TOWARDS A UNIFIED COSMOLOGY

by     Reginald O. Kapp

PART V - APPENDICES

Appendix D - Origin of the Planetary System


D.l: A Star's Gain and Loss of Mass
Continuous origin is a diffuse process. As its rate per unit volume is everywhere constant, this rate is also low; hence clearly observable effects cannot be expected over short distances. To be pronounced they must occur in extragalactic space. This explains why the best evidence in support of continuous origin is provided by the spiral nebulae.

Continuous extinction, on the other hand, has its highest rate per unit volume where matter is most concentrated, namely in the heavenly bodies. Its observable effects are localized. This is why the best evidence in support of it is provided by gravitation.

But gravitation cannot be the only manifestation of continuous extinction. One of its implications is that the earth and the other planets, together with their satellites, are continuously losing mass by extinction. If Tm is the half-life of matter, the masses of these bodies must have been twice their present values Tm years ago and will have half their present values Tm years hence. Such a change must have observable consequences.

It must be noted that the sun's mass cannot have experienced a change at the same rate. For while the sun is assumed to be losing mass by extinction, it is experiencing two additional changes. It is losing further mass by radiation and it is gaining mass by the capture of hydrogen from its surroundings. The net result may be either a loss or a gain. It would not be easy to estimate which in fact it is.

What the planets gain by capture must be negligible. Its great mass makes the sun a successful competitor for new matter. The sun's domain is large compared with that of any planet; in other words a particle that finds itself at only a comparatively short distance from a planet does not fall on to the planet but on to the sun. Hence the sun may have been either larger or smaller in former times than it is now, but the planets must have been larger. If their loss of mass were known, it could be used to evaluate Tm.

If any observed facts showed that there has been no loss of mass, one would have to conclude that Tm was infinite; which would be to abandon the hypothesis of continuous extinction. But I do not know of any such facts. On the contrary, wherever I turn I find facts that seem to give a finite value to Tm They are all consistent with a value less, perhaps substantially less than 109 years.

These facts belong to several disciplines, including astronomy and biology. They are facts that have hitherto defied a satisfactory explanation. But then those who tried to explain them took it for granted that the half-life of matter was infinite. I propose to show in this appendix and in Appendix E that there is a good prospect of finding explanations of many baffling facts if one adopts the hypothesis of continuous extinction.

A detailed attempt at solving riddles that have been presented by facts in these various disciplines would be beyond the scope of the present study. It will have to be left to the combined efforts of various experts. I shall be content here to give no more than a bare outline of some further conclusions that are reached on the basis of Symmetrical Impermanence. I am not claiming to give the result of an exhaustive study but, instead, some tentative conclusions. I am hoping that they are at least near to the truth and that they will suggest rewarding fields of research for some astro-physicists and biologists. What is presented in these Appendices D and E might aptly be regarded as outlines of some research programmes.

D.2: The Double Star Theory of Planetary Formation
In the past many theories have been invented in order to explain the origin of the planetary system. Each in turn has proved defective in some particular, has been abandoned and replaced by another one. One of these theories was developed a few years ago in sundry detail by F. Hoyle. It is based on a suggestion that seems first to have been made by Lyttleton. Hoyle has put forward a quite different theory since then; so he presumably has found the earlier one to be untenable, as indeed it is. But its core had so much to recommend it that the earlier theory ought not, I think, to be wholly abandoned. There is a case for retaining the core and correcting the errors. I propose to show here that symmetrical impermanence provides a means of doing this. Hoyle's earlier theory was as follows.

At one time the sun had a companion with which it formed one of the numerous double stars that are known to astronomy. Both stars consisted mainly of hydrogen, as most stars are known to. In both the pressure and temperature were such that the hydrogen was being slowly converted into helium, as it still is in the sun. The large quantity of energy released by this conversion deep in the interior of the stars appeared as radiation and exerted sufficient pressure on the gas particles above them to push them outwards against gravity. The internal radiation acted, in other words, like air in a balloon and kept both stars inflated. It is generally accepted today that this does happen in all stars in which the rate of helium synthesis is between certain upper and lower limits, and is still happening in the sun. The diameter of the sun would be a small fraction of its present value if the helium factory did not maintain a considerable internal pressure of radiation.

