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 ${\displaystyle N\gg 10.}$

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.^[1]

Connection with fluid dynamics

Stellar dynamics also has connections to the field of plasma physics.^[2] 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, ${\displaystyle (L/V,M/N)}$

• ${\displaystyle \sim (10^{-8}{\text{pc))/500{\text{km/s)),1M_{\odot }/10^{55}=m_{p})}$ at stellar surfaces,
• ${\displaystyle \sim (10^{-4}{\text{pc))/10{\text{km/s)),0.1M_{\odot }/10^{54}\sim m_{p})}$ around Sun-like stars or km-sized stellar black holes,
• ${\displaystyle \sim (10^{-1}{\text{pc))/100{\text{km/s)),10M_{\odot }/10^{56}\sim m_{p})}$ 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

${\displaystyle 1000{\text{pc))/1{\text{km/s))=1000{\text{Myr))={\text{HubbleTime))/14.}$

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 ${\displaystyle (L/V,M/N)}$

• ${\displaystyle \sim (\mathrm {10pc/10km/s} ,1000M_{\odot }/1000)}$ for M13 Star Cluster,
• ${\displaystyle \sim (\mathrm {100kpc/100km/s} ,10^{11}M_{\odot }/10^{11})}$ for M31 Disk Galaxy,
• ${\displaystyle \sim (\mathrm {10Mpc/1000km/s} ,10^{14}M_{\odot }/10^{77}=m_{\nu })}$ 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,

$${\displaystyle m_{i}{\frac {d^{2}\mathbf {r_{i)) }{dt^{2))}=\sum _{i=1 \atop i\neq j}^{N}{\frac {Gm_{i}m_{j}\left(\mathbf {r} _{j}-\mathbf {r} _{i}\right)}{\left\|\mathbf {r} _{j}-\mathbf {r} _{i}\right\|^{3))}.}$$

${\displaystyle m_{i))$

${\displaystyle m_{j))$

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.^[1]

Concept of a gravitational potential field

Stellar dynamics involves determining the gravitational potential of a substantial number of stars. The stars can be modeled as point masses whose orbits are determined by the combined interactions with each other. Typically, these point masses represent stars in a variety of clusters or galaxies, such as a Galaxy cluster, or a Globular cluster. Without getting a system's gravitational potential by adding all of the point-mass potentials in the system at every second, stellar dynamicists develop potential models that can accurately model the system while remaining computationally inexpensive.^[3] The gravitational potential, ${\displaystyle \Phi }$, of a system is related to the acceleration and the gravitational field, ${\displaystyle \mathbf {g} }$ by:

$${\displaystyle {\frac {d^{2}\mathbf {r_{i)) }{dt^{2))))=\mathbf {\vec {g)) =-\nabla _{\mathbf {r_{i)) }\Phi (\mathbf {r_{i)) ),~~\Phi (\mathbf {r} _{i})=-\sum _{k=1 \atop k\neq i}^{N}((\frac {Gm_{k)){\left\|\mathbf {r} _{i}-\mathbf {r} _{k}\right\|)),}$$

${\displaystyle \rho }$

$${\displaystyle \Phi (\mathbf {r} )=-\int {G\rho (\mathbf {R} )d^{3}\mathbf {R} \over \left\|\mathbf {r} -\mathbf {R} \right\|))$$

$${\displaystyle \nabla ^{2}\Phi =4\pi G\rho .}$$

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

Consider an analytically smooth spherical potential

$${\displaystyle {\begin{aligned}\Phi (r)&\equiv \left(-V_{0}^{2}\right)+\left[{r^{2}-r_{0}^{2} \over 2r_{0}^{2)),~~1-{r_{0} \over r}\right]_{\max }\!\!\!\!V_{0}^{2}\equiv \Phi (r_{0})-{V_{e}^{2}(r) \over 2},~~\Phi (r_{0})=-V_{0}^{2},\\\mathbf {g} &=-\mathbf {\nabla } \Phi (r)=-\Omega ^{2}rH(r_{0}-r)-{GM_{0} \over r^{2))H(r-r_{0}),~~\Omega ={V_{0} \over r_{0)),~~M_{0}={V_{0}^{2}r_{0} \over G},\end{aligned))}$$

