We now begin exploration of the mathematics of our cosmological onion model.

Since there is substance, there must be the concept of mass density. Lacking valid observational figures of the mass density of the universe, we use 16 hydrogen molecules per cubic meter. (We will later see that the exact figure of the mass density of the universe in our model is not of critical concern.) This is about 3 times the upper limit of the critical mass density of another contemporary theory (Note 1 - Appendix VII-4). According to it, if the universe ever had a mass density above 5 hydrogen molecules per cubic meter, the “big crunch” would happen. However, in the onion model, it is shown that the big crunch has no chance of happening even with a figure 3 times as high.

Let us have a hollow shell made up of homogeneous materials, like that shown as C in Fig. 5-1. The shell is isolated from any massive body by a significant distance. An object like A placed inside that hollow shell will then not experience gravitational influence from anything, including the shell. Indeed, the thickness of the shell C is irrelevant; it can range from being very thin to infinitely thick. The situation will change if there is also another object like B placed inside the shell, as shown in Fig. 5-2. The two objects, A and B, will exert a gravitational influence on each other, but the shell, regardless of its thickness, has no gravitational influence upon either.

The large collection of hydrogen of the spherical region of 0.1 light year radius in Fig. 4, and any hydrogen molecule in the thinner region will behave in exactly the same manner as the two objects A and B inside the hollow shell C in Fig. 5-2. They will interact through gravitation while the entire space outside of the thinner region will be comparable to hollow shell of infinite thickness. By allowing enough time, the large collection of hydrogen of the denser region will eventually pull the single hydrogen particle into itself.

According to Newton's law of universal gravitation, the calculated gravitational force between the denser sphere and the hydrogen molecule is 4.96x10^-47  NT, and the corresponding acceleration for that molecule moving toward the sphere is 0.49x10^-20 m/sec. Both values are extremely small, but not zero. By allowing enough time, the particle will ultimately be forced to relocate.

If a molecule in the thinner region can be forced to move towards the denser spherical region, we can imagine that molecules within the denser region will also constantly experience a gravitational force pulling them towards the mass center of the region. In effect, the material body of this region as a whole will always have a tendency to gravitationally contract. The contraction of the entire material body consistently forces the neighboring particles to come closer and closer to each other, yielding more space for more material particles to enter the thinner region from areas farther away.

We will refer to this denser region as a condensed ball.

The radiation of heat produced due to the contraction of the condensed ball also contributes to the movement of particles in the neighboring region. Coupled with the unidirectional gravitational force, more particles are recruited and thus join the condensed ball, further escalating the mass quantity as well as its density. This, in turn, increases its gravitational force upon the neighboring region, and thus attracts more particles. The cycle will repeat itself with an increased amount of substance, strength and frequency. With each contraction of the condensed ball, as well as the bombardment of particles from some distance away, more and more heat is generated.

Given enough time, sufficient (but not all) heat energy converted from the gravitational contractions would have escaped from the condensed ball allowing it to build up its mass quantity beyond any limit. Let us take a look at what would happen if the condensed ball were to have a mass density comparable to that of our Sun.

Let   be the overall uniform mass density of the universe,
be the mass density of the condensed ball, (For purpose of illustration only, this figure is numerically assumed to be 7.8x10³kg/m³, i.e., the Sun's mass density.),
be the radius of the hollow spherical dome that is formed because of the gradual depletion of material caused by gravitational influence from the condensed ball.; In areas beyond what  can describe, the mass density retains the uniform density of  (Fig. 6).
be the radius of the condensed ball.

Starting from the Onion Model, the universe is assumed consisted of an indefinite number of concentric layers of the same substance. Each layer has the same mass density as the others and is of the same thickness, . The outer radius of the n-th layer, or the inner radius of the (n+1)-th layer, in space is R_u,n= n( ) , where n   can be any positive integer. (Fig. 7). When the substance in the n-th layer is recruited by the condensed ball, the hollow dome will expand its wall up to the radius of .
With the above hypothesis, we can calculate the total potential energy available from all substances up to the n-th layer as they all accreted to the central area of the “onion”.  Our calculations show [For detailed calculations, see Appendix VII-1(a)]:

 

Under both situations, whatever amount of heat was produced by gravitational accretion, would be accumulated within the condensed ball. Then, it should be reasonable to assume all gravitationally generated heat would be utilized to heat up the condensed ball.

