an area where the information from observation is erroneously interpreted. This is one of the two most misunderstood areas in astronomy (aside from cosmology, which belongs in a somewhat different category), and it is significant that the other badly confused area, the realm of the quasars and associated phenomena, is another victim of the same basic error: a misunderstanding of the cause of the extremely high density of such objects as white dwarfs and quasars.
The wrong conclusion as to the nature of the very dense state of matter leads to an equally wrong conclusion as to the ultimate destiny of the stars that attain this state: the conclusion that they must, in the end, sink into oblivion as black dwarfs, cold, lifeless remnants that play no further part in the activity of the universe. This is the basis for the assumption, already discussed, that the white dwarfs must have evolved from the red giants. Extension of this line of thought then leads to the conclusion that, except for “freaks,” the stars of the high density classes should line up in some kind of an evolutionary sequence. As previously noted, the position of the planetary nebulae in the CM diagram has been interpreted by the astronomers as indicating that they are the first products of the unidentified hypothetical process that carries the red giants into the white dwarf region. It then follows that the central stars of the planetary nebulae must evolve into the ordinary white dwarfs.
Shklovsky regards this as incontestable. “There can be no question,” he says, “but that the stable object into which the nucleus of a planetary nebula evolves should be a white dwarf.”143 But even this essential step in the hypothetical evolutionary course runs into difficulties. Aller and Liller give us this assessment of the situation:
Our evidence indicates that they [the central stars of the planetary nebulae] evolve into white dwarfs, but we do not yet know whether they represent an intermediate stage for most stars or not. Neither do we know from what specific kinds of stars they may evolve.144
This problem persists all the way down the line. The theorists not only have difficulty in explaining how the planetaries evolve from the red giants, and how the ordinary white dwarfs evolve from the planetaries; they are also confronted with the problem of how to account for the existence of a variety of high density objects for which their evolutionary sequence has no place. The novae, for instance, must fit into the picture somehow. But there does not seem to be any place for them in the astronomers’ version of the evolutionary path. “Nova outbursts are too rare to be a typical stage in stellar evolution,”145 says Robert P. Kraft. Because of the lack of any explanation consistent with the accepted theories of stellar evolution, there is a rather general tendency to dismiss the novae and related objects, the cataclysmic variables, as aberrations. For example, one astronomy textbook offers this comment:
Very little is known about the reason for a nova’s outburst. It appears that something has gone wrong with the process of nuclear energy generation in the star.146
Development of the theory of the universe of motion now shows that the planetaries and the ordinary white dwarfs follow parallel, rather than sequential, evolutionary paths. All of these dwarf stars enter the observable region along a critical temperature line at the left of the CM diagram, and move downward and to the right along parallel lines as they cool (evolutionary stage 3). On reaching the temperature at which a transition to motion in space takes precedence over further cooling of the atoms moving in time, a temperature that is determined by the stellar mass, each star converts to motion in space. This change takes it upward on the CM diagram (evolutionary stage 4). The general nature of the conversion process is the same for all of these stars, but the specific character of the observable results depends on the magnitudes of the factors involved. Our next objective will be to examine the details of this process.
As successive portions of the intermediate speed matter of which the two classes of white dwarf stars are composed cross the unit speed boundary in their continuing loss of thermal energy, they form local concentrations of gas—bubbles, we may say—with particle speeds in the range below unity. Because of the inverse density gradient in the interior of the white dwarf star, these gas bubbles move downward to the center, the location of lowest density, and accumulate there. Some interchange takes place between the gas and the surrounding intermediate speed matter, tending to convert part of the gas back to intermediate speeds, but this interchange is slower than the oppositely directed movement across the unit boundary that produces the gas in the outer regions. A gas pressure therefore builds up at the center of the star. When this pressure is high enough, the compressed gas breaks through the overlying material, and the very hot matter from the interior is exposed briefly at the surface of the star, increasing its luminosity by a factor that may be as high as 50,000. The star also becomes a x-ray emitter. The significance of this emission will be discussed in Chapter19.