According to Hoyle's double star theory the time came when a large proportion of the hydrogen in the sun's companion began to be exhausted, while the sun still had enough of the factory's raw material, hydrogen, to maintain internal pressure of radiation and has enough even now for a long while to come.

Failing to maintain its previous internal radiation the sun's companion collapsed under the influence of its own weight. Thereby the nuclei of the remaining hydrogen and of the helium that was present were forced into very close association. The pressure and temperature were sufficient to cause the synthesis of all the heavier elements. The energy required for this synthesis was largely derived from the potential energy released in the star's collapse. After the synthesis the star consisted to a smaller extent of hydrogen and helium. Most of its substance was the same as is now found in the planets.

Hoyle proceeded from there to some further assumptions. He said rightly that in collapsing the angular velocity of the sun's companion increased. By the principle of conservation of angular momentum it would have to do so. But Hoyle went on to assert that the accompanying increase in centrifugal force sufficed to exceed the force of gravity, that wisps of gas were thrown off from the star, that these subsequently condensed into the planets, that centrifugal force thereafter propelled the parent star out of the sun's neighbourhood. It is these further assumptions that need to be corrected.

The core of the theory carries conviction and would make it far superior to any rival if it were consistent with known facts in other respects. Other theories have assumed that the planets were formed out of the sun's substance. But of the elements up to uranium, with a nucleus that carries 92 charges, hydrogen and helium are the only ones that occur with any abundance in the sun. Of the others there are, at most, only traces. If the planets had been formed out of the sun's substance, one would have to find a reason why the substances are so different. One might succeed in inventing an hypothesis by which the synthesis of the heavier elements had been formed after they left the sun. But no satisfactory reason why this should happen has ever been found, and the hope of finding one seems remote. The conditions needed for converting lighter into heavier elements are now well understood and they require a combination of very high pressure and temperature. It is impossible to believe that they can have occurred in the comparatively small bodies that formed the planets.

Alternately, one might perhaps postulate some filtering process by which the heavier elements would be extracted from the sun and the lighter ones left behind. Ingenuity can accomplish much. But it must be allied with disciplined thought if it is to achieve anything of lasting value. The notion that all the elementary substances existed in the parent body from which the planets were formed, and in roughly the same relative quantities, is less fantastic than any other that seems to have been put forward. It deserves to be tested for its consistency with other known facts. Many doubts and queries arise. Let each of them be considered in turn.

A reason has to be found why the two companion stars should have had such different fates. What circumstances could have left the sun with enough hydrogen to go on synthesizing helium for many thousands of millions of years while its companion's store was exhausted at an early date? One possible answer would be that the sun's companion used up its hydrogen more quickly than the sun did, another is that it had less hydrogen to use up.

Advocates of the earlier double star theory have chosen the former explanation. Their view is that for some reason the helium synthesis was greatly increased at a certain moment in the star's life. It is reasonable to suppose that this happened, for the synthesis can become an unstable process and would, if it increased above a certain critical rate, get out of hand. The subsequent collapse is assumed to have been correspondingly catastrophic and to have occurred in a very short period of time. The potential energy of position possessed by the substance of the star in its inflated condition was converted into heat at the surface, where it would have caused a very great increase of brightness.

It is believed by some that a supernova is the result of this kind of catastrophic collapse of a star, and so advocates of the double star theory believe that the substance of the sun's companion was converted into the heavier elements during the process that is manifest as a supernova.

The second explanation, namely that there was less hydrogen to begin with, provides a less dramatic history but is more consistent with Symmetrical Impermanence. It would seem equally well to account for production of the heavier elements.

If the sun's companion were somewhat less massive than the sun, it will have lost to the sun in competition for hydrogen from the surroundings. While the sun could replenish its store as fast as it was being consumed, the smaller companion could not do so. Eventually the moment must then have arrived when there was not enough fuel to maintain the amount of radiation needed to inflate the star. Consequently it collapsed. According to this explanation, the pressure after collapse would be the same as for the history in which the hydrogen was being used up in a burst of extravagance, and the temperature also could hardly be very different. One should expect, with the assumption of gradual consumption, the combination of pressure and temperature to account equally satisfactorily for the synthesis of the heavier elements. Tentatively I should therefore like to put forward the theory of hydrogen starvation in preference to the current one of an orgy of spending as an explanation of the exhaustion of the hydrogen store. But a decision can only be reached by a specialist in this branch of astrophysics. Whether the hydrogen consumption and collapse were slow or rapid, the collapse must have been accompanied by a great increase in the rotational speed of the star. For by the principle of conservation of angular momentum the angular speed increases as the moment of gyration decreases. During collapse, therefore, the centrifugal force on the substance of the star must have increased to a very high value.

