Stellar dynamics

Stellar dynamics is the branch of astrophysics which describes in a statistical way the collective motions of stars subject to their mutual gravity. The essential difference from celestial mechanics is that the number of body <math> N \gg 10. </math>

thumb|Slingshot of a test body in a two-body potential

thumb|N-particles in quasi-periodic motion in the phase space (x, mv) of an essentially static potential

Typical galaxies have upwards of millions of macroscopic gravitating bodies and countless number of neutrinos and perhaps other dark microscopic bodies. Also each star contributes more or less equally to the total gravitational field, whereas in celestial mechanics the pull of a massive body dominates any satellite orbits.

Connection with fluid dynamics

Stellar dynamics also has connections to the field of plasma physics. The two fields underwent significant development during a similar time period in the early 20th century, and both borrow mathematical formalism originally developed in the field of fluid mechanics.

In accretion disks and stellar surfaces, the dense plasma or gas particles collide very frequently, and collisions result in equipartition and perhaps viscosity under magnetic field. We see various sizes for accretion disks and stellar atmosphere, both made of enormous number of microscopic particle mass,

<math> \sim (10^{-8}\text{pc}/500\text{km/s},1M_\odot/10^{55}=m_{p}) </math> at stellar surfaces,
<math>\sim (10^{-4}{\text{pc/10{\text{km/s,0.1M_{\odot }/10^{54}\sim m_{p}) </math> around Sun-like stars or km-sized stellar black holes,
<math> \sim (10^{-1}{\text{pc/100{\text{km/s,10M_{\odot }/10^{56}\sim m_{p}) </math> around million solar mass black holes (about AU-sized) in centres of galaxies.

The system crossing time scale is long in stellar dynamics, where it is handy to note that

<math> 1000\text{pc}/1\text{km/s} = 1000 \text{Myr} = \text{HubbleTime}/14. </math>

The long timescale means that, unlike gas particles in accretion disks, stars in galaxy disks very rarely see a collision in their stellar lifetime. However, galaxies collide occasionally in galaxy clusters, and stars have close encounters occasionally in star clusters.

As a rule of thumb, the typical scales concerned (see the Upper Portion of P.C.Budassi's Logarithmic Map of the Universe) are <math> (L/V,M/N) </math>

<math> \sim (\mathrm{10pc/10km/s},1000M_\odot/1000) </math> for M13 Star Cluster,
<math> \sim (\mathrm{100kpc/100km/s}, 10^{11}M_{\odot }/10^{11}) </math> for M31 Disk Galaxy,
<math> \sim (\mathrm{10Mpc/1000km/s},10^{14}M_{\odot }/10^{77}=m_\nu) </math> for neutrinos in the Bullet Clusters, which is a merging system of N = 1000 galaxies.

Connection with Kepler problem and 3-body problem

At a superficial level, all of stellar dynamics might be formulated as an N-body problem

by Newton's second law, where the equation of motion (EOM) for internal interactions of an isolated stellar system of N members can be written down as,

<math display="block">m_i\frac{d^{2} \mathbf{r_i{dt^{2 = \sum_{i=1 \atop i \ne j}^N \frac{G m_i m_j \left(\mathbf{r}_j - \mathbf{r}_i\right)}{\left\| \mathbf{r}_j - \mathbf{r}_i\right\|^3}. </math>

Here in the N-body system, any individual member, <math>m_i</math> is influenced by the gravitational potentials of the remaining <math>m_j</math> members.

