Editing Stellar dynamics (section)

=== Dynamical friction time vs Crossing time in a virialised system ===
Consider a Mach-1 BH, which travels initially at the sound speed <math> \text{ς} = V_0 </math>, hence its Bondi radius <math> s_\bullet </math> satisfies 
<math display="block"> {GM_\bullet \sqrt{\ln\Lambda} \over s_\bullet} = V_0^2 = \text{ς}^2 = { 0.4053 G M_\odot (N-1) \over R}, </math>
where 
the sound speed is <math>  \text{ς} = \sqrt{  4 G M_\odot (N-1) \over \pi^2 R} </math> 
with the prefactor <math> {4 \over \pi^2} \approx {4 \over 10}=0.4</math> fixed by the fact that for a uniform spherical cluster of the mass density <math> \rho = n M_\odot \approx {M_\odot (N-1) \over 4.19 R^3} </math>, half of a circular period is the time for "sound" to make a oneway crossing in its longest dimension, i.e., 
<math display="block">
2t_{\text{ς}} \equiv 2t_{\text{cross}} \equiv {2R \over \text{ς}} = \pi \sqrt{R^3 \over G M_\odot (N-1)} \approx (0.4244 G \rho)^{-1/2}. </math>
It is customary to call the "half-diameter" crossing time <math> t_{\text{cross}} </math> the dynamical time scale.

Assume the BH stops after traveling a length of  <math>l_\text{fric} \equiv \text{ς} t_\text{fric} </math> with its momentum <math> M_\bullet V_0=M_\bullet \text{ς}</math> deposited to <math> {M_\bullet \over M_\odot} </math> stars in its path over <math>l_\text{fric}/(2R)</math> crossings, then 
the number of stars deflected by the BH's Bondi cross section per "diameter" crossing time is
<math display="block">  
N^\text{defl} = { ({M_\bullet \over M_\odot}) } {2R \over l_\text{fric}}  = 
N {\pi s_\bullet^2 \over \pi R^2} = N \left({M_\bullet \over  0.4053 M_\odot N}\right)^2 \ln\Lambda. </math>

More generally, the Equation of Motion of the BH at a general velocity <math> \mathbf{V}_\bullet </math> in the potential <math> \Phi </math> of a sea of stars can be written as  
<math display="block">  
 -{d\over dt} (M_\bullet V_\bullet) - M_\bullet \nabla \Phi  \equiv {(M_\bullet V_\bullet) \over t_\text{fric}} = 
\overbrace{ N \pi s_\bullet^2 \over \pi R^2}^{N^\text{defl}} {(M_\odot V_\bullet) \over 2t_\text{ς}} = { 8 \ln\Lambda' \over N t_\text{ς}} M_\bullet V_\bullet, </math>
<math>{\pi^2 \over 8} \approx 1 </math> and the Coulomb logarithm modifying factor <math> {\ln\Lambda' \over \ln\Lambda} \equiv \left[{\pi^2 \over 8}\right]^2 \left[(1+ {V_\bullet^2 \over \text{ς'}^2})\right]^{-2} (1+{M_\odot \over M_\bullet}) \le \left[{\text{ς'} \over V_\bullet}\right]^4 \le 1 </math> discounts friction on a supersonic moving BH with mass <math> M_\bullet \ge M_\odot </math>.  As a rule of thumb, it takes about a sound crossing <math> t_\text{ς'} </math> time to "sink" subsonic BHs, from the edge to the centre without overshooting, if they weigh more than 1/8th of the total cluster mass. Lighter and faster holes can stay afloat much longer.