Within a relatively short time (astronomically speaking) the small amount of matter brought to the surface by the outburst cools, and the star gradually returns to its original status. A white dwarf is inconspicuous and, since the first observed events of this kind could not be correlated with previously identified objects, they were thought to involve the formation of entirely new stars. As a result, the inappropriate term nova was applied to this phenomenon.
From the foregoing description it is apparent that the nova process is periodic. As soon as one gas accumulation is ejected, the compressive and thermal forces in the interior of the star begin working toward development of a successor. Inasmuch as the gravitational forces operating within the star are gradually expanding it toward the condition of equilibrium for motion in space represented by the spatial main sequence (that is, they are drawing the constituent atoms closer together in time), the resistance to the gas pressure that builds up in the center of the star decreases as the star moves through this stage of its existence. The decreasing resistance shortens the time interval between explosions. The first event of this kind may not occur for a very long time after the beginning of the observable life of the star, but as the star approaches closer to the point of full conversion to motion in space the time interval decreases, and a number of novae have repeated within the last 100 years.
Novae are relatively infrequent phenomena, and observationally difficult because of the relatively short duration of the active period, and the rapid changes that take place during this time. Meaningful information about them is consequently limited. The theoretical conclusions with respect to this stage of the evolution of the stars on the dwarf side of the main sequence can therefore be compared with observation only to a very limited extent. We will have to be content, in most cases, with a showing that the theoretical findings are not inconsistent with what has been observed.
Two of the brightest novae, T Coronae Borealis and RS Ophiuchi, are in the class known as recurrent novae, having repeated three or four times during the period in which they have been subject to observation. This is another name that is not very appropriate, as some novae of the more common “classical” type have also been observed to repeat their outbursts, and theoretical considerations indicate that all will eventually repeat many times. T Coronae is estimated to have a mass of 2.1 solar units,147 which puts it, and presumably RS Ophiuchi, in the class of the larger white dwarfs, those that were formerly the central stars of planetary nebulae. This large mass is consistent with the high luminosity of the two novae that have been mentioned.
The nature of the nova process is the same regardless of the size of the star that is involved. In all cases there is a pressure build-up that eventually breaks through the overlying layers of the star. But there are differences in the rate of pressure increase, and in the weight of matter through which the confined gas must force its way in order to escape, and the variability in these factors results in major differences in the character of the outbursts from different classes and sizes of stars. In the white dwarfs of the larger (planetary) class, the luminosity and temperature changes required to move a star from the point on the evolutionary line where it begins its final transition to motion in space to the appropriate main sequence position on the line segment BC are relatively small, averaging about three magnitudes, and they are accomplished quite rapidly. This accounts for the short interval between the outbursts of these stars.
On the other side of the dividing line the situation is quite different. The first stars of the smaller class, the ordinary white dwarfs, not only enter the observable region at a much lower luminosity, but undergo a greater decrease in luminosity and temperature as they cool, so that when they arrive at the point where they are ready to begin the transition from motion in time to motion in space they have a long way to go, as Figure 21 clearly indicates. The time between outbursts is correspondingly long. On the other hand, the magnitude of the outburst is not related to the amount of energy decrease involved in the transition, but to the size of the star, which determines the resistance to escape of the confined gas. Even the largest of the novae produced by ordinary white dwarfs are therefore less violent than those of the T Coronae class, although their range of magnitudes is greater. Initially they repeat only at very long intervals, too long for more than one event to have occurred during the time that observations of these phenomena have been carried on.
The observers classify novae as slow, fast, or very fast, depending on the rate at which the luminosity develops and returns to normal. Aside from details of the spectra, which are not being covered in this work, available quantitative information about these objects includes the maximum and minimum luminosity, together with the difference between the two: the total luminosity range. The distances to the novae are not known, and the absolute magnitudes are therefore unavailable. The most significant luminosity measurement is the total range, which is independent of the distance, except to the extent that there has been absorption of light in passing through the intervening matter. Table III compares the ranges of the group of novae tabulated by McLaughlin (reference148) with the assigned classifications and with the number of days required for the luminosity to decline seven magnitudes, a rough check on the validity of the classification.