D.3: Disruption by Centrifugal Force
The effect of a centrifugal force that reached such a high value needs to be thought out with care. It is fatally easy to jump to the conclusion that the star would necessarily break into fragments like a bursting flywheel and fling the pieces away to a great distance. But the truth is that this would not happen to a gaseous star. From what I have read and heard said I gather, nevertheless, that there is some misapprehension about this, so I make no apology for presenting here some elementary mechanics.

A gaseous star consists of a collection of separate particles. In the outer layers these are molecules of the gas and there is no cohesion between them. They have momentum and are subjected to the forces of impact and gravitation. Centrifugal force acts on them in the same way as it would act on loose stones resting on a smooth surface. As a step towards under- standing the effect of centrifugal force on the gas particles, let us therefore first consider what the effect would be if the star were solid and an isolated stone were lying on its horizontal surface.

The star is supposed to be rotating faster and faster. There is thus a tangential acceleration of its surface and the stone, having inert mass, would be left behind if the surface on which it rests were perfectly smooth. But this need not worry us for the moment. Let the surface be rough enough to entrain the stone by friction. It will then share the acceleration of the surface of the star.

As the angular velocity increases, the apparent weight of the stone decreases. The moment arrives eventually when the pressure between the stone and the surface on which it rests becomes zero. If the angular velocity of the star does not increase further beyond this value, the stone will continue to retain its position on the ground by virtue of its inertia. If the speed of rotation increases still further, the stone will, however, not be further entrained by friction; for there is no friction. It will continue to move with its former velocity, which is now less than that of the surface on which it has been resting. To an observer on the star it will appear to slip over the surface in the opposite direction to that of rotation. But so long as it is free from restraint nothing will cause the stone to rise from the surface.

Now let it be supposed that there is restraint. Let the stone be held in position by a wire anchored to a bolt in a rock. By this means the stone is prevented from slipping backwards as the star accelerates. It is made to participate in the rotation of the star, whatever this may be, for so long as the wire holds. When the speed of rotation exceeds the value at which gravity is exactly balanced, the stone will, by virtue of its inertia, tend to move with constant velocity in the straight line that forms a tangent to the surface of the star instead of following the curved path taken by the rock to which the wire is attached. As the distance between the tangent and the circumference increases, the stone will be seen to rise. It will do so until the wire prevents it from rising further. The wire will then be stretched vertically upwards with the stone at its end. It will look like a captive balloon.

If the speed of rotation increases further without limit, the centrifugal force on the stone will eventually exceed the tensile strength of the wire. The stone will then continue to move tangentially by virtue of its inertia. As the distance between tangent and circumference increases, the stone will seem to be rising vertically. But it will now be free from centrifugal force and again subject to gravity. Superimposed on the tangential move- ment at constant velocity there will be a radial movement downwards with the acceleration of gravity. It is not difficult to show that the vector sum of these movements, if the force exceeds a certain critical value, is an orbit around the star.

The diameter of the orbit depends on the excess of centrifugal force over gravity at the moment when the wire breaks and this, in turn, depends on the tensile strength of the wire. As has already been shown, the stone does not leave the ground if there is no wire.

Gases and liquids have no tensile strength. Particles of either are like stones not secured by a wire. Centrifugal force could never throw off puffs of gas or drops of water from a rotating star, however fast it rotated. For any substance to leave the surface there must be cohesion, the equivalent of a securing wire. The tensile strength of a solid substance has to be reached so that the material that breaks off may have sufficient momentum to leave the star and pursue an independent orbit.

This conclusion is not inconsistent with the double star theory but amplifies it. If the planets came out of a companion of the sun, they can only have done so after the companion had changed from a gaseous to a solid state. The star's substance may have been plastic enough to be deformable, but it must have had cohesion.

