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The Dynamics of
an Exploding System of Particles
Subject to Gravitational Attraction


The notion of the nature of the Universe held by modern civilization has undergone some remarkable changes in the past century. Up until 1925 it was universally believed that our Milky Way galaxy was the entire Universe. In 1925 Edwin Hubble and his associates published articles which established that the observed nebula were outside of the Milky Way and were galaxies on their own. For the full story see The Birth of Modern Cosmology.

In 1915 Albert Einstein published his General Theory of Relativity that involved tensor equations for the Universe. He included a term which implied the Universe is stable in scale.

Later spectral measurement established that contrary to Einstein the galaxies were moving away from each other at high speeds. Astronomers traced back the paths of the galaxies and concluded that they all originated in some cosmic Big Bang.

Later astronomers discovered that theoretically there are agglomerations of mass with gravitation so intense that not even light can escape from them. They came to be called Black Holes.

This meant that The Big Bang could not have been an explosion within the Universe but had to have been an inflation in the space of the Universe. The world before The Big Bang would have been the ultimate Black Hole.

To visualize what this means consider a two dimensional world located on a spherical surface. The masses of this world interact within the this spherical surface but not through the extra dimension involved in the curvature of the world. But through some unknown mechanism the radius of the sphere can jncrease without having to overcome the gravitational attraction between the masses of the world. This seems to be the wildest speculation.

More recently astronomers have discovered evidence that the rate of expansion of the Universe is increasing. As a preliminary to investigating the dynamics of such an expansion here the dynamics of particles from an explosion which attract each other is investigated.

The gravitational attraction between galaxies slows their rate of expansion. The dynamics of this is of interest. What is given below is an analysis of the dynamics of two equal particles blown apart at time t=0.


Let m be the mass of one of the particle and x is its distance from the initial explosion. The separation distance between the two particles is 2x. The equation of the dynamics one particle is then

m(d²x/dt²) = −Gm²/(2x)² = −(G/4)m²/x²
which reduces to
(d²x/dt²) = −(G/4)/x²

Multiply both sides of the above equation by (dx/dt) to obtain

(dx/dt)(d²x/dt²) = −(G/4);[(dx/dt)/x²]
which is equivalent to
d/dt[½(dx/dt)²] = (Gm/4)d(1/x)/dt

Integrate both sides of the above equation from 0 to t to obtain

½v² − ½v0² = (Gm/4)(1/x − 1/x0)
and thus
v² = v0² − (Gm/2)(1/x0) + (Gm/2)(1/x)
and of course
v(t) = [ v0² − (Gm/2)(1/x0) + (Gm/2)(1/x)]1/2

This equation gives the shape of the relationship between velocity and the distance from the site of the explosion, as shown below.

It also gives the limit of v as t->∞ providing that (v0² − (Gm/2)(1/x0)>0; i.e.,

limt->∞ = limx->∞ = [v0² − (Gm/2)(1/x0]½

That is to say, if the explosion gives the particles sufficient energy they will never return to the site of the explosion and travel away from each other forever.

Let (v0² − (Gm/2)(1/x0) be denoted as E. The dynamics of the location x of one particle are given by the differential equation

(dx/dt) = [E + (Gm/2)(1/x)]½
or in differential form
dx/[E + (Gm/2)(1/x)]½ = dt

The integration of the LHS of the above equation from x0 to x and the RHS from 0 to t gives

F(x) − F(x0) = t


F(x) = ∫0x[E + (Gm/2)(1/z)]½dz

Here is t plotted versus F(x):

and here x versus F−1(t)

(To be continued.)


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