${\displaystyle V_{e}(r)}$

${\displaystyle r_{0))$

${\displaystyle {\sqrt {2))V_{0))$

We can fix the normalisation ${\displaystyle V_{0))$ by computing the corresponding density using the spherical Poisson Equation

$${\displaystyle G\rho ={d \over 4\pi r^{2}dr}{r^{2}d\Phi \over dr}={d(GM) \over 4\pi r^{2}dr}={3V_{0}^{2} \over 4\pi r_{0}^{2))H(r_{0}-r),}$$

$${\displaystyle M(r)={r^{2}d\Phi \over Gdr}=\int _{0}^{r}dr\int _{0}^{\pi }(rd\theta )\int _{0}^{2\pi }(r\sin \theta d\varphi )\rho _{0}H(r_{0}-r)=\left.M_{0}x^{3}\right|_{x={r \over r_{0))}.}$$

Hence the potential model corresponds to a uniform sphere of radius ${\displaystyle r_{0))$, total mass ${\displaystyle M_{0))$ with

$${\displaystyle {V_{0} \over r_{0))\equiv {\sqrt {4\pi G\rho _{0} \over 3))={\sqrt {GM_{0} \over r_{0}^{3))}.}$$

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.^[4]

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

$${\displaystyle (v/c)^{2}\ll 10^{-4))$$

${\displaystyle v\sim 3-3000}$

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,

$${\displaystyle Q^{\text{Eddington))=((\sigma _{e} \over 4\pi m_{H}c}{L\odot \over r^{2)) \over {GM_{\odot } \over r^{2))}={1 \over 30,000},}$$

${\displaystyle 30M_{\odot ))$

${\displaystyle L_{\bullet }/M_{\bullet ))$

${\displaystyle Q^{\text{Eddington))=1}$

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

$${\displaystyle s\leq s_{\text{Loss))={\frac {6GM_{\bullet )){c^{2))))$$

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 ${\displaystyle \theta \ll 1}$ have small angular momentum per unit mass

$${\displaystyle J\equiv rv\sin \theta \leq J_{\text{loss))={\frac {4GM_{\bullet )){c)).}$$

${\displaystyle s_{\text{Loss))}$

The effective potential

$${\displaystyle \Phi _{\text{eff))(r)\equiv E-((\dot {r))^{2} \over 2}={J^{2} \over 2r^{2))+\Phi (r),}$$

${\displaystyle {\frac {6GM_{\bullet )){c^{2))))$

${\displaystyle J\leq {\frac {4GM_{\bullet )){c)).}$

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

$${\displaystyle \Phi (r)=-{(4GM_{\bullet }/c)^{2} \over 2r^{2))\left[1+{3(6GM_{\bullet }/c^{2})^{2} \over 8r^{2))\right]-{\frac {GM_{\bullet )){r))\left[1-{(6GM_{\bullet }/c^{2})^{2} \over r^{2))\right].}$$

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,^[5] i.e.,

$${\displaystyle (1-1.5)\geq Q^{\text{tide))\equiv {GM_{\odot }/R_{\odot }^{2} \over [GM_{\bullet }/s_{\text{Hill))^{2}-GM_{\bullet }/(s_{\text{Hill))+R_{\odot })^{2}]},~~~s_{\text{Hill))\rightarrow R_{\odot }\left({(2-3)GM_{\bullet } \over GM_{\odot ))\right)^{1 \over 3},}$$

For typical black holes of ${\displaystyle M_{\bullet }=(10^{0}-10^{8.5})M_{\odot ))$ the destruction radius

$${\displaystyle \max[s_{\text{Hill)),s_{\text{Loss))]=400R_{\odot }\max \left[\left({M_{\bullet } \over 3\times 10^{7}M_{\odot ))\right)^{1/3},{M_{\bullet } \over 3\times 10^{7}M_{\odot ))\right]=(1-4000)R_{\odot }\ll 0.001\mathrm {pc} ,}$$

Radius of sphere of influence

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

$${\displaystyle 1\sim {\sqrt {\ln \Lambda ))\equiv {\frac {V^{2}/2}{G(M_{\bullet }+m)/s_{\bullet ))},}$$