D4: Can Centrifugal Force ever cause a Star to throw off Fragments?
It does not, however, save the double star theory to do no more than to postulate that the sun's companion became solid before it hurled the planets off into the surrounding space. There are several further serious difficulties, one of which is that any star, solid or gaseous, would not be disrupted by centrifugal force, it would not hurl fragments into space, so long as it retained its original mass. From the views that have been expressed by some I have had to conclude that this is not understood as generally as it needs to be. So I make no apology for discussing here some elementary facts concerning the relation between cooling, contraction, cohesion, rotation, centrifugal force and gravity.

Gravity is a centripetal force and its direction in a star is opposite to that of centrifugal force. A particle anywhere within a rotating star is subjected to both forces and the direction of its movement depends on which force exceeds the other in magnitude. If gravity is the greater and the particle is unrestrained, it will move towards the centre. If centrifugal force is the greater, it will move away from the centre.

Consider a particle of mass m in a star of density σ and rotating with an angular velocity ω. Let the distance of the particle from the centre of the star be r. The mass, m within radius r is (4/3)πσr3 and the gravitational force towards the centre of the star is

Fg = (4/3)πGσmr

The centrifugal force is

Fc = ω2 mr

When these forces balance one has

ω2 = (4/3) πGσ .. (Da)

This shows that the angular velocity at which centrifugal force equals gravity depends only on the density of the star. If this is uniform from the centre outwards, and if all parts of the star have the same angular velocity, a balance in one place means a balance in every place. When the angular momentum defined by equation (Da) has been reached, there is no tendency for any particle, large or small and wherever situated, to move outwards towards the circumference or inwards towards the centre. The motion of particles is determined only by random collisions.

If in such a star there is pressure of radiation, its effect is equivalent to a reduction in the value of G. It does not alter the form of the equation.

When the angular velocity has reached the limiting value defined by equation (Da) it can never increase further so long as gravity alone is the cause of contraction. For the star will cease to contract. No particle can weigh on others so as to press them deeper into the star's interior. There is, indeed, no predominant vertical pressure, either downwards or upwards. There is nothing to hold particles tightly against each other. The star could be cooled down to within a few degrees absolute and its component parts would still fail to rest on each other, as grains of sand do when piled into a heap. The squeezing together of particles can only occur if ω is smaller than defined by equation (Da). And disintegration can occur only if ω is larger.

Bearing this in mind one should not expect a cooling star to solidify if its angular velocity were as high as is defined by equation (Da), for the molecules would then not be squeezed into the condition of interlocking that characterizes the solid state. Hence the double star theory can hardly be saved unless one postulates a low value for the angular momentum of the sun's companion. This must have been low enough for gravitation to predominate over centrifugal force when it was solidifying.

To save the double star theory one must assume, let me repeat, that the angular velocity of the sun's companion was low enough after its collapse for gravity significantly to exceed centrifugal force. If it was not so there would have been no compressive forces within the star; nuclei would not have been brought close enough together for the heavier elements to be synthesized. But if gravity exceeded centrifugal force no fragments of the star would be thrown off. So one has also to assume that, subsequent to solidification, centrifugal force significantly exceeded gravity. For this to have happened the angular velocity of the star must have increased very greatly after the star had been compacted into a solid mass. Could this have happened?

The answer is, of course, that according to traditional views it could not. In the absence of an external couple the angular momentum of the star must be conserved and so an increase in angular velocity must be accompanied by contraction. But traditional views seem to permit only one cause of contraction. This is cooling coupled with gravitational force.

The temperature coefficient of solids is not great and so thermal contraction cannot be great after the star has solidified, even when there is a centripetal pull on the component particles. But it has to be remembered that this pull ceases as soon as centrifugal force balances gravity. Thermal contraction, however great, cannot lead to an excess of centrifugal force over gravity. To save the double star theory a different cause of contraction must be found. The contraction must be so considerable that a strong preponderance of gravity over centrifugal force is replaced by a sufficient preponderance of centrifugal force over gravity to overcome the tensile strength of the material of which the star is made and to cause fragments to be torn off and buried away.

