Editing Stellar dynamics (section)

=== Relation between friction and relaxation ===
Clearly that the dynamical friction of a black hole is much faster than the relaxation time by roughly a factor <math> M_\odot / M_\bullet </math>, but these two are very similar for a cluster of black holes, 
<math display="block">  N^\text{fric} ={t_\text{fric} \over t_\text{ς}} \rightarrow  {t_\text{relax} \over t_\text{ς}} = N^\text{relax} \sim {(N-1) \over 10-100},  ~ \text{when}~ {M_\bullet \rightarrow m \leftarrow M_\odot}.  </math>

For a star cluster or galaxy cluster with, say,  <math> N=10^3, ~ R=\mathrm{1 pc-10^5 pc}, ~ V=\mathrm{1 km/s - 10^3 km/s }</math>,  we have <math> t_{\text{relax}} \sim 100 t_\text{ς}\approx 100 \mathrm{Myr} -10 \mathrm{Gyr} </math>. Hence encounters of members in these stellar or galaxy clusters are significant during the typical 10 Gyr lifetime.

On the other hand, typical galaxy with, say, <math> N=10^6 - 10^{11} </math> stars, would have a crossing time <math> t_\text{ς} \sim {1 \mathrm{kpc} - 100 \mathrm{kpc} \over 1  \mathrm{km/s} - 100 \mathrm{km/s}} \sim 100 \mathrm{Myr} </math> and their relaxation time is much longer than the age of the Universe.  This justifies modelling galaxy potentials with mathematically smooth functions, neglecting two-body encounters throughout the lifetime of typical galaxies.  And inside such a typical galaxy the dynamical friction and accretion on stellar black holes over a 10-Gyr Hubble time change the black hole's velocity and mass by only an insignificant fraction
<math display="block"> \Delta \sim {M_\bullet \over 0.1 N M_\odot} {t \over t_\text{ς}}  \le  {M_\bullet \over 0.1\% N M_\odot}  </math>

if the black hole makes up less than 0.1% of the total galaxy mass <math> N M_\odot \sim 10^{6-11}M_\odot</math>.  Especially when <math> M_\bullet \sim M_\odot </math>, we see that a typical star never experiences an encounter, hence stays on its orbit in a smooth galaxy potential.

The dynamical friction or relaxation time identifies collisionless vs. collisional particle systems. Dynamics on timescales much less than the relaxation time is effectively collisionless because typical star will deviate from its initial orbit size by a tiny fraction <math> t/t_{\text{relax}} \ll 1 </math>.  They are also identified as systems where subject stars interact with a smooth gravitational potential as opposed to the sum of point-mass potentials.  The accumulated effects of two-body relaxation in a galaxy can lead to what is known as [[Mass segregation (astronomy)|mass segregation]], where more massive stars gather near the center of clusters, while the less massive ones are pushed towards the outer parts of the cluster.