Editing Stellar dynamics (section)

== Gravitational dynamical friction ==
Consider the case that a heavy black hole of mass <math> M_\bullet</math> moves relative to a background of stars in random motion in 
a cluster of total mass <math> (N  M_\odot)</math> with a mean number density <math display="block"> n \sim (N-1)/(4\pi R^3/3) </math> within a typical size <math> R </math>.

Intuition says that gravity causes the light bodies to accelerate and gain momentum and kinetic energy (see slingshot effect). By conservation of energy and momentum, we may conclude that the heavier body will be slowed by an amount to compensate. Since there is a loss of momentum and kinetic energy for the body under consideration, the effect is called dynamical friction.

After certain time of relaxations the heavy black hole's kinetic energy should be in equal partition with the less-massive background objects. The slow-down of the black hole can be described as
<math display="block"> -{M_\bullet \dot{V}_\bullet }   = {M_\bullet V_\bullet \over t_\text{fric}^\text{star} } , </math>
where <math> t_\text{fric}^\text{star}  </math> is called a dynamical friction time.

=== 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.

=== More rigorous formulation of dynamical friction ===

The full [[Dynamical friction#Chandrasekhar dynamical friction formula|Chandrasekhar dynamical friction formula]] for the change in velocity of the object involves integrating over the phase space density of the field of matter and is far from transparent.

It reads as
<math display="block">{M_\bullet d (\mathbf{V}_\bullet) \over dt} = -{M_\bullet \mathbf{V}_\bullet \over t_\text{fric}^\text{star} } = - {m \mathbf{V}_\bullet ~ n(\mathbf{x}) d\mathbf{x}^3 \over dt} \ln\Lambda_\text{lag}, </math>
where 
<math display="block"> ~~ n(\mathbf{x}) dx^3 = dt V_{\bullet} (\pi s_\bullet^2)  n(\mathbf{x}) = dt n(\mathbf{x}) |V_{\bullet}| \pi \left[{G(m+M_\bullet) \over |V_{\bullet}|^2/2}\right]^2 </math>
is the number of particles in an infinitesimal cylindrical volume of length <math>|V_{\bullet} dt| </math> and a cross-section <math> \pi s_\bullet^2 </math> within the black hole's sphere of influence.

Like the "Couloumb logarithm" <math> \ln\Lambda </math> factors in the contribution of distant background particles, here the factor <math> \ln(\Lambda_\text{lag}) </math> also 
factors in the probability of finding a background slower-than-BH particle to contribute to the drag.  The more particles are overtaken by the BH, the more particles drag the BH, and the greater is <math> \ln(\Lambda_\text{beaten}) </math>. Also the bigger the system, the greater is <math> \ln\Lambda </math>.

A background of elementary (gas or dark) particles can also induce dynamical friction, which scales with the mass density of the surrounding medium, <math> m~ n</math>; the lower particle mass m is compensated by the higher number density n.  The more massive the object, the more matter will be pulled into the wake.

Summing up the gravitational drag of both collisional gas and collisionless stars, we have  
<math display="block"> M_\bullet {d ( \mathbf{V}_{\bullet}) \over M_\bullet dt} = 
-  4\pi \left[{GM_\bullet \over |V_{\bullet}|}\right]^2 \mathbf{\hat{V}}_{\bullet} (\rho_\text{gas} \ln\Lambda_\text{lag}^{gas} + m n_\text{*} \ln\Lambda_\text{lag}^{*}).~~</math>
Here the "lagging-behind" fraction for gas <ref>{{cite journal |last1=Ostriker |first1=Eva |title=Dynamical Friction in a Gaseous Medium |journal=The Astrophysical Journal |year=1999 |volume=513 |issue=1 |page=252 |doi=10.1086/306858 |arxiv=astro-ph/9810324 |bibcode=1999ApJ...513..252O |s2cid=16138105 |url=https://ui.adsabs.harvard.edu/abs/1999ApJ...513..252O/abstract}}</ref> and for stars are given by 
<math display="block"> \begin{align}
\ln\Lambda_\text{lag}^{gas}(u) & =  \ln~ { \left[{1+u\over \lambda}\right]^{1 \over 2} \left[{|1-u|\over \lambda}\right]^{H[u-\lambda-1]-H[1-\lambda-u] \over 2}  \over  \exp{ [u+\lambda,1]_\min^2 - [u-\lambda,1]_\min^2  \over 4 \lambda} }, 
\\
& \approx \ln \left[ {\sqrt{ (u^3 - 1)^2 + \lambda^3 } + u^3 -1 \over  \sqrt{1+\lambda^3}-1 } \right]^{1 \over 3}, ~~ u \equiv {|V_\bullet| t \over \text{ς'} t}, ~~ \lambda \equiv({s_\bullet \over \text{ς'}t}) \\
{\ln\Lambda_\text{lag}^{*} \over \ln\Lambda} & \equiv \int_{0}^{|m V_{\bullet}|} \!\!\!\! { (4\pi p^2 dp) e^{-{p^2 \over 2 (m \sigma)^2}}\over (\sqrt{2\pi} m \sigma)^3 }  \left.\right|_{p=m |v|} \approx { |\mathbf{V}_{\bullet}|^3 \over |\mathbf{V}_{\bullet}|^3 + 3.45 \sigma^3 }, \\
\ln\Lambda &= \int{d\mathbf{x_1}^3 ~2 Heaviside[{n(\mathbf{x_1}) \over n(\mathbf{x})} - 1 - {M_\bullet \over N M_\odot} ] \over (s_\bullet^2 + |\mathbf{x_1}-\mathbf{x}|^2)^{3 \over 2} } \approx \ln\sqrt{1+\left({0.123 N M_\odot \over M_\bullet}\right)^2 }, \end{align}</math> 
where we have further assumed that the BH starts to move from time <math> t=0</math>; the gas is isothermal with sound speed <math> \text{ς} </math>; the background stars have of (mass) density <math> m n(\mathbf{x}) </math> in a [[Maxwell distribution]] of momentum <math>p=m v </math> with a [[Gaussian distribution]] velocity spread <math> \sigma </math> (called velocity dispersion, typically <math> \sigma \le \text{ς} </math>).

Interestingly, the  <math> G^2 (m+M_\bullet) (m n(\mathbf{x})) </math> dependence suggests that dynamical friction is from the gravitational pull of by the wake, which is induced by the [[gravitational focusing]] of the massive body in its two-body encounters with background objects.

We see the force is also proportional to the inverse square of the velocity at the high end, hence the fractional rate of energy loss drops rapidly at high velocities.
Dynamical friction is, therefore, unimportant for objects that move relativistically, such as photons. This can be rationalized by realizing that the faster the object moves through the media, the less time there is for a wake to build up behind it.  Friction tends to be the highest at the sound barrier, where <math> \ln\Lambda_\text{lag}^{gas}\left.\right|_{u=1} =\ln {\text{ς'}t \over s_\bullet } </math>.