If I were called upon to invent an ad hoc hypothesis to explain so great a contraction, I should not attempt the task. I should instead remember Newton's 'hypotheses non fingo'. But an ad hoc hypothesis is not necessary. The contraction is an inference from Symmetrical Impermanence.

D.5: The Effect of Continuous Extinction on Centrifugal Force
The sun's companion, it is assumed in the revised double star theory, was the smaller partner and gained less and less mass by capture as its relative size decreased more and more and its competitive capacity relative to the sun decreased correspondingly. In the course of time the sun's companion therefore lost mass and volume and continued to do so after its solidification. So long as gravity exceeded centrifugal force the loss was accompanied by contraction without limit. The density remained roughly constant while the radius decreased. Any given particle with mass m moved from a radius r to a radius r1,less than r. In conserving its angular momentum it acquired an angular velocity ω1 greater than the previous value ω.

This could continue only until the condition defined by equation (Da) was reached. Thereafter extinctions resulted in a reduction in the value of σ with no corresponding change in the value of r and ω. The consequence would be a decrease in gravity for a particle in any given position. The star would become too flimsy to enable its own gravity to hold it together at the speed at which it was spinning.

One should then expect fairly large chunks to break off, particularly if the star was still semi-plastic and not yet very strong. These would form orbits at no great distance. But if cooling and consolidation continued, one would expect subsequent broken fragments to decrease in size. By the time these were thrown off the star would have acquired a greater rotational speed and so the smaller fragments would be hurled to greater distances and have larger orbits.

In this account of possible events I have ignored the retarding effect of tides. I think this is justified. For the assumption is that the sun's com- panion was of a mass comparable with that of the sun at the time when it underwent this succession of adventures. On so massive a star the retarding effect of tides would be small in comparison with the accelerating effect of extinctions.

D.6: After the Planets had been thrown off
I have mentioned some of the grave objections that must be urged against the original double star theory and have shown how they can be met, without the invention of ad hoc hypotheses, but only by inference from Symmetrical Impermanence. But there remains one further objection. It is quite as serious as those already mentioned and is raised by the simple question: Where is the parent star now ? The double star theory cannot be accepted unless at least this question can find a satisfactory answer. A body of the size of the sun cannot be hidden.

The only answer that I have come across is that the sun's companion, after throwing off the planets, was itself propelled far away into outer space. It has even been suggested that it can still be identified as a faint and distant star. But I have failed to find any attempt to account for the propelling force.

This could not have been generated by the helium synthesis nor by the synthesis of the heavier elements. However violent the happenings may have been during these processes the forces acted from within the star; they could not have exerted a force on it from without.

Centrifugal force is equally unable to account for the removal of the star. It is, after all, a matter of elementary mechanics that centrifugal force acts from and not on the centre of gravity of a rotating body. Centrifugal force cannot displace the centre of gravity.

It is thus evident that the double star theory, as so far presented, is quite unsatisfactory. It accounts for the materials of which the planets are formed and this is its great merit. But it leads one to expect a star of comparable size to that of the sun within the solar system and does not account for our failure to observe it. Advocates of the theory assert that this star was removed but do not explain the removal. To do so they would have to invent an additional ad hoc hypothesis. This would have to show how a force was exerted on the sun's companion from somewhere, how this force carried the star off to a distant place, and how it did so without interfering with the sun. The force would have to be shown to have operated after the planets had been thrown off. To explain the removal of the sun's companion we have to postulate something that acted from without, like a billiard cue.

Given sufficient ingenuity one would, no doubt, be able to invent some possible hypothesis. But even that would not save the double star theory. It would still need to be explained why the planets were left behind. It must not be forgotten that, according to the double star theory, the planets would at the time not have been circling the sun but have had their orbits around the star from which they had been torn. They would have been within the gravitational domain of this star. Where the star went, its planets would go too. This, again, is a matter of simple mechanics. So further ingenuity would be required in order to invent a process that would, as it were, pick the planets off their parent star and transfer them to the sun before the parent star was propelled away.

As hitherto presented, the double star theory is therefore quite unacceptable. And yet it is the only one that explains satisfactorily why the earth and other planets contain all the elements. May it be true after all, in spite of its apparent absurdities?

D.7: Jupiter in a New Light
Let me list the modifications to the original double star theory that I have suggested above:
The sun's companion was not larger than the sun but smaller.

