Editing Stellar dynamics (section)

=== A Spherical-Cow Summary of Continuity Eq. in Collisional and Collisionless Processes ===

Having gone through the details of the rather complex interactions of particles in a gravitational system, it is always helpful to zoom out and extract some generic theme, at an affordable price of rigour, so carry on with a lighter load.

First important concept is "gravity balancing motion" near the perturber and for the background as a whole 
<math display="block"> \text{Perturber Virial} \approx {GM_\bullet \over s_\bullet} \approx V_\text{cir}^2 \approx \langle V \rangle^2 \approx \overline{\langle V^2 \rangle} \approx \sigma^2 \approx \left({R \over t_\text{ς}}\right)^2 \approx c_\text{ς}^2 \approx {G (N m) \over R} \approx \text{Background Virial}, </math> 
by consistently ''omitting'' all factors of unity <math> 4\pi </math>, <math>\pi</math>, <math> \ln \text{Λ} </math> etc for clarity, ''approximating'' the combined mass <math> M_\bullet + m \approx M_\bullet </math> and 
being ''ambiguous'' whether the ''geometry'' of the system is a thin/thick gas/stellar disk or a (non)-uniform stellar/dark sphere with or without a boundary, and about the ''subtle distinctions'' among the kinetic energies from the local [[Circular rotation speed]] <math> V_\text{cir}</math>, radial infall speed <math> \langle V \rangle </math>, globally isotropic or anisotropic random motion <math> \sigma </math> in one or three directions, or the (non)-uniform isotropic [[Sound speed]] <math> c_\text{ς} </math> to ''emphasize of the logic'' behind the order of magnitude of the friction time scale.

Second we can recap ''very loosely summarise'' the various processes so far of collisional and collisionless gas/star or dark matter by [[Spherical cow]] style ''Continuity Equation on any generic quantity Q'' of the system:
<math display="block"> {d Q\over dt} \approx {\pm Q \over  ({l \over c_\text{ς} }) }, ~\text{Q being mass M, energy E, momentum (M V), Phase density f, size R,  density} {N m \over {4\pi \over 3} R^3}...,  </math> 
where the <math>\pm </math> sign is generally negative except for the (accreting) mass M, and the [[Mean free path]] <math> l = c_\text{ς} t_\text{fric} </math> or the friction time <math> t_\text{fric} </math> can be due to direct molecular viscosity from a physical collision [[Cross section (physics)|Cross section]], or due to gravitational scattering (bending/focusing/[[Sling shot]]) of particles; generally the influenced area is the greatest of the competing processes of [[Bondi accretion]], [[Tidal disruption]], and [[Loss cone]] capture,
<math display="block"> s^2 \approx \max\left[\text{Bondi radius}~ s_\bullet, \text{Tidal radius}~s_\text{Hill}, \text{physical size}~ s_\text{Loss cone}\right]^2. </math>

E.g., in case Q is the perturber's mass <math> Q = M_\bullet </math>, then we can estimate the [[Dynamical friction]] time via the (gas/star) Accretion rate 
<math display="block"> \begin{align}\dot{M}_\bullet=& {M_\bullet \over t_\text{fric} }  \approx \int_0^{s^2} d(\text{area}) ~(\text{background mean flux}) \approx  s^2 (\rho c_\text{ς})  \\
\approx & \frac{\text{Perturber influenced cross section}~(s^2)}{\text{background system cross section}~(R^2) }  \times 
\frac{\text{background mass}~(N m)}{\text{crossing time}~t_\text{ς} \approx {R \over c_\text{ς}} \approx {1 \over \sqrt{G (N m) \over R^3}  \sim \sqrt{G \rho} \sim \kappa } } \\
\approx & {G M_\bullet \over G t_\text{ς} }  {G M_\bullet \over G (Nm) } \approx  (\rho c_\text{ς}) \left({G M_\bullet \over c_\text{ς}^2 }\right)^2  ,~~\text{if consider only gravitationally focusing,} \\
\approx & {M_\bullet \over N t_\text{ς} },~~\text{if for a light perturber} M_\bullet \rightarrow m = M_\odot \\
\rightarrow & 0, ~~\text{if practically collisionless}~~N \rightarrow \infty,
\end{align}</math> 
where we have applied the relations motion-balancing-gravity.

In the limit the perturber is just 1 of the N background particle, <math> M_\bullet \rightarrow m </math>, this friction time is identified with the (gravitational) [[Relaxation time]].  Again all [[Coulomb logarithm]] etc are suppressed without changing the estimations from these qualitative equations.

For the rest of Stellar dynamics, we will consistently work on ''precise'' calculations through primarily ''Worked Examples'', by neglecting gravitational friction and relaxation of the perturber, working in the limit <math> N \rightarrow \infty </math> as approximated true in most galaxies on the 14Gyrs Hubble time scale, even though this is sometimes violated for some clusters of stars or clusters of galaxies.of the cluster.<ref name=":2" />

A concise 1-page summary of some main equations in Stellar dynamics and [[Accretion disc]] physics are shown here, where one attempts to be more rigorous on the qualitative equations above.

[[File:GAPbrief.pdf|thumb|Stellar dynamics Key concepts and equations]]