Editing Dynamical friction (section)

==Chandrasekhar dynamical friction formula==
The full 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. The Chandrasekhar dynamical friction formula reads as
<math display="block">\frac{d\mathbf{v}_M}{dt} = -16 \pi^2 (\ln \Lambda) G^2 m (M+m) \frac{1}{v_M^3}\int_0^{v_M}v^2 f(v)  d v \mathbf{v}_M</math>
where
* <math>G</math> is the [[gravitational constant]]
* <math>M</math> is the mass under consideration
* <math>m</math> is the mass of each star in the star distribution
* <math> {v_M} </math> is the velocity of the object under consideration, in a frame where the center of gravity of the matter field is initially at rest
* <math> \mbox{ln}(\Lambda) </math> is the "[[Coulomb collision|Coulomb logarithm]]"
* <math>f(v)</math> is the number density distribution of the stars

The result of the equation is the gravitational acceleration produced on the object under consideration by the stars or celestial bodies, as acceleration is the ratio of velocity and time.

===Maxwell's distribution===

A commonly used special case is where there is a uniform density in the field of matter, with matter particles significantly lighter than the major particle under consideration i.e., <math>M\gg m</math> and with a [[Maxwell–Boltzmann distribution|Maxwellian distribution]] for the velocity of matter particles i.e.,
<math display="block">f(v) = \frac{N}{(2\pi\sigma^2)^{3/2}}e^{-\frac{v^2}{2\sigma^2}}</math>
where <math>N</math> is the total number of stars and <math>\sigma</math> is the dispersion. In this case, the dynamical friction formula is as follows:<ref>{{Citation | last = Merritt | first = David | author-link = David Merritt | title = Dynamics and Evolution of Galactic Nuclei | publisher = [[Princeton University Press]] | date = 2013 | bibcode = 2013degn.book.....M | isbn = 9781400846122 | url = https://openlibrary.org/works/OL16802359W/Dynamics_and_Evolution_of_Galactic_Nuclei }}</ref>

<math display="block"> \frac{d\mathbf{v}_M}{dt} = -\frac{4\pi \ln (\Lambda) G^2  \rho M}{v_M^3}\left[\mathrm{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2}\right]\mathbf{v}_M</math>
where
* <math> X = v_M/(\sqrt{2} \sigma) </math> is the ratio of the velocity of the object under consideration to the modal velocity of the Maxwellian distribution. 
* <math> \mathrm{erf}(X) </math> is the [[error function]].
* <math> \rho= mN </math> is the density of the matter field.

In general, a simplified equation for the force from dynamical friction has the form

<math display="block">F_\text{dyn} \approx C \frac{G^2 M^2 \rho}{v^2_M}</math>

where the [[dimensionless]] numerical factor <math> C </math> depends on how <math>v_M</math> compares to the velocity dispersion of the surrounding matter.<ref>{{Citation
  | last1 = Carroll | first1 = Bradley W. 
  | last2 = Ostlie | first2 = Dale A. 
  | title = An Introduction to Modern Astrophysics
  | publisher = [[Weber State University]]
  | date = 1996
  | bibcode = 1996ima..book.....C 
 | isbn = 0-201-54730-9 }}</ref>
But note that this simplified expression diverges when <math> v_M \to 0 </math>; caution should therefore be exercised when using it.