Editing Stellar dynamics (section)

== Gravitational encounters and relaxation ==
Stars in a stellar system will influence each other's trajectories due to strong and weak gravitational encounters. An encounter between two stars is defined to be strong/weak if their mutual potential energy at the closest passage is comparable/minuscule to their initial kinetic energy. Strong encounters are rare, and they are typically only considered important in dense stellar systems, e.g., a passing star can be sling-shot out by binary stars in the core of a globular cluster.<ref name=":2">{{Cite book | title=Galaxies in the Universe|last1=Sparke|first1=Linda|author-link=Linda Sparke|last2=Gallagher|first2=John | publisher=Cambridge |year=2007|isbn=978-0521855938|location=New York|pages=131}}</ref> This means that two stars need to come within a separation,
<math display="block"> s_* = {G M_\odot + G M_\odot \over  V^2/2} = { 2 \over 1.5}{G M_\odot \over \text{ς}^2} = {3.29 R \over N-1},</math>
where we used the Virial Theorem, "mutual potential energy balances twice kinetic energy on average", i.e., "the pairwise potential energy per star balances with twice kinetic energy associated with the sound speed in three directions", 
<math display="block"> 1 \sim Q^\text{virial} \equiv {\overbrace{2K}^{(N M_\odot) V^2} \over |W|} = {N M_\odot\text{ς}^2 + N M_\odot\text{ς}^2 + N M_\odot\text{ς}^2
\over {N (N-1) \over 2} {G M_\odot^2 \over  R_{pair} } },</math>
where the factor <math>N (N-1)/2 </math> is the number of handshakes between a pair of stars without double-counting, the mean pair separation <math>  R_\text{pair} ={\pi^2 \over 24} R \approx 0.411234 R</math> is only about 40\% of the radius of the uniform sphere.
Note also the similarity of the <math>  Q^\text{virial} \leftarrow \rightarrow 
\sqrt{\ln\Lambda}. </math>

=== Mean free path ===
The mean free path of strong encounters in a typically <math> (N-1) = 4.19 n R^3 \gg 100 </math> stellar system is then
<math display="block">  l_\text{strong} = {1 \over  (\pi s_*^2)n } \approx {(N-1) \over 8.117} R \gg R ,</math>
i.e., it takes about <math> 0.123 N </math> radius crossings for a typical star to come within a cross-section <math> \pi s_*^2 </math> to be deflected from its path completely.  Hence the mean free time of a strong encounter is much longer than the crossing time <math> R/V </math>.

=== Weak encounters ===
Weak encounters have a more profound effect on the evolution of a stellar system over the course of many passages. The effects of gravitational encounters can be studied with the concept of [[Relaxation (physics)|relaxation]] time.  A simple example illustrating relaxation is two-body relaxation, where a star's orbit is altered due to the gravitational interaction with another star.

Initially, the subject star travels along an orbit with initial velocity, <math>\mathbf{v}</math>, that is perpendicular to the [[impact parameter]], the distance of closest approach, to the field star whose gravitational field will affect the original orbit. Using Newton's laws, the change in the subject star's velocity, <math>\delta \mathbf{v}</math>, is approximately equal to the acceleration at the impact parameter, multiplied by the time duration of the acceleration.

The relaxation time can be thought as the time it takes for <math>\delta \mathbf{v}</math> to equal <math>\mathbf{v}</math>, or the time it takes for the small deviations in velocity to equal the star's initial velocity. The number of "half-diameter" crossings for an average star to relax in a stellar system of <math>N</math> objects is approximately 
<math display="block">{t_\text{relax} \over t_\text{ς}} = N^{\text{relax}} 
\backsimeq \frac{0.123(N-1)}{\ln (N-1)} \gg 1</math>
from a more rigorous calculation than the above mean free time estimates for strong deflection.

The answer makes sense because there is no relaxation for a single body or 2-body system. A better approximation of the ratio of timescales is <math> \left.\frac{N'}{\ln \sqrt{1+ N'^2}}\right|_{N'=0.123(N-2)}</math>, hence the relaxation time for 3-body, 4-body, 5-body, 7-body, 10-body, ..., 42-body, 72-body, 140-body, 210-body, 550-body are about 16, 8, 6, 4, 3, ..., 3, 4, 6, 8, 16 crossings.  There is no relaxation for an isolated binary, and the relaxation is the fastest for a 16-body system; it takes about 2.5 crossings for orbits to scatter each other.  A system with <math> N \sim  10^2 - 10^{10} </math> have much smoother potential, typically takes <math> \sim \ln N' \approx (2-20) </math> weak encounters to build a strong deflection to change orbital energy significantly.

=== Relation between friction and relaxation ===
Clearly that the dynamical friction of a black hole is much faster than the relaxation time by roughly a factor <math> M_\odot / M_\bullet </math>, but these two are very similar for a cluster of black holes, 
<math display="block">  N^\text{fric} ={t_\text{fric} \over t_\text{ς}} \rightarrow  {t_\text{relax} \over t_\text{ς}} = N^\text{relax} \sim {(N-1) \over 10-100},  ~ \text{when}~ {M_\bullet \rightarrow m \leftarrow M_\odot}.  </math>

For a star cluster or galaxy cluster with, say,  <math> N=10^3, ~ R=\mathrm{1 pc-10^5 pc}, ~ V=\mathrm{1 km/s - 10^3 km/s }</math>,  we have <math> t_{\text{relax}} \sim 100 t_\text{ς}\approx 100 \mathrm{Myr} -10 \mathrm{Gyr} </math>. Hence encounters of members in these stellar or galaxy clusters are significant during the typical 10 Gyr lifetime.

On the other hand, typical galaxy with, say, <math> N=10^6 - 10^{11} </math> stars, would have a crossing time <math> t_\text{ς} \sim {1 \mathrm{kpc} - 100 \mathrm{kpc} \over 1  \mathrm{km/s} - 100 \mathrm{km/s}} \sim 100 \mathrm{Myr} </math> and their relaxation time is much longer than the age of the Universe.  This justifies modelling galaxy potentials with mathematically smooth functions, neglecting two-body encounters throughout the lifetime of typical galaxies.  And inside such a typical galaxy the dynamical friction and accretion on stellar black holes over a 10-Gyr Hubble time change the black hole's velocity and mass by only an insignificant fraction
<math display="block"> \Delta \sim {M_\bullet \over 0.1 N M_\odot} {t \over t_\text{ς}}  \le  {M_\bullet \over 0.1\% N M_\odot}  </math>

if the black hole makes up less than 0.1% of the total galaxy mass <math> N M_\odot \sim 10^{6-11}M_\odot</math>.  Especially when <math> M_\bullet \sim M_\odot </math>, we see that a typical star never experiences an encounter, hence stays on its orbit in a smooth galaxy potential.

The dynamical friction or relaxation time identifies collisionless vs. collisional particle systems. Dynamics on timescales much less than the relaxation time is effectively collisionless because typical star will deviate from its initial orbit size by a tiny fraction <math> t/t_{\text{relax}} \ll 1 </math>.  They are also identified as systems where subject stars interact with a smooth gravitational potential as opposed to the sum of point-mass potentials.  The accumulated effects of two-body relaxation in a galaxy can lead to what is known as [[Mass segregation (astronomy)|mass segregation]], where more massive stars gather near the center of clusters, while the less massive ones are pushed towards the outer parts of the cluster.

=== 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]]