Some general conclusions can be drawn from this information. Theoretically the earliest outbursts of the largest novae should be the fastest, and should have the maximum magnitude range, since these largest stars are at the bottom of
|T Cr B*||8.6||VF||300|
|* “recurrent”||** “repeated outburst”|
the white dwarf evolutionary band. Both the rate of luminosity change and the magnitude range should decrease as the white dwarf star ages. The mass does not change significantly. The slowest novae with the smallest magnitude range should therefore be those in which the stars are at the low end of the nova size range, and also near the end of their nova stage. In between these two extremes, the magnitude range is determined by the size and age of the nova. Average range may indicate either an old large nova, or a young small one, or one that is near average in both respects.
The information available from observation gives only a very general indication of how well the novae conform to this theoretical pattern, but the little that is available is clearly in agreement with the theory. Most of the novae with large magnitude ranges are in the very fast category, and there is a general trend toward the next lower rating, the “fast” class, as the range decreases. Only one nova with a range greater than 11 magnitudes is definitely classified as fast. Below this magnitude level, the fast group outnumbers the very fast by about three to one. This is consistent with the theoretical conclusion that the earliest outbursts of the largest novae should have the maximum magnitude range, that this range should be less for the smaller novae, and that in all cases the range should decrease as time goes on and the outbursts are repeated.
While the slow novae are not concentrated at the lower end of the list as strongly as the very fast are concentrated at the upper end, there is a definite increase in the proportion of slow novae as the magnitude range decreases. If we omit the two stars of the larger class (identified as recurrent), the proportion of slow novae in the group with magnitude range 9.0 or below is 64 percent. In the group with ranges above this level it is only 24 percent. In between the extremes there are some relatively slow novae that are quite high on the list, and some of the very fast class that are quite low. Theoretically, the latter should be quite small stars and the former quite large. This cannot be confirmed observationally, as matters now stand.
The novae that are observed to have repeated are at the low end of the list; that is, they have the lowest magnitude ranges. Of course, these are not the only novae in the list that have repeated; they are merely the white dwarfs that are approaching the end of the nova stage of their existence, and are repeating their outbursts at short enough intervals to have had at least two within the time interval during which observations have been made. Their position at the low end of the list is another agreement with the theory.
At this point we need to take into account the fact that the possible speeds in the intermediate speed range do not constitute a continuous succession of values, but are confined to eight distinct levels, a characteristic of this speed range that we have already had occasion to recognize in applications such as the explanation of the relation known as Bode’s Law. As noted earlier, four of the eight speed levels are on the spatial side of the dividing line, and correspond to identifiable locations in equivalent space. In the planetary stars, which are in the gaseous state throughout, the particles moving at the different speed levels are well mixed, and there is a continuous density gradient from the outer to the inner regions. Here the build-up of pressure and eventual escape of the compressed gas follows essentially the same pattern irrespective of the size of the star. The situation in the ordinary white dwarf stars is quite different because the outer shell of these stars is in the condensed gas state. In this state, as in a liquid, matter of different densities stratifies. The outer shell is therefore not homogeneous, but consists of a number of layers, initially four. Since the white dwarf aggregate is a time structure, rather than a space structure, the gas bubbles in the center of the star (space structures) remain in the aggregate, rather than separating from it. Thus they accumulate in the lowest of the four layers, and are confined by the weight of the three overlying denser layers.
Smaller stars of the same surface temperature have lower internal temperatures, and at some point in the mass range the fourth speed level is vacant. The compressed gas in the central regions of these smaller stars is located in the third level, and is subject to the weight of only two overlying denser levels. This very substantial reduction in the weight that is confining the gas results in a corresponding reduction in the pressure that has to be built up before the compressed gas can break through. It can be expected, therefore, that at some definite level of white dwarf mass the nova type of outburst will be replaced by a different type of eruptive behavior in which the outbursts are more frequent but less violent.