In practice, except for in the highest performance computer simulations, it is not feasible to calculate rigorously the future of a large N system this way. Also this EOM gives very little intuition. Historically, the methods utilised in stellar dynamics originated from the fields of both classical mechanics and statistical mechanics. In essence, the fundamental problem of stellar dynamics is the N-body problem, where the N members refer to the members of a given stellar system. Given the large number of objects in a stellar system, stellar dynamics can address both the global, statistical properties of many orbits as well as the specific data on the positions and velocities of individual orbits. The gravitational potential, <math>\Phi</math>, of a system is related to the acceleration and the gravitational field, <math>\mathbf{g}</math> by:

<math display="block">\frac {d^{2}\mathbf {r_{i }{dt^{2}=\mathbf {\vec {g =-\nabla _{\mathbf {r_{i }\Phi (\mathbf {r_{i ),~~\Phi (\mathbf {r} _{i})=-\sum _{k=1 \atop k\neq i}^{N}{\frac {Gm_{k{\left\|\mathbf {r} _{i}-\mathbf {r} _{k}\right\|}, </math>

whereas the potential is related to a (smoothened) mass density, <math>\rho </math>, via the Poisson's equation in the integral form

<math display="block"> \Phi(\mathbf {r}) = - \int {G \rho(\mathbf{R}) d^3\mathbf{R} \over \left\|\mathbf {r}-\mathbf {R} \right\|} </math>

or the more common differential form

<math display="block">\nabla^2\Phi = 4\pi G \rho. </math>

An example of the Poisson Equation and escape speed in a uniform sphere

Consider an analytically smooth spherical potential

<math display="block"> \begin{align}

\Phi(r) & \equiv \left(-V_0^2\right) + \left[{r^2 -r_0^2 \over 2r_0^2}, ~~ 1 -{r_0 \over r} \right]_{\max} \!\!\!\! V_0^2 \equiv \Phi(r_0)-{V_e^2(r) \over 2}, ~~\Phi(r_0) = - V_0^2 , \\

\mathbf{g} &= -\mathbf{\nabla} \Phi(r) = -\Omega^2 r H(r_0 - r) - { G M_0 \over r^2}H(r-r_0), ~~\Omega={V_0 \over r_0}, ~~M_0 = {V_0^2 r_0 \over G},\end{align}</math>

where <math> V_e(r) </math> takes the meaning of the speed to "escape to the edge" <math> r_0</math>, and <math>\sqrt{2}V_0 </math> is the speed to "escape from the edge to infinity". The gravity is like the restoring force of harmonic oscillator inside the sphere, and Keplerian outside as described by the Heaviside functions.

We can fix the normalisation <math> V_0 </math> by computing the corresponding density using the spherical Poisson Equation

where the enclosed mass

<math display="block"> M(r) = {r^2 d\Phi \over G dr} = \int_0^{r} dr \int_0^{\pi} (r d\theta) \int_0^{2 \pi} (r \sin\theta d\varphi) \rho_0 H(r_0-r) = \left. M_0 x^3\right|_{x={r \over r_0.</math>

Hence the potential model corresponds to a uniform sphere of radius <math> r_0 </math>, total mass <math> M_0 </math> with

<math display="block"> {V_0 \over r_0} \equiv \sqrt{4\pi G \rho_0 \over 3} = \sqrt{G M_0 \over r_0^3}. </math>

Key concepts

While both the equations of motion and Poisson Equation can also take on non-spherical forms, depending on the coordinate system and the symmetry of the physical system, the essence is the same:

The motions of stars in a galaxy or in a globular cluster are principally determined by the average distribution of the other, distant stars. The infrequent stellar encounters involve processes such as relaxation, mass segregation, tidal forces, and dynamical friction that influence the trajectories of the system's members.

Relativistic Approximations

There are three related approximations made in the Newtonian EOM and Poisson Equation above.

SR and GR

Firstly above equations neglect relativistic corrections, which are of order of

<math display="block"> (v/c)^2 \ll 10^{-4} </math> as typical stellar 3-dimensional speed, <math> v \sim 3-3000 </math> km/s, is much below the speed of light.

Eddington Limit

Secondly non-gravitational force is typically negligible in stellar systems. For example, in the vicinity of a typical star the ratio of radiation-to-gravity force on a hydrogen atom or ion,

<math display="block"> Q^\text{Eddington} = { {\sigma_e \over 4\pi m_H c} {L\odot \over r^2} \over {G M_\odot \over r^2} } = {1 \over 30,000}, </math>

hence radiation force is negligible in general, except perhaps around a luminous O-type star of mass <math> 30M_\odot</math>, or around a black hole accreting gas at the Eddington limit so that its luminosity-to-mass ratio <math> L_\bullet / M_\bullet </math> is defined by <math> Q^\text{Eddington} =1 </math>.

