Editing Stellar dynamics (section)

=== 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>.