This theoretical expectation is fulfilled observationally. While the white dwarf stars that reach the main sequence at the higher luminosities are observed as novae during the conversion from motion in time to motion in space, those that are smaller, and less luminous in their final state, follow what may be described as a nova pattern in miniature, with less violent outbursts and much shorter periods, ranging from about a year downward. These small scale novae, of which SS Cygni and U Geminorum are the type stars, are classified together with the true novae, the recurrent novae, and certain nova-like variables, as cataclysmic variables.
The magnitude at which the change in behavior occurs is a critical level that, on the basis of the considerations previously discussed, should be related to the other critical levels of the CM diagram by integral numbers of natural (compound) units. We have identified the difference between points B and C on the spatial main sequence, 2.8 magnitudes, as one such natural unit. The explanation just given for the transition from nova to a less violent type of outburst suggests that it should take place one natural unit below the upper limit of the normal, or “classical” novae, which coincides with the lower limit of the planetary stars at point B on the diagram, magnitude 4.6. This puts the boundary between the two types of cataclysmic variables at 7.4 magnitudes on the spatial main sequence. The corresponding mass is 0.65 solar units. Thus the ordinary white dwarf in the range above 0.65 solar masses follows the nova pattern in its conversion to motion in space, while those in the range immediately below 0.65 solar masses are SS Cygni variables in the conversion stage.
We have identified the novae as white dwarfs that have component particles with speeds in all four of the levels on the (equivalent) spatial side of the dividing line, and the SS Cygni variables as white dwarfs in which the component speeds are limited to three of these four levels. On the basis of the same considerations that are applicable to the novae, the magnitude range of the SS Cygni stars, after conversion to motion in space, should be between 7.4 and 10.2, and the mass range should be from 0.65 to 0.40 solar units. Inasmuch as there are white dwarfs with still smaller masses, it follows that there should exist a class of these stars in which component speeds occupy only two levels. Since this leaves only one level overlying the one in which the gas is accumulating, it can be expected that the gas bubbles in these stars will break out at a relatively early stage before they reach any substantial size. The observations indicate that this third theoretical class of cataclysmic variable can be identified with the flare stars. These are theoretically stars of from 0.40 to 0.25 solar masses, with main sequence magnitudes, after conversion to motion in space, in the range from 10.2 to 13.0.
The 0.25 lower limit of the mass of the flare stars leaves room for some white dwarfs with only one speed level, as the minimum mass of the ordinary white dwarfs is somewhat lower, probably around 0.20. There is no significant amount of resistance to escape of gas from these stars, other than the viscosity of the condensed gas through which it has to make its way, but since the gas comes out in the form of bubbles, it is probable that there are visible flares from these stars similar to those from the two-level class, known to the astronomers as UV Ceti stars. The flare stars are not usually included in the classification of cataclysmic variables but they share the distinguishing characteristic of those variables, periodic outbursts of very energetic matter, and differ mainly in the magnitude of the outbursts. As indicated in the preceding paragraphs, there is a specific place for them in the pattern of the cataclysmic variables that has been derived from the theory of the universe of motion. the pattern that applies also to the novae and the SS Cygni stars.
Turning now to a consideration of the information that is available from observation, we find that the average magnitudes reported by the observers are within the theoretical limits, but these limits are so wide that the agreement with observation is not very significant. The mean absolute magnitude of the SS Cygni stars is reported as 7.5+0.7 (reference 149) and that of the UV Ceti flare stars as 13.1 (reference 150). The cataclysmic variable stage begins at about magnitude 16, the low limit of the ordinary white dwarfs, and extends to the level of the galactic main sequence, 0.8 magnitudes above the limiting magnitudes on the undisplaced basis as noted above. Since the conversion process accelerates, the average position of these variable stars, as observed, should be well below the midpoint of the magnitude range. The 13.1 magnitude reported for the UV Ceti stars is consistent with this prediction. The 7.5 magnitude of the SS Cygni stars is too high, near the upper end of the theoretical range, but it is likely that this value includes a large contribution from the after effects of the interior heat released during the outbursts.
