…Continued from Section 1

How Nature has Formulated

the Hubble’s Law

―Section  2―

5.  A Model of Substance Distribution

5-I.    An Onion Model

All the mechanical development aforementioned is based on an assumption that those materials ever contributing to such movement all start their vigorous movement from one point but later propagate with equal chance in all directions. Therefore, it should be reasonable for us to assume a mathematical model in which material distribution in space is consisted of numerous concentric spherical shells of homogeneous substance wrapping around a single point. The radius of all these shells, or layers, can be any value from zero to infinity, while all layers have the same thickness ΔR.   Physical properties of all shells, or layers, are identical.  Boundary between any two layers does not physically exist; it has only mathematical significance.  Larger layers wrapping over smaller layers, similar to what an onion shows, this model will be called an onion model.  (Fig. 8)

5-II.   The work of Gravity

At where is substance, there must be the concept of mass density.  Lacking valid figures as the mass density of the universe, let’s use 16 hydrogen molecules per cubic meter as the mass density.  We will later see that the exact figure of the mass density of the universe in our model is not of critical concern.  The figure we hereby assume is about 3 times as much as the upper limit of the critical mass density from another contemporary theory (Page 55, The New Physics, edited by Paul Davis, published by Cambridge University Press, 1989).  According to that theory, if the universe ever had a mass density above 5 hydrogen molecules per cubic meter, a “Big Crunch” would happen.

Suppose there is a spherical region of radius of 0.1 light-years.  This region is to have a mass density equivalent to 17 hydrogen molecules per cubic meter.  This extra one molecule per cubic meter for this region must have been ‘stolen’ from the neighboring region.  The neighboring region now has a smaller mass density.   Let’s assume this neighboring region has a density of 15 molecules per cubic meter and just wraps around the denser region like a thin spherical shell.  If the denser and the thinner regions together maintained the same average mass density, i.e., (16 hydrogen molecules)/m3 , the thickness of this thin shell would be 0.026 light-years (Fig. 9a).

A molecule in the thinner region would always feel a unidirectional pulling force from the denser region.  The situation for this molecule is very similar to a problem that many of us are already familiar with.  Let us briefly review this problem in the following.

Let us have a hollow shell made up of homogeneous materials like what is shown as C in Fig. 9b.  The shell is isolated from any massive body by a significant distance.   Object A inside that hollow shell then will not experience gravitational influence from anything, not even from shell C.  The variation of thickness of shell C, from very thin to infinity, has no gravitational influence on A.  Object A stays gravitationally free until another object B is placed inside shell C, as shown in the lower figure in Fig. 9b.  These two objects, A and B, will exert a gravitational influence on each other, but the shell, regardless of its thickness, still has no gravitational influence upon either of these two objects.

In comparison, the entire big collection of molecules of the spherical region of radius of 0.1 light-years in Fig 9a can be regarded as object A in Fig. 9b.  Any molecule staying in the thinner region of the thickness of 0.026 light-years can be regarded as object B.  The entire infinite space outside the thinner region can be regarded as shell C.  By allowing enough time, the big collection of hydrogen in the denser region will eventually pull the single hydrogen particle to join its conglomeration. Meanwhile, all molecules within the denser region also constantly experience a gravitational force pulling them towards the mass center of this region, more and more closely condensing with each other.

Governed by the gravitational equilibrium that Newton visualized, materials in this infinitive shell should stay at the same place all the time.  However, the infinite region is not a solid shell. The gravitational defect experienced by the molecules of this infinite shell but bordering at the thinner region cannot forever keeps the same fidelity as those molecules further beyond.  Being a loose particle at the border of the shell, any of them always feels a force pulling it toward the denser region just like another ball B.  This allows the denser region to incessantly recruit new particles  from the infinite shell.  So the denser region has better and better chance to condense with more mass.  Its ever escalating gravitational force seems unstoppable.

From now on, we will call this dense region a condensed ball.

Although extremely trivial, heat is generated each time a particle joins the denser region, or now the condensed ball. Such heat, when escaping, would more or less excite the movement of particles in the neighboring region. This in turn helps the condensed ball even to accelerate in recruiting more neighboring particles with its gravitational influence.  Gradually, after excessively long time, the joining of the particles escalates its manner from feather like drifting to forceful bombardment. Of course, by the time bombardment appears, the thickness of the thinner region would have been far exceeding 0.026 light-years; and particles of the infinite shell but bordering the thinner region would have also been replaced by chunks that look like patches of clouds.