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

$${\displaystyle V_{\odot }={\sqrt {2GM_{\odot }/R_{\odot ))}=615\mathrm {km/s} }$$

$${\displaystyle V\sim {\sqrt {2G(NM_{\odot })/R))\ll \mathrm {300km/s} }$$

Connections between star loss cone and gravitational gas accretion physics

First consider a heavy black hole of mass ${\displaystyle M_{\bullet ))$ is moving through a dissipational gas of (rescaled) thermal sound speed ${\displaystyle {\text{ς'))}$ and density ${\displaystyle \rho _{\text{gas))}$, then every gas particle of mass m will likely transfer its relative momentum ${\displaystyle mV_{\bullet ))$ to the BH when coming within a cross-section of radius

$${\displaystyle s_{\bullet }\equiv {(GM_{\bullet }+Gm){\sqrt {\ln \Lambda )) \over (V_{\bullet }^{2}+{\text{ς'))^{2})/2},}$$

${\displaystyle t_{\text{fric))}$

${\displaystyle s_{\bullet ))$

$${\displaystyle {M_{\bullet } \over t_{\text{Bondi))^{gas))={\sqrt ((\text{ς'))^{2}+V_{\bullet }^{2))}(\pi s_{\bullet }^{2})\rho _{\text{gas))=4\pi \rho _{\text{gas))\left[{(GM_{\bullet })^{2} \over ({\text{ς'))^{2}+V_{\bullet }^{2})^{3 \over 2))\right]\ln \Lambda ,~~{\text{ς'))\equiv \sigma {\sqrt {1+\gamma ^{3} \over 2(9/8)^{2/3))}\approx [{\text{ς)),\gamma \sigma ]_{\text{max)),}$$

${\displaystyle \gamma }$

${\displaystyle {\text{ς'))}$

${\displaystyle {\dot {M))_{\bullet }\approx \pi \rho _{\text{gas)){\text{ς))\left[{(GM_{\bullet }) \over {\text{ς))^{2))\right]^{2))$

${\displaystyle \gamma =5/3}$

${\displaystyle {\dot {M))_{\bullet }\approx 4\pi \rho _{\text{gas)){\text{ς))\left[{(GM_{\bullet }) \over {\text{ς))^{2))\right]^{2))$

${\displaystyle \gamma =1}$

Coming back to star tidal disruption and star capture by a (moving) black hole, setting ${\displaystyle \ln \Lambda =1}$, we could summarise the BH's growth rate from gas and stars, ${\displaystyle {M_{\bullet } \over t_{\text{Bondi))^{gas))+{M_{\bullet } \over t_{\text{loss))^{*))}$ with,

$${\displaystyle {\dot {M))_{\bullet }={\sqrt ((\text{ς'))^{2}+V_{\bullet }^{2))}mn(\pi s_{\bullet }^{2},\pi s_{\text{Hill))^{2},\pi s_{\text{Loss))^{2})_{\text{max)),~~s_{\bullet }\approx {(GM_{\bullet }+Gm) \over (V_{\bullet }^{2}+{\text{ς'))^{2})/2},}$$

Gravitational dynamical friction

Consider the case that a heavy black hole of mass ${\displaystyle M_{\bullet ))$ moves relative to a background of stars in random motion in a cluster of total mass ${\displaystyle (NM_{\odot })}$ with a mean number density

$${\displaystyle n\sim (N-1)/(4\pi R^{3}/3)}$$

${\displaystyle R}$

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

$${\displaystyle -{M_{\bullet }{\dot {V))_{\bullet ))={M_{\bullet }V_{\bullet } \over t_{\text{fric))^{\text{star))},}$$

${\displaystyle t_{\text{fric))^{\text{star))}$

Dynamical friction time vs Crossing time in a virialised system

Consider a Mach-1 BH, which travels initially at the sound speed ${\displaystyle {\text{ς))=V_{0))$, hence its Bondi radius ${\displaystyle s_{\bullet ))$ satisfies

$${\displaystyle {GM_{\bullet }{\sqrt {\ln \Lambda )) \over s_{\bullet ))=V_{0}^{2}={\text{ς))^{2}={0.4053GM_{\odot }(N-1) \over R},}$$

${\displaystyle {\text{ς))={\sqrt {4GM_{\odot }(N-1) \over \pi ^{2}R))}$

${\displaystyle {4 \over \pi ^{2))\approx {4 \over 10}=0.4}$

${\displaystyle \rho =nM_{\odot }\approx {M_{\odot }(N-1) \over 4.19R^{3))}$

$${\displaystyle 2t_{\text{ς))\equiv 2t_{\text{cross))\equiv {2R \over {\text{ς))}=\pi {\sqrt {R^{3} \over GM_{\odot }(N-1)))\approx (0.4244G\rho )^{-1/2}.}$$