Exhaustion of the hydrogen in the sun's companion was not the consequence of a sudden rise in consumption but of the lack of replenishment that one would infer from Symmetrical Impermanence if the sun's companion were the smaller partner and could not compete successfully with the sun for hydrogen.

Synthesis of the heavier elements did not occur during a catastrophic rise in activity but more slowly and as a result of the close packing of nuclei after collapse of the star.

The parent star consisted of a solid core surrounded by a very thick envelope of gas. The planets were not formed from this envelope but from the solid core. In their passage through the envelope they took some of this with them. Therewith they acquired an atmosphere, though not relatively as voluminous a one as that of the parent star. Their specific weights were lower than that of the parent.

No planets broke off until the sun's companion had lost considerable mass by extinction and had therefore acquired a greatly increased angular velocity.

This history is admittedly hypothetical and will have to be studied critically by experts not only in astrophysics but also in other branches of science. I have already said that I am presenting here a programme for research and not a conclusion that has already been fully tested.

Among the questions that have to be put to specialists are:
Would synthesis of the heavier elements occur if the sun's companion collapsed comparatively slowly, as postulated in my revised theory?
Would the synthesized substances acquire a density similar to that of the earth?
What is the quantitative relation between the tensile strength of the substance thrown off and the distance to which it would be flung?
Could planets have acquired sufficient angular velocity to throw off moons and have been subsequently slowed by tides down to their present angular velocity?
Or could the moons too have been thrown off by the parent star and subsequently become attached to this or that planet?

For the moment I can only hope that answers to these and some other questions will prove to be consistent with the revised double star theory, quantitatively as well as qualitatively. On the assumption that it is so, some further conclusions follow from Symmetrical Impermanence. These are not hypotheses any longer; they are inferences from continuous extinction.

Being smaller than the sun the companion star would continue to lose mass by extinction after it had thrown off the planets. The smaller it got, moreover, the more it would fail in competition with the sun to obtain adequate replenishment by capturing hydrogen from outer space. The discrepancy between the masses of the two stars would thus become greater and greater as time went on. The companion star would dwindle in size and the smaller it became the less hydrogen it would capture until it captured virtually none and its size would be wholly a function of the half-life of matter. Given time enough one would infer that the companion star would dwindle to a mass that was but a small fraction of the mass of the sun.

For a while the shrinking companion would continue to be the centre of the circling planets. But it would also be becoming more and more itself a planet of the sun. As its mass decreased, the orbits of its planets would also become less and less like ellipses; the sun would exercise an increasingly pronounced distortion. It would, as it were, begin to lure the planets away from their parent star. When this became very small it would lose all hold on its offspring. The planets would then edge bit by bit towards orbits with the sun as centre and would finally move in these. The effect of the companion star would be only to prevent the orbits from being perfect ellipses. But even this effect would diminish with time.

At a certain date the mass of the parent of the planets would be about one-thousandth that of the sun. This, I suggest, is the date at which we have now arrived. The sun's companion was not propelled away. It is still in the solar system and we can see it on any clear night. It is a large golden sphere and is called Jupiter.

If this theory is correct it throws some light on the constitution of Jupiter and the other big planets. Their density is much lower than that of the earth. But if all the planets are offspring of Jupiter this latter must have a solid core consisting of much the same substances as the earth and with a density that does not differ greatly from that of the earth. From that it has to be concluded that Jupiter is surrounded by a very thick envelope of lighter material, which may be liquid or gaseous and may consist in its outer layers largely of hydrogen.

On my theory of the origin of the planetary system one should expect fragments thrown off from the parent star to carry a relatively small amount of liquid or gaseous substance from that star away with them; for while being buried through the atmosphere of the parent star they would leave most of it behind. But some atmosphere would cling to the fragments by virtue of their gravitational field, though it would be but a small fraction of their mass.

One should expect this fraction to be greater the greater the mass of the fragment. The other large planets besides Jupiter should, therefore, be expected to have a much thicker liquid and gaseous envelope than the earth and, therefore, to have a smaller density. One should expect Jupiter itself to have the thickest envelope and lowest density of all. Observation confirms this expectation (with the exception of Saturn).

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