Loss cone

Thirdly a star can be swallowed if coming within a few Schwarzschild radii of the black hole. This radius of Loss is given by <math display="block"> s \le s_\text{Loss} = \frac{6 G M_\bullet}{c^2} </math>

The loss cone can be visualised by considering infalling particles aiming to the black hole within a small solid angle (a cone in velocity).

These particle with small <math> \theta \ll 1</math> have small angular momentum per unit mass <math display="block"> J \equiv r v \sin\theta \le J_\text{loss} = \frac{4G M_\bullet}{c}.</math> Their small angular momentum (due to ) does not make a high enough barrier near <math> s_\text{Loss} </math> to force the particle to turn around.

The effective potential

<math display="block"> \Phi_\text{eff}(r) \equiv E- {\dot{r}^2 \over 2} = {J^2 \over 2r^2} + \Phi(r) , </math> is always positive infinity in Newtonian gravity. However, in GR, it

nosedives to minus infinity near <math> \frac{6 G M_\bullet}{c^2} </math> if <math> J \le \frac{4G M_\bullet}{c}. </math>

Sparing a rigorous GR treatment, one can verify this <math> s_\text{loss}, J_\text{loss} </math> by computing the last stable circular orbit, where the effective potential is at an inflection point <math> \Phi_\text{eff}(s_\text{loss})=\Phi'_\text{eff}(s_\text{loss})=0 </math> using an approximate classical potential of a Schwarzschild black hole

<math display="block"> \Phi(r) = - {(4G M_\bullet/c)^2 \over 2r^2} \left[1+{3 (6 G M_\bullet/c^2)^2 \over 8 r^2 }\right] - \frac{G M_\bullet}{r} \left[1 - {(6G M_\bullet/c^2)^2 \over r^2}\right]. </math>

Tidal disruption radius

A star can be tidally torn by a heavier black hole when coming within the so-called Hill's radius of the black hole, inside which a star's surface gravity yields to the tidal force from the black hole, i.e.,

<math display="block"> (1-1.5) \ge Q^\text{tide}

\equiv { G M_\odot /R_\odot^2 \over [G M_\bullet/s^2_\text{Hill} - G M_\bullet/(s_\text{Hill}+R_\odot)^2] }, ~~~

s_\text{Hill} \rightarrow R_\odot \left({ (2-3) GM_\bullet \over GM_\odot}\right)^{1 \over 3}, </math>

For typical black holes of <math> M_\bullet = (10^0-10^{8.5}) M_\odot</math> the destruction radius <math display="block"> \max[s_\text{Hill}, s_\text{Loss}] = 400R_\odot \max\left[\left({M_\bullet \over 3 \times 10^7 M_\odot}\right)^{1/3}, {M_\bullet \over 3 \times 10^7 M_\odot}\right] = (1-4000) R_\odot \ll 0.001 \mathrm{pc},</math> where 0.001pc is the stellar spacing in the densest stellar systems (e.g., the nuclear star cluster in the Milky Way centre). Hence (main sequence) stars are generally too compact internally and too far apart spaced to be disrupted by even the strongest black hole tides in galaxy or cluster environment.

