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of a Central Gravitational Attraction |
The dynamics of a body falling toward a mass M a distance r away from it is given by the equation
where G is the gravitational constant.
Suppose the body is initially at rest at a distance of r_{0}. If the above equation is multiplied by v=(dr/dt) and integrated, taking into account the initial conditions the result is
If this equation is divided by r_{0} on the left and equivalently muliplied by (r_{0})^{1/2}/(r_{0})^{3/2}) on the right and y substituted for r/r_{0} the result is
A further substitution of 1/y-1=z and y=1/(z+1) gives
Now let z^{1/2}=q, and hence z=q². The above equation then takes the form
Indefinite integration of this equation gives
where C is an arbitrary constant.
The variable r ranged for r_{0} to 0 so y ranged from 1 to 0. The variable z=1/y-1 then ranged from 0 to ∞ and likewise for q=z^{1/2}.
For an integration with respect to q from q=0 to q=∞ the first term, q/(q²+1), vanishes at both the upper and lower limits. The second term, tan^{-1}(q) has the value π/2 for q=∞ and 0 for q=0. The value of the integral is −π/2 because of the inversion of the limits in going from y to q. Let t_{f} be the time at which r goes to 0. Then
For a comparison consider a particle moving under a constant acceleration g. In time t it travels a distance ½gt². Thus the time t_{c} required to travel a distance r_{0} under constant acceleration is
The value of g is GM/r_{0}² so the above expression reduces to
This leads to the interesting result that
That is to say, the time which it takes a particle to reach the center of the gravitational attraction is 78.54% of the time it would take to reach it under constant acceleration equal to its initial acceleration.
If the mass is distributed uniformly throughout a spherical cloud the value of M depends upon r_{0}. In fact
where ρ is the mass density.
When this expression is substituted for M in the equation for t_{f} the result is
Thus t_{f} is independent of r_{0}. This means all elements of the spherical cloud collapse to the center at exactly the same time! Of course this would be prevented by the build up of material and pressure in the process of collapse.
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