Other data on the smaller classes of cataclysmic variables are scarce. Unlike the novae, which are spectacular, but rare because of the long intervals between outbursts, the SS Cygni stars are dim and hard to detect. It is reported that about 100 of them have been located, but only a few of them have been studied in detail. These are found to have a “period-amplitude relation whereby the stars of longer period show the more violent outbursts,”151 thus continuing the pattern of the true novae, noted earlier. The maximum observed magnitude range is near 6 magnitudes, about one magnitude below the minimum of the true novae indicated in Table III.
Very little is known about the properties of the flare stars, aside from those that they share with the other cataclysmic variables. A. H. Joy describes them as “extremely faint M-type dwarfs” in which the “light curve rises to maximum in a few seconds or minutes of time and declines to normal in less than a half hour.”152 These light curves “are similar in form to the light-curves of novae,”153 an observation that supports the theoretical identification of the flare stars as junior members of the group headed by the novae.
The heterogeneous group of stars known as the nova-like variables do not constitute a separate class, but are members of the classes already identified, with some special characteristics that distinguish them from the type stars of their respective classes. For instance, R. Aquarii and similar stars differ from SS Cygni mainly in that in SS Cygni both components of the binary system are dwarfs. whereas R. Aquarii combines a red giant and a hot blue dwarf.154 Z Andromedae is the prototype of a group of stars that undergo outbursts of about three magnitudes, and “combine the features of a low temperature red giant and a hot bluish B star which is probably a subdwarf.”155 The terms applied to the dwarf components in these quoted descriptions of the binary systems are appropriate for the white dwarf members of cataclysmic variable pairs. A “blue dwarf” is simply a hot white dwarf. while a “subdwarf” is a dwarf star below the spatial main sequence in the area in which the cataclysmic variables are theoretically located. As noted previously, a combination of a red giant and white dwarf is not unusual; it is an early evolutionary stage that in time evolves into the more familiar combination of main sequence star and white dwarf
A dwarf nova, like all cataclysmic variables (novae, recurrent novae, dwarf novae, and novalike variables) is a close binary system in which the primary component is a white dwarf. The secondary is a normal star.156
The current tendency is to ascribe the explosive behavior to this binary nature of the system. “The sudden outbursts” of the SS Cygni stars, says Burnham, “are undoubtedly connected in some way with the duplicity of the system, but the exact details are uncertain.”157 Notwithstanding the use of the word “undoubtedly” in this statement, our findings are that the binary nature of the cataclysmic variables, which we confirm, has no connection with their explosive behavior. This is why the astronomers have not been able to explain how their hypothetical process operates. These systems are binary because they originate in supernova explosions powerful enough to accelerate some of their products to intermediate speeds, and the white dwarf member of the binary system is explosive because the intermediate speed component of the supernova products goes through an explosive stage on its way back to the normal speeds of the material sector. The theoretical conclusions agree with the observed fact—the binary nature of these objects—but they disagree with the prevailing assumption as to the nature of the process responsible for the explosive outbursts.
The situation with respect to the location of the cataclysmic variables on the CM diagram is similar. We deduce from the theory that these objects are on the way from the white dwarf status to positions on the spatial main sequence, and therefore occupy intermediate positions. The observers agree as to the positions.