Radius of sphere of influence

A particle of mass <math> m </math> with a relative speed V will be deflected when entering the (much larger) cross section <math> \pi s^2_\bullet </math> of a black hole. This so-called sphere of influence is loosely defined by, up to a Q-like fudge factor <math> \sqrt{\ln\Lambda} </math>,

<math display="block"> 1 \sim \sqrt{\ln\Lambda} \equiv \frac{V^2/2}{G (M_\bullet + m)/s_\bullet},</math>

hence for a Sun-like star we have,

<math display="block"> s_\bullet = {G (M_\bullet +M_\odot) \sqrt{\ln\Lambda} \over V^2/2 } \approx {M_\bullet \over M_\odot} {V^2_\odot \over V^2} R_\odot >[s_\text{Hill}, s_\text{Loss}]_{max} = (1-4000) R_\odot, </math>

i.e., stars will neither be tidally disrupted nor physically hit/swallowed in a typical encounter with the black hole thanks to the high surface escape speed <math display="block"> V_\odot =\sqrt{2 G M_\odot/R_\odot} = 615\mathrm{km/s} </math> from any solar mass star, comparable to the internal speed between galaxies in the Bullet Cluster of galaxies, and greater than the typical internal speed <math display="block"> V \sim \sqrt{2 G (N M_\odot)/R} \ll \mathrm{300 km/s} </math> inside all star clusters and in galaxies.

Connections between star loss cone and gravitational gas accretion physics

First consider a heavy black hole of mass <math> M_\bullet</math> is moving through a dissipational gas of (rescaled) thermal sound speed <math> \text{ς'} </math> and density <math> \rho_\text{gas} </math>, then every gas particle of mass m will likely transfer its relative momentum <math> m V_\bullet </math> to the BH when coming within a cross-section of radius <math display="block"> s_\bullet \equiv {(G M_\bullet+ G m) \sqrt{\ln\Lambda} \over (V_\bullet^2+\text{ς'}^2)/2},</math>

In a time scale <math> t_\text{fric} </math> that the black hole loses half of its streaming velocity, its mass may double by Bondi accretion, a process of capturing most of gas particles that enter its sphere of influence <math> s_\bullet </math>, dissipate kinetic energy by gas collisions and fall in the black hole. The gas capture rate is

<math display="block"> {M_\bullet \over t_\text{Bondi}^{gas} }

=\sqrt{\text{ς'}^2 + V_\bullet^2}(\pi s_\bullet^2) \rho_\text{gas}

=4\pi \rho_\text{gas} \left[ {(G M_\bullet)^2 \over (\text{ς'}^2 + V_\bullet^2)^{3 \over 2} } \right] \ln\Lambda, ~~ \text{ς'} \equiv \sigma \sqrt{1+ \gamma^3 \over 2 (9/8)^{2/3 \approx [\text{ς} , \gamma \sigma]_\text{max}, </math>

where the polytropic index <math> \gamma </math> is the sound speed in units of velocity dispersion squared, and the rescaled sound speed <math>\text{ς'} </math> allows us to match the Bondi spherical accretion rate, <math> \dot{M}_\bullet \approx \pi \rho_\text{gas} \text{ς} \left[ {(G M_\bullet) \over \text{ς}^2} \right]^2 </math> for the adiabatic gas <math>\gamma=5/3</math>, compared to <math> \dot{M}_\bullet \approx 4\pi \rho_\text{gas} \text{ς} \left[ {(G M_\bullet) \over \text{ς}^2} \right]^2 </math> of the isothermal case <math> \gamma=1 </math>.

Coming back to star tidal disruption and star capture by a (moving) black hole, setting <math> \ln \Lambda =1 </math>, we could summarise the BH's growth rate from gas and stars,

<math> {M_\bullet \over t_\text{Bondi}^{gas} } + {M_\bullet \over t_\text{loss}^{*} } </math> with,

<math display="block"> \dot{M}_\bullet

=\sqrt{\text{ς'}^2 + V_\bullet^2} m n (\pi s_\bullet^2, \pi s_\text{Hill}^2 , \pi s_\text{Loss}^2)_\text{max}, ~~s_\bullet \approx {(G M_\bullet+ G m) \over (V_\bullet^2+\text{ς'}^2)/2}, </math>

because the black hole consumes a fractional/most part of star/gas particles passing its sphere of influence.

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.,

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

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

-{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 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 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>.

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. This means that two stars need to come within a separation,

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 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, 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, 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. Both approaches separate themselves from the kinetic theory of gases by introducing long-range forces to study the long term evolution of a many particle system. In addition to the Vlasov equation, the concept of Landau damping in plasmas was applied to gravitational systems by Donald Lynden-Bell to describe the effects of damping in spherical stellar systems.