${\displaystyle t_{\text{cross))}$

Assume the BH stops after traveling a length of ${\displaystyle l_{\text{fric))\equiv {\text{ς))t_{\text{fric))}$ with its momentum ${\displaystyle M_{\bullet }V_{0}=M_{\bullet }{\text{ς))}$ deposited to ${\displaystyle {M_{\bullet } \over M_{\odot ))}$ stars in its path over ${\displaystyle l_{\text{fric))/(2R)}$ crossings, then the number of stars deflected by the BH's Bondi cross section per "diameter" crossing time is

$${\displaystyle N^{\text{defl))={({M_{\bullet } \over M_{\odot )))}{2R \over l_{\text{fric))}=N{\pi s_{\bullet }^{2} \over \pi R^{2))=N\left({M_{\bullet } \over 0.4053M_{\odot }N}\right)^{2}\ln \Lambda .}$$

More generally, the Equation of Motion of the BH at a general velocity ${\displaystyle \mathbf {V} _{\bullet ))$ in the potential ${\displaystyle \Phi }$ of a sea of stars can be written as

$${\displaystyle -{d \over dt}(M_{\bullet }V_{\bullet })-M_{\bullet }\nabla \Phi \equiv {(M_{\bullet }V_{\bullet }) \over t_{\text{fric))}=\overbrace {N\pi s_{\bullet }^{2} \over \pi R^{2)) ^{N^{\text{defl))}{(M_{\odot }V_{\bullet }) \over 2t_{\text{ς))}={8\ln \Lambda ' \over Nt_{\text{ς))}M_{\bullet }V_{\bullet },}$$

${\displaystyle {\pi ^{2} \over 8}\approx 1}$

${\displaystyle {\ln \Lambda ' \over \ln \Lambda }\equiv \left[{\pi ^{2} \over 8}\right]^{2}\left[(1+{V_{\bullet }^{2} \over {\text{ς'))^{2)))\right]^{-2}(1+{M_{\odot } \over M_{\bullet )))\leq \left[((\text{ς')) \over V_{\bullet ))\right]^{4}\leq 1}$

${\displaystyle M_{\bullet }\geq M_{\odot ))$

${\displaystyle t_{\text{ς'))}$

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

$${\displaystyle {M_{\bullet }d(\mathbf {V} _{\bullet }) \over dt}=-{M_{\bullet }\mathbf {V} _{\bullet } \over t_{\text{fric))^{\text{star))}=-{m\mathbf {V} _{\bullet }~n(\mathbf {x} )d\mathbf {x} ^{3} \over dt}\ln \Lambda _{\text{lag)),}$$

$${\displaystyle ~~n(\mathbf {x} )dx^{3}=dtV_{\bullet }(\pi s_{\bullet }^{2})n(\mathbf {x} )=dtn(\mathbf {x} )|V_{\bullet }|\pi \left[{G(m+M_{\bullet }) \over |V_{\bullet }|^{2}/2}\right]^{2))$$

${\displaystyle |V_{\bullet }dt|}$

${\displaystyle \pi s_{\bullet }^{2))$

Like the "Couloumb logarithm" ${\displaystyle \ln \Lambda }$ factors in the contribution of distant background particles, here the factor ${\displaystyle \ln(\Lambda _{\text{lag)))}$ 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 ${\displaystyle \ln(\Lambda _{\text{beaten)))}$. Also the bigger the system, the greater is ${\displaystyle \ln \Lambda }$.

A background of elementary (gas or dark) particles can also induce dynamical friction, which scales with the mass density of the surrounding medium, ${\displaystyle m~n}$; 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

$${\displaystyle M_{\bullet }{d(\mathbf {V} _{\bullet }) \over M_{\bullet }dt}=-4\pi \left[{GM_{\bullet } \over |V_{\bullet }|}\right]^{2}\mathbf {\hat {V)) _{\bullet }(\rho _{\text{gas))\ln \Lambda _{\text{lag))^{gas}+mn_{\text{*))\ln \Lambda _{\text{lag))^{*}).~~}$$

$${\displaystyle {\begin{aligned}\ln \Lambda _{\text{lag))^{gas}(u)&=\ln ~{\left[{1+u \over \lambda }\right]^{1 \over 2}\left[{|1-u| \over \lambda }\right]^{H[u-\lambda -1]-H[1-\lambda -u] \over 2} \over \exp {[u+\lambda ,1]_{\min }^{2}-[u-\lambda ,1]_{\min }^{2} \over 4\lambda )),\\&\approx \