Editing Stellar dynamics (section)

== Connections between star loss cone and gravitational gas accretion physics ==
First consider a heavy black hole of mass <math> M_\bullet</math> is moving through a dissipational gas of (rescaled) thermal sound speed <math> \text{ς'} </math> and density <math> \rho_\text{gas} </math>, then every gas particle of mass m will likely transfer its relative momentum <math> m V_\bullet </math> to the BH when coming within a cross-section of radius <math display="block"> s_\bullet \equiv {(G M_\bullet+ G m) \sqrt{\ln\Lambda} \over (V_\bullet^2+\text{ς'}^2)/2},</math> 
In a time scale <math> t_\text{fric} </math> that the black hole loses half of its streaming velocity, its mass may double by Bondi accretion, a process of capturing most of gas particles that enter its sphere of influence <math> s_\bullet </math>, dissipate kinetic energy by gas collisions and fall in the black hole.  The gas capture rate is  
<math display="block"> {M_\bullet \over t_\text{Bondi}^{gas} }
=\sqrt{\text{ς'}^2 + V_\bullet^2}(\pi s_\bullet^2)  \rho_\text{gas}  
=4\pi \rho_\text{gas} \left[ {(G M_\bullet)^2 \over (\text{ς'}^2 + V_\bullet^2)^{3 \over 2} } \right] \ln\Lambda, ~~  \text{ς'} \equiv \sigma \sqrt{1+ \gamma^3 \over 2 (9/8)^{2/3}} \approx [\text{ς} , \gamma \sigma]_\text{max},  </math> 
where the polytropic index <math> \gamma </math> is the sound speed in units of velocity dispersion squared, and the rescaled sound speed <math>\text{ς'} </math> allows us to match the Bondi spherical accretion rate, <math> \dot{M}_\bullet \approx \pi \rho_\text{gas} \text{ς} \left[ {(G M_\bullet) \over \text{ς}^2} \right]^2 </math> for the adiabatic gas <math>\gamma=5/3</math>, compared to <math> \dot{M}_\bullet \approx 4\pi \rho_\text{gas} \text{ς} \left[ {(G M_\bullet) \over \text{ς}^2} \right]^2 </math> of the isothermal case <math> \gamma=1 </math>.

Coming back to star tidal disruption and star capture by a (moving) black hole, setting <math> \ln \Lambda =1 </math>, we could summarise the BH's growth rate from gas and stars,   
<math> {M_\bullet \over t_\text{Bondi}^{gas} } + {M_\bullet \over t_\text{loss}^{*} } </math> with,  
<math display="block"> \dot{M}_\bullet 
=\sqrt{\text{ς'}^2 + V_\bullet^2} m n (\pi s_\bullet^2, \pi s_\text{Hill}^2 , \pi s_\text{Loss}^2)_\text{max}, ~~s_\bullet \approx {(G M_\bullet+ G m) \over (V_\bullet^2+\text{ς'}^2)/2}, </math> 
because the black hole consumes a fractional/most part of star/gas particles passing its sphere of influence.