From their luminosities, which are similar to the sun on the average, we are forced to conclude that they [the novae] are small superdense stars somewhat like white dwarfs, but not so extreme.78 (D. B. McLaughlin) Virtually all known post-nova stars are objects of the same peculiar type, hot bluish subdwarfs of small radius and high density, apparently intermediate between the main sequence stars and the true white dwarfs.158 (R. Burnham) The prenova is below the main sequence intermediate between white dwarf and main sequence stars.159 (E. Schatzman)
Where our findings differ from astronomical theory is in the direction of the evolution of the cataclysmic variables. The prevailing astronomical opinion is that the evolutionary direction is down the CM diagram from some location above the main sequence, generally identified as the red giant region, toward the white dwarf stage. The white dwarf is seen as the last form in which the less massive stars are observable, the penultimate stage on the way to extinction as black dwarfs. This leaves the planetaries and the cataclysmic variables dangling without any clearly identified role. Shklovsky calls them “freaks.”
Our analysis now shows that here again the astronomers’ evolutionary sequence is upside down. The white dwarfs of both classes (planetaries and ordinary white dwarfs) enter the field of observation at the left of the CM diagram from an unobservable condition analogous to that of the earliest of the protostars that eventually enter the diagram in the red giant region. Just as these giants move to the left and down the diagram to the equilibrium positions on the spatial main sequence, the white dwarfs move to the right and up the diagram to reach similar equilibrium positions on that sequence. The upward movement takes place in the cataclysmic variable stage.
As the foregoing survey of the results of observation of the cataclysmic variables indicates, existing empirical knowledge is much too limited to provide a clear picture of these objects. But each of the isolated bits of information currently available fits into the general pattern derived from the theory of the universe of motion. While the theoretical pattern of behavior conflicts to some extent with current astronomical thought, it is really not accurate to say that the results of this present investigation contradict the astronomers’ theory of the cataclysmic variables, because, aside from the rather vague idea of a giant star “shedding mass” and moving toward the hypothetical black dwarf status along an unspecified route, the astronomers have no theory of these objects. “Severe problems remain,”142 in arriving at an understanding, says H. M. Van Horn. A. H. Joy describes the situation with respect to the stars of the SS Cygni class in this manner:
The general problem of the SS Cygni stars is so complicated that little progress has been made toward its solution… No satisfactory explanation of the novalike outbursts which occur at semi-regular intervals in the variable stars of this class has been proposed, and their relationship to other groups has yet to be determined.160
Gallagher and Starrfield give us a similar assessment of the current state of knowledge with respect to the novae.
It is clear that there are few problems relating to the novae that we may consider as solved, and many phenomena for which we have yet even to identify the nature of the underlying physical processes.161
The cause of nova outbursts is not likely to be identified directly by observation. At best we can only hope to arrive at an idea of the cause by devising hypotheses, calculating their consequences, and comparing the expected results with the observed facts.162
The development in this work, based on deductions from the postulates that define the universe of motion, has now provided the kind of a complete and consistent theory of the cataclysmic variables that has heretofore been lacking. In the course of this development we have identified the three basic errors that have diverted astronomical thinking about the white dwarfs into the wrong channels: (1) the assumption that conversion of hydrogen into heavier elements is the energy production process in the stars, (2) the assumption that speeds in excess of that of light are impossible, and (3) the assumption that the white dwarf is a dying star. Correction of these errors and application of the physical principles governing motion at speeds greater than light, derived in the preceding volumes of this work, have arrived at a logical and consistent theory of the entire class of cataclysmic variables.
These results show that Shklovsky’s characterization of the cataclysmic variables as “freaks” is totally wrong. These stars (and the planetaries as well) are in the direct line of one of the two coordinate branches of the stellar evolutionary cycle. They are all white dwarfs, differing only in the properties that are affected by the particular evolutionary stage at which each type of object makes its appearance, and they all go through the same general processes of cooling to a critical temperature level and then converting from motion in time to motion in space. Meanwhile the companions of these white dwarfs are going through the successive stages of the giant evolutionary cycle. The two stars of each of these binary systems are at comparable evolutionary stages, regardless of the difference in their properties, and they ultimately arrive at the same kind of a gravitationally and thermally stable state. When they’re relatively short excursion away from the main sequence is ended, both of the partners will settle down for another long stay in that equilibrium condition.