Given enough time, the condensed ball would have opportunity to build up its mass quantity beyond any limit.  Let us take a look at what would happen if the condensed ball could ever collect so much mass such that its mass density became comparable to that of our sun today.

According to the onion model, the universe is assumed consisting of an infinite number of concentric layers of the same substance.  When mass shed from the inner wall of the infinite shell in Fig 9a and accreted on the condensed ball, we could consider materials just pealed themselves from the wall and landed on the ball successively layer after layer.

Due to the poor thermal conductivity of hydrogen, the heat yielded through the accretion of each layer could not  instantly and homogeneously spread over the entire condensed ball. Instead, only part of the condensed ball, particularly the surface part immediately receiving accretion, was mixed with the oncoming layer, and thus, reached a mutual temperature between this part and the newly arrival.  It should also be reasonable to assume that equal amounts of substance from the condensed ball mixed with equal amounts of substance from the oncoming layers during each accretion.

With this heat generation and transferring manner in mind, we can calculate how many layers of materials are needed from the infinite shell to have enough heat generated through accretion and to have a nuclear fusion reaction triggered. While the layer number is only fictitious, what we actually pursue is the radius of the volume that would have contained the amount of materials to satisfy the critical temperature requirement for hydrogen fusion to start.     Assuming the universe’s overall mass density being ρu=16 hydrogen molecules/m3, and ρB=1.42×103kg/m3 for the sun’s mass density, the anticipated radius is calculated as

Ru,n=4.78×1018 m

The spherical volume represented by this radius should be comparable to the volume occupied by the Milky way, which is a disk of radius of 3×1020 m.    This calculation only serves to tell us that gravitational accretion of enough hydrogen may render a possibility to ignite a fusion reaction of this material.  We must not forget that we have simplified the calculation by using a higher figure of mass density in the space, and 100% heat conversion of the potential energy.  If we consider a lower mass density for the space and a heat conversion efficiency of less than 100 %, Ru,n should be larger than what we calculated.

After the first ball’s explosion, the future nuclear reaction out of other virginal bodies no longer needs to rely on material accretion to get ignited.  The explosion of the first ball would send out many burning torches in various directions. All these torches by themselves are infernos of nuclear reaction, they could just ignite explosion out of any virginal ball of enough mass when landing on it.

Now we need to consider if the speed of the torches produced through an explosion was high enough to overcome the gravitational pull of the condensed ball.  If not, the torches cannot fly away from where they are born.  Calculation shows that the torches had velocity many hundred times bigger than the escape velocity exerted by the condensed ball.

With the onion model, we can also mathematically speculate the time needed to trigger the fusion reaction after accretion began. Calculation shows that as n , the number of onion layers, approaches higher and higher, the time needed is

By taking ρu=16 × 3.321× 10-27 kg/m3 , we have

t=2.88 × 1017 sec,  or 9.13× 109 years.

The significance of this numeric figure regarding time is as follows:

(1)    The time needed for accretion before explosion occurs is determined only by the homogeneous mass density of the universe and the universal gravitational constant.  Even though calculations point out that the amount of mass to trigger the first big explosion is also determined by the specific heat of hydrogen and the final mass density of the condensed ball, this will not affect the amount of time needed for accretion. If more mass is needed, the condensed material will have to recruit more substance from the surrounding. This in turn would enable the ball, at the time of having accumulated more masses, to exert a stronger gravitational influence upon those masses on the surface of the hollow dome. The stronger force will give the falling masses a higher acceleration for a faster trip towards the condensed ball.  The result is that the overall amount of time needed for the entire accretion would end up being the same.

(2)  According to the contemporary believing, the universe has an age of 13.5 billion years.  However, using the assumed mass density of (16 hydrogen molecules)/m3, our calculations show that accretion alone may have taken almost 10 billion years.  Some contemporary theories think a density of 5 hydrogen molecules/m3 would cause the universe to end itself with a so-called Big Crunch. But, obviously, our calculation shows that even a density of 3 times more than that value would have nothing to do with any Big Crunch.   As we continue our exploration in this article, we would see that an infinitive universe can never ever have anything like Big Bang or Big Crunch. If we use a lower mass density for our calculation, the accretion time will be even further extended; heat generated by potential energy out of gravitational accretion would have more time to escape from the condensed ball, too!

5-III.   A Picture of Dynamic Development

If we assume a 3-D coordinate system x’-y’-z’ for every virginal condensed ball with its (-z’) pointing toward the ASP and its origin coincides with the mass center of the ball, and if we further assume that each ball, after its explosion, would evenly send out one torch along each of these directions: +x’, -x’, +y’, -y’, +z’ and -z’,  it is apparent that 5 out of the 6 torches are out-going objects leaving the ASP.  With so many torches rushing out, and so many of those tangentially moving torches from various virginal balls being in opposite directions, collisions between materials are unavoidable.  The collisions are mainly:

1)  Between torches and torches.  This is further divided into two groups:

a) The out-going torches, carrying various velocities on different but generally out-going directions.  They will combine into some bigger chunks and continue the out-going tendency after the collision.

b) The out-going torches from the n-th and lower shells being hit by the in-going torches from the (n+1)-th and higher shells. Remember, before torches are produced, condensed virginal balls of the same size should have evenly distributed in each layer, or shell, while retaining the overall mass density of the universe for every shell.  So, depending on the individual case, the resultant material gathering out of such a collision may be an in-going chunk moving toward the ASP while others may be out-going and moving away from the ASP.  However, this group of collision would make the overall material flow to have only 2/3 out of it genuinely moving outward, although 5 out of every six newly born torches are out-going torches.

2)  Between the torches and those virginal materials that have been further away from the ASP.  It is this type of combination that offers all the energy leading to all astonishing results in the sky.  But before we further look at this kind of collision, we need more preparation for further illustration.

Collision causes momentum combination.  Before we analyze the results of the various possible combinations of momentum, let us concentrate on one typical case.

Suppose in an isolated space, there are 3 blocks A, E and F (Fig. 10).  In relation to the reference frame x-o-y, block A of mass m1  is moving to the right with speed v1 , while blocks E and F, both of mass m2 , are motionless.  Blocks E and F are tied together by a weightless but highly compressed spring between them.  The tension of the spring is set up according to m2 such that when the tie breaks, and if only E and F are involved, blocks E and F will fly away from each other at speed 2vc with respect to each other, or at vc with respect to the frame x-o-y. The spring would break if block A lands on E.  When A lands on E and the spring breaks, F would have received some momentum from A and naturally move to the right at speed higher than vc with respect to frame x-o-y.  The combination of A and E together, however, may move to the left or to the right, depending on their ultimate momentum combination. There is a mass center with respect to the entire mass system including A, E, and F together.  This mass center has a speed with respect to x-o-y because of A’s movement and this speed will not change before or after block A causes the spring to break.

Now, in another system that is of 3-D and called x’-y’-z’, let’s have a block arrangement that is slightly different from what is shown in Fig 10.    In this system, we are to have 4 more blocks being of the same property as E and F in Fig 10.  They are closely packed with Blocks E and in this 3-D frame.    The mass center of all 6 blocks together coincides with the origin of the x’-y’-z’ frame and each one of these blocks motionlessly lies on each of the axes +x’, +y’, +z’, -x’, -y’, -z’.  We further assume that a spontaneous explosion will cause all these block to fly away from each other, with one of each flying in the following directions: +x’, +y’, +z’, -x’, -y’, -z’.    A block similar to block A in Fig 10 but on the -z’ axis of this 3-D frame is approaching the pact of these 6 blocks.

If we include a certain spherical volume in space with its center at the initial big explosion, or ASP, we could regard this volume to have been consisted of numerous block systems as mentioned above and each of these system carries a 3-D coordinate frame x’-y’-z’.  When the surface of the entire spherical volume is overwhelmed with new explosions in manners described in the above few paragraph, we can have the total momentum produced by all the explosions out of the entire spherical volume summed up as:

Ptotal =P0 + Pi

=Pr + Pt + Pi

where  Po is the out-going momentum that represents the sum of Pr and P;

Pr  is the centrifugal momentum with respect to the ASP,

P is the tangential momentum;

Pi  is the in-going momentum pointing at the ASP.

Through such calculations we will come up with

Ptotal  ≅ 0.32Mnvc

where  Mn is the total explosive mass contained in the spherical volume we are studying,

vc is the natural speed of unit mass enabled by the nuclear reaction out of the same unit mass of materials.

The positive value of the equation  Ptotal  ≅ 0.32Mnvc tells us that, as a consequence of all the explosions adding together, substances contained in the spherical volume have extra momentum to fly away from the center of the sphere.  In other words, (1) a spherical volume containing materials that are exploding is bound to expand, therefore, (2) inevitably, as long as there is fuel stocked outside of that volume, the fire region will have its expanding capability spread and occupy more and more space.

Because out-going torches always outnumber the in-going torches, between the surfaces of all concentric spheres, each spherical surface of higher n number will receive more torches per unit area than that of the spherical surface of smaller n number. Subsequently, virginal condensed balls at layer of higher n number will have higher chance to be hit by torches that have combined with more pronounced amount of mass.   So, as n   increases, before any collision has occurred on each virginal ball, it is increasingly possible to have m1>m2 , where, in comparison with Fig 10, m1 is the mass of block A,    and m2  is the mass of block E and F in Fig 10.  If E and F are to represent the size of virginal balls, statistically, these balls should be universally about the same size all over in the heavens. In a 3-D system x’-y’-z’, we can simply regard the combined mass of the 6 motionless blocks to be 6m2 together.  After enough generations of explosion and mass combination, we can always have bigger and bigger block A, or to have m1>6m2, when n reaches certain value.

It is true that, with respect to each of the x’-y’-z’ frame, the newly born tangential torches moving on +x’, -x’, +y’, -y’ will have the same speed enabled by nuclear reaction no matter on what layer the torches are produced.  However, as we pointed out previously, with respect to the ASP, many torches have combined their mass and been moving in all directions including the more or less tangential direction. Acting like block A in Fig 10 but with ever escalating momentum on the more or less tangential direction, these combined torches can make some of the newly born tangential torches to fly at speed far higher than the natural speed enabled by fusion reaction.  The torches acting like block A in the sky have contributed not only their ever escalating out-going momentum, but  also their population that is incandescently multiplied.  Gradually, in-going flow of materials in sky with respect to the ASP disappeared in history.

With the help of computer data generating and charting, calculation according to the above derivation shows that the material flow of those out-going torches have their speed increase at a fairly linear rate with respect to the distance from the ASP.  This outcome matches what is found as the Hubble’s phenomenon. This linear rate between speed and distance is shown in the two charts at the end of this web page.  Again, this so called rate is never a constant number!

The same calculation also gives us another interesting outcome.   If there is always only one out-going torch to ignite the explosion of one condense ball in a series of explosions, the ultimate speed increasing of the later outgoing torch will soon reach a limit.  This limit is typically represented by a speed value of 1.347vc and is reached at n=8 in our calculation.  However, this speed limit would be broken upon the arrival of those out-going torches that have combined into more mass with stronger momentum.    Nevertheless, each new generation of explosion must tend to send out some singly cruising torches on the radial line to pioneer the outgoing material flow.   If at some point the speed limit has been broken, some ever newer outgoing torches down the road in the next few waves pioneering the flow will reach a new speed limit again.  This new speed limit can be broken upon the arrival of another crowd of some even more massive torches traveling at even higher speed.   In other words, because of the ever renewed speed limit, every so often during the course of the celestial materials moving away from the ASP, material flow must display some rippling pattern, presenting as layers in our observation.

5-IV Distribution of Radiating Energy

Just like mechanical energy, we will postulate that the radiating energy released per unit quantity of virginal material is finite and denote it with ΔER  .  Since the mass density of the virginal materials is the same in all layers of our onion model, we can say that the radiating energy released per unit surface area from any layer of the onion model is also the same.  Let this energy density released per unit surface area from any layer be denoted as ρR . When the same radiating energy intensity from the n-th spherical layer reaches its center, the energy intensity will be reduced to ρR ÷[(n-1)ΔR]2 , where ΔR is the thickness of each of the smaller concentric layers contained within the n-th layer. The total surface area of the inner surface of the n-th layer is 4π[(n-1)ΔR]2 . Therefore, the total radiating energy from the n-th layer received by an observer at the center is ρR.   So, regardless of the radius, the total energy that an observer receives from any single layer is the same, which is 4πρR..  If the observer is standing at a distance L away from the center, calculation shows that he would receive the same total energy from each single layer, which is, again, 4πρR..  However, he would also feel the heat from one side is more intense than from the other side.

The above deduction is for one layer.  Now we will analyze the resulting intensity of radiating energy from multiple layers and see what happens if all of them are accumulated at a single location.  Again, let's begin at the center of all concentric layers.

In Fig. 11, an observer will receive radiating energy through an aperture of finite angular size.  This aperture of finite angular size keeps the same constant ratio between an observed area of any layer and the entire surface area of the same spherical layer.   If the energy he receives from one whole layer is constant regardless of the radius of the layer, a finite aperture would enable him to receive the same finite portion of such energy from any layer, in spite of the numerical order number of that layer.  In other words, it does not matter whether the layer belongs to the n-th, or the (n+1)-th, or the (n+k)-th order, he receives the same amount of energy through this aperture from each layer.

The next reasonable assumption for us to make is that the intensity of the radiating energy released by each layer is proportional to the time rate that virginal material is converted, and such a time rate is proportional to the quantity of virginal materials remaining.

Since the number of layers can be of infinity and the number of layers hosting explosion increases with acceleration against time, we naturally wonder whether or not all intensity from all layers added together will also increase beyond any limits when the sum of them reaches the observer.  Calculations show us a negative answer.  This simply means that the total radiating energy received by an observer at the center of all concentric layers, i.e., the ASP, will not exceed a certain value, regardless of how quickly the explosion wave front progresses and how many and how fast the virginal layers are unlocking their energy, provided that the homogeneous mass density in space is kept constant everywhere.  Now, of course, we have come to a point to see how Olbers’ paradox cannot present itself as a refutation against Newton’s idea of infinite space for the universe.  Simply, in any conic section of finite angular area that infinitely penetrates deep into the space containing the virginal bodies, any depth (D in Fig. 11) from which the radiating energy is released must be finite.   No matter how fast the luminous depth can increase itself,  it can only always be a limited section in this bottomless cone.  Olbers must have infinite depth occupied by perpetually shining objects in the cone to refute Newton's idea.

If we place the observer at a location other than the ASP, his observations will lose the isotropic character.  However, with the increase of the order number of layers joining the releasing of energy, i.e., with the expansion of the fire region, his observations will gradually gain back their isotropic character.

All this tells us that:

(1)   No matter where an observer is located within the fire region, the radiating energy will reach him with a value bound by a certain upper and lower limit.

(2)   If he is placed closer to the concentric center, i.e., the ASP, the overall radiating energy will reach him with a lower value (bound by a finite bottom limit) but also with a more isotropic character.

5-V  More Conclusions

.

Besides the conclusion presented at the beginning of this article, we are going to add a few more in the following:

1.  An “Ever -Expanding” Universe

Since the explosion wave front keeps a faster and faster pace moving away from the ASP, it is possible that someday we can no longer detect any of those explosion that have been too remote from us but traveling at speed so extraordinarily high that it would be even beyond our comprehension.  When their radiating signal reaches us, the signal from any direction would have been too weak for us to observe.  Indeed, we cannot exclude that this has been happening to us now.

2.  Gas Presentation

Throughout our calculations, we have assumed complete collisions between matters.  Reality should deviate far from this orderly pattern, such as: (1) Some virginal bodies have not condensed enough when some torches hit. (2) By some random chance, some condensed materials just never get hit by anything that is mighty enough to ignite an explosion. (3) When a virginal ball does explode, part of it is pushed so far away that it is stripped of the chance to participate in any further reaction. These possibilities allow a great amount of virginal materials to remain intact and observable to us today.  With our knowledge today, we cannot identify this virginal materials as anything else but hydrogen.

3.   “Heat Death”

With the onion model, we can see that heat death inevitably happens locally.  Statistically it should have started from the center of the fire region, or the ASP, and slowly spreads out.  Although it takes a long time for energy at a location to be exhausted, any local amount of energy is finite, limited. Sooner or later, the infinite amount of time must deplete the finite amount of energy.  This phenomenon will eventually spread to our neighborhood, no matter where we are in relation to the ASP.  In other words, “heat death” is locally inevitable; however, it can never conquer the entire, limitless universe.   (Fig. 12 -a, -b; 13-a, -b, -c, -d, -e, -f, -g, -h,

-j)

4.  Cosmic background noise.

Generally, the releasing time and location of the radiating energy are locally random, but they are governed by such a big picture: the further away from the ASP such releasing occurs, the later is the releasing time.  As such, at any particular location where bundles of the propagating energy from various directions have reached, these bundles of energy would all superimpose on each other.  The intensity resulted by such superimposition is stronger at some locations than at some others.  Therefore, space in the university should be “sliced” by various temperatures into different zones; each zone about the same temperature may easily occupy a space dimension of light-years.  The same characteristics should also happen to cosmic background noise.  We can hardly rely on the earth’s traveling speed to compare between these zones. However, these zones do travel themselves and comb through our vicinity.

5. Solar System

Each explosion tossed out torches of various sizes, and these torches also randomly recombined in their flying, resulting in objects of various nature.  We should not be surprised that members of our solar system are the results of the reorganization and recombination of these torches.  Originated from different locations in the formidable space of the universe,  these members of different background and age joined each other by some random chance, organizing themselves into a stable heavenly family under the balance between gravity and rotation.  For a more detailed illustration on this point of view, please refer to the article Different Views on the Formation of Solar System, by Cameron Rebigsol. (Available soon)

6.  The Milky Way is highly likely locating near where the ASP is in history.

The symmetrical characters in our astronomical observation strongly suggest to us that the Milky Way, thus the solar system as well, is highly likely at a location near where can be described as the ASP in our onion model.  The distribution of age of heavenly objects and their movement pronouncedly suggest that we are at the center of all this symmetry.  If not, celestial object movement alone would present to us a different picture.  For explanation, in Fig. 14, an observer is assumed to be off the center by ¼ of the radius of the total fire region.

Glossary

ASP:  The acronym of Assumed Single Point.  This is a hypothetical point where the very first explosion was assumed to have taken place.

Centrifugal direction:  A direction pointing away from the ASP.

Centripetal direction:  A direction pointing towards the ASP.

Condensed ball: A body contains virginal materials that has gravitationally accreted, and is ready for explosion when certain conditions meet.

Mean-shell (only found in PDF copy): A shell that has concentrated all the materials of any particular layer in the onion model.  Each of the mean-shell geometrically divides each of such layers such that the portion of a layer outside of the mean-shell has the same equal volume as that of the portion of the same layer inside the mean-shell.

Onion model: Assuming that the virginal materials in the universe are homogeneously distributed all over, we can imagine the universe to have been consisted of numerous concentric spherical layers of the same materials.  All layers are of the same thickness and of the same mass density, wrapping around one geometrical point one layer after another in a pattern similar to an onion.  No actual physical existence of boundary can be found between the layers.  The devise of such boundary is only for the purpose of calculation.

Primary coordinate frame: An imaginary 3-D coordinate system named as x-y-z, with its origin O to be the same as the ASP.

Radial line: A ray that has the ASP as its starting point.

Tangential direction: A direction that is perpendicular to a radial line.

Virginal material: Material that has never undergone any explosion and thus its internal energy of high explosive power has never been tapped into.

Virginal object: An object that contains only the virginal material.

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Computer Data (Available only in PDF copy) and Charts

<<<<<<     >>>>>>

Chart 1 below is plotted by computer according to the above data. It shows the relationship between speed of celestial objects and their distance with respect to some geometrical point in the space, which is the ASP (Assumed Single Point) for our calculation based on the onion model.  While “n” stands for the order of layers in the onion model, it should equivalently suggests distance in space; V(nol) is the speed for the overall celestial objects in general.

Chart 2 below is plotted in large scale of order number n.  Each unit increment of  n (=1) signifies the distance  (measured from the ASP) upon which a speed limit is broken by the newly born torches whose birth is forced by the arrival of the crowd of more massive torches traveling at higher speed.  This chart shows the near perfect linear relationship between the speed and distance like what is shown in Chart 1for the overall celestial objects.

Back to Section 1

PDF copy is available     (Only PDF copy shows detailed calculation support)

For a more straight forward visualization on this subject, please also visit our video series of The ORIGIN OF THE UNIVERSE—ITS ETERNITY, beginning with the following link