Imagine a huge number of small discrete particles forming a spherical cloud.

My assumptions:-

  1. They are stationary at $t=0$.
  2. The particles are isolated, i.e. there is no external influence due to any other bodies.
  3. The mass density/distribution is the same everywhere.
  4. The particles interact only through their mutual gravitational attractions.

This scenario is quite analogous to chopping off the Earth in tiny pieces and removing every heavenly body near Earth.

Now my concern is about the way how the particles will move and interact and eventually collapse after $t=0$.

I tried to work this out, but I am stuck because of the complexity(or rather the beauty) of this system. This is what I am talking about. If we find the acceleration due to gravitational forces on a spherical layer at a distance $r$ from the centre of mass of the system at $t=0$, then it turns out to be,

$$ a=\frac{G \left(\frac{Mx^3}{R^3}\right) }{x^2} $$

where $a$ is the acceleration of the layer, $M$ is the total mass of all the particles, $R$ is the radius of the system of particles and $G$ is the gravitational constant.

Clearly from the above equation, we can see that,

$$a\propto x$$

This means that the farther is the layer, the greater is its acceleration. And that means that the layers which are far away will collapse faster and soon enough they will get closer to the centre than the layers which were initially beneath them. So now the farthest layers is changed. And thus the new farthest layer will again accelerate faster and again the farthest layer would change. And as you can see this is perpetual and complex.

How do I analyse this system, both qualitatively and quantitatively?

Source of inspiration :- https://brilliant.org/problems/a-mechanics-problem-by-jason-hu-2/

  • $\begingroup$ Does R mean the distance of the farthest particle from center? $\endgroup$ – Ibraheem Moosa Nov 16 at 8:40
  • $\begingroup$ arxiv.org/abs/1101.0601 $\endgroup$ – safesphere Nov 16 at 8:43
  • $\begingroup$ @IbraheemMoosa Yes $\endgroup$ – Dhruv Maroo Nov 16 at 16:01
  • $\begingroup$ Search term: "mean-field approximation." Useful when you expand to the case where the initial velocities are random/thermal rather than zero. $\endgroup$ – rob 4 hours ago

Think of one particle of a layer. It is moving in simple harmonic motion around the center, since

$$a\propto x$$

Now think of the whole layer. The whole layer is moving together. If you can analyze how one particle of this layer moves than you know how the whole layer moves.

If you understand how each layer moves then you understand how the whole sphere of particles are moving.


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