Editing Stellar dynamics (section)

== Radius of sphere of influence ==
A particle of mass <math> m </math> with a relative speed V will be deflected when entering the (much larger) cross section <math> \pi s^2_\bullet </math> of a black hole. This so-called sphere of influence is loosely defined by, up to a Q-like fudge factor <math> \sqrt{\ln\Lambda} </math>, 
<math display="block"> 1 \sim \sqrt{\ln\Lambda} \equiv \frac{V^2/2}{G (M_\bullet + m)/s_\bullet},</math>
hence for a Sun-like star we have, 
<math display="block"> s_\bullet = {G (M_\bullet +M_\odot) \sqrt{\ln\Lambda} \over V^2/2 } \approx {M_\bullet \over M_\odot} {V^2_\odot \over V^2} R_\odot  >[s_\text{Hill}, s_\text{Loss}]_{max} = (1-4000) R_\odot, </math>
i.e., stars will neither be tidally disrupted nor physically hit/swallowed in a typical encounter with the black hole thanks to the high surface escape speed <math display="block"> V_\odot =\sqrt{2 G M_\odot/R_\odot} = 615\mathrm{km/s} </math> from any solar mass star, comparable to the internal speed between galaxies in the Bullet Cluster of galaxies, and greater than the typical internal speed <math display="block"> V \sim \sqrt{2 G (N M_\odot)/R} \ll \mathrm{300 km/s} </math> inside all star clusters and in galaxies.