A nice property of f(t,x,v) is that many other dynamical quantities can be formed by its moments, e.g., the total mass, local density, pressure, and mean velocity. Applying the collisionless Boltzmann equation, these moments are then related by various forms of continuity equations, of which most notable are the Jeans equations and Virial theorem.

Probability-weighted moments and hydrostatic equilibrium

Jeans computed the weighted velocity of the Boltzmann Equation after integrating over velocity space <math display="block">

{1 \over \rho_p } \int\! \left\{\mathbf{v}_p {d [f_p m_p]\over dt} - \langle{\mathbf{v\rangle_p {d [f_p m_p]\over dt}\right\} d^3\mathbf{v} = 0, </math> and obtain the Momentum (Jeans) Eqs. of a <math>^p</math>opulation (e.g., gas, stars, dark matter):

<math display="block">\begin{align}

\overbrace{ \left({\partial \over \partial t}+\sum_{j=1}^{3} \langle{v_j^p}\rangle {\partial \over \partial x_j}\right) \langle{v_i^p}\rangle}^{\dot{\langle{v}\rangle}_i^p} &

\underbrace{=}_{EoM}

\overbrace{-\partial \Phi(t,\mathbf{x})\over \partial x_i}^{g_i\sim O(-GM/R^2)} ~~

\underbrace{-}^\text{pressure}_\text{balance}~~\sum_{j=1}^{3} {\partial \over \rho^p \partial x_j}

\overbrace{[\underbrace{\rho^p(t,\mathbf{x})}_{\int_\infty\!\!\!\!m_p f_p d^3\mathbf{v \underbrace{\sigma_{ji}^p(t,\mathbf{x})}_{O(c_s^2)}]}^{\int\limits_\infty\!\! d\mathbf{v}^3 (\mathbf{v}_j-\langle{v}\rangle^p_j) (\mathbf{v}_i-\langle{v}\rangle^p_i)m_pf_p }

- {\underbrace{\langle{v_i^p}\rangle \overbrace{[\dot{m}_p/m_p]}^{1/t|^\text{fric}_{\text{visc}~m_p=M_\text{gas_\text{snow.plough, \\

0& = -{\partial \Phi(t,\mathbf{x})\over \partial x_i} -{\partial (n \sigma^2 ) \over n \partial x_i}, ~~\text{hydrostatic isotropic velocity, no flow and friction }.\end{align}

</math>

The general version of Jeans equation, involving (3 x 3) velocity moments is cumbersome.

It only becomes useful or solvable if we could drop some of these moments, especially drop the off-diagonal cross terms for systems of high symmetry, and also drop net rotation or net inflow speed everywhere.

The isotropic version is also called

Hydrostatic equilibrium equation where balancing pressure gradient with gravity; the isotropic version works for axisymmetric disks as well, after replacing the derivative dr with vertical coordinate dz. It means that we could measure the gravity (of dark matter) by observing the gradients of the velocity dispersion and the number density of stars.

Applications and examples

Stellar dynamics is primarily used to study the mass distributions within stellar systems and galaxies. Early examples of applying stellar dynamics to clusters include Albert Einstein's 1921 paper applying the virial theorem to spherical star clusters and Fritz Zwicky's 1933 paper applying the virial theorem specifically to the Coma Cluster, which was one of the original harbingers of the idea of dark matter in the universe. The Jeans equations have been used to understand different observational data of stellar motions in the Milky Way galaxy. For example, Jan Oort utilized the Jeans equations to determine the average matter density in the vicinity of the solar neighborhood, whereas the concept of asymmetric drift came from studying the Jeans equations in cylindrical coordinates.

Stellar dynamics also provides insight into the structure of galaxy formation and evolution. Dynamical models and observations are used to study the triaxial structure of elliptical galaxies and suggest that prominent spiral galaxies are created from galaxy mergers.