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Let's ignore the dark matter legend and stay with Keplerian physics.

Assuming that there is a cloud with $N$ stationary particles with the same size uniformly distributed in a sphere and they condense to form a galaxy.

$$N>>10^{\text{many}}$$

Some particles come to the center. Some will escape. Is there any estimation that how much percentage of particles remain in the galaxy and how much will escape to the infinite space?

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To quote Binney and Tremaine Galactic Dynamics, 2nd et p. 556:

From time to time an encounter gives a star enough energy to escape from the stellar system. Thus there is a slow but irreversible leakage of stars from the system, so stellar systems gradually evolve towards a final state consisting of only two stars in a Keplerian orbit, all the others having escaped to infinity.

So the fraction remaining is $2/N$. This assumes random motions, which is fairly plausible.

The time to evaporation is on the order of $t_{evap}\approx \frac{14 N}{\log(N)}t_{crossing}$ where $t_{crossing}$ is the typical time to cross the cloud of particles, $\langle r\rangle /\langle v\rangle$.

For actual galaxy models one has to take collision cross sections with stars and the central black hole into account, I think about 10% of the stars tend to accrete.

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  • $\begingroup$ The quoted claim is strange. How can the authors know that all gravitational particle systems end in that kind of state? Why couldn't there be more pairs of stars in Keplerian orbits, or even more complex motion of the system where it does not evaporate all to infinity? Are they just extrapolating based on computer simulations? $\endgroup$ – Ján Lalinský Oct 21 '18 at 23:33
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    $\begingroup$ The claim is somewhat statistical. It follows from that the measure of bounded solutions to 3-body, 4-body, and n-body problems seem to have measure zero. So while a collapse to a set of neat stable pairs is possible, the probability is negligible. This is a mathematical result; simulations will typically show extremely long lived transients. $\endgroup$ – Anders Sandberg Oct 22 '18 at 13:40

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