Due to the poor thermal conductivity of hydrogen, the heat yielded through the accretion of each layer could not have been instantly and evenly spread over the entire condensed ball to allow a homogenous temperature. Instead, only part of the condensed ball, particularly the surface part immediately receiving accretion, would be mixed with the oncoming layer to reach a mutual temperature. It should also be reasonable to assume that equal amounts of substance from the condensed ball would mix with equal amounts of substance from the oncoming layers during each accretion.

With this manner of accretion and heating in mind, and with the potential energy formula deduced in Appendix VII-1(a), we can proceed to calculate the radius of the spherical volume that would have yielded so much mass as to have contributed enough gravitational energy to have triggered a nuclear fusion reaction. Calculations shown in Appendix VII-1(b) lead us to the formula for the radius of such a volume, which is
 , where  is the specific heat of hydrogen, T_n  is the mutual temperature of the amount of substance which includes (and only includes) the substance of the n-th oncoming layer and the equal amount of substance from the condensed ball.

With the assumed =16  hydrogen molecules /m³ , and =7.8x10 kg/m³  , the anticipated radius is calculated as
R_u,n = 3.58x10¹⁸ m.
 

The spherical volume represented by this radius should be comparable to the volume occupied by Milky way, which is a disk of radius 3x10²⁰ m .  This calculation only serves to tell us that gravitational accretion of enough hydrogen may make it possible to ignite a fusion reaction of the material. We must not forget that we have simplified the hypothetical calculation by using a higher figure of mass density in the space, and in 100% conversion from potential energy to heat. If we consider a lower mass density for the space and less than 100% conversion between the energy, R_u,n should be larger than what we calculated.

Now let’s see if the speed of the torches produced through an explosion was high enough for them to overcome the gravitational pull of the condensed ball and be ejected out to space.

The total mass contained in the spherical region represented by R_u,n   is

 This figure is comparable to the Sun's mass of 0.97x10³⁰kg .  Using the Sun's density as a yardstick, we can estimate the radius r_B of the condensed ball. The Sun’s mass density is  7.84kg/m³. Thus,.
 

With the mass and size obtained above for the condensed ball, the escape velocity v_B at the surface is calculated as

Hydrogen fusion reactions can provide huge amounts of kinetic energy such that each quantity of mass equivalent to a proton can attain a velocity in the order of 10⁴km/sec   This is a magnitude much higher than the escape velocity required at the surface of the condensed ball as we have just calculated. This means that the gravitational force generated by the condensed ball is not enough to stop the torches from being ejected by the explosion. It takes a stronger gravitational force to increase the escape velocity and lock up the exploding matter. Stronger gravitational force in turn needs more substance. The requirement of more substance accretion, however, will in turn end up with an ignition of an earlier explosion. In short, the local conditions have destined explosions to occur before the gravitational force could ever concentrate enough strength to prevent the post-explosion matter from permanently flying away.

With the Onion Model, we can also mathematically speculate about the time needed to trigger the fusion reaction after accretion had begun. Such a calculation is exhibited in Appendix VII-1(c).  The formula there shows that as n , the number of onion layers, approaches infinity, the time needed is.
By taking =16  hydrogen molecules /m³  , we have t=2.88x10¹⁷sec=9.13x10⁹years.
The significance of this number regarding time is as follows:

(1 The equation tells us that 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 in Appendix VII-1(b) point out that the amount of mass, hence the size of the hollow dome, needed to trigger the first big explosion is 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, then it implies that there is not enough condensed material to start an explosion. Then the condensed material will have chance to recruit more substance; more substance together will exert a stronger gravitational influence onto those masses which have just been stripped from the wall of the hollow dome of an ever-increasing radius. The stronger force will give the falling masses a higher initial speed and a higher acceleration towards the condensed ball. The result is that the amount of time needed for accretion will be the same. Although we have dropped some terms to simplify calculations, the order of magnitude of the approximations would not deviate much.

(2)According to contemporary theory, the universe has existed for a time period between 14 and 20 billion years. However, using the assumed mass density of 16 hydrogen molecules per cubic meter, our calculations show that accretion alone may have taken almost 10 billion years. Please note, the assumed mass density has been increased to 3 times as much as the upper limit of the critical mass density which is believed to cause the so-called Big Crunch in some contemporary theories. If we use a lower mass density for our calculation, the accretion time will be even further extended.

After the first big explosion, there would be a long time during which substances continued to fly towards the ASP. A wide area around the ASP would have become a huge burning field. The continuous movement of materials towards the ASP would help shorten the distance between each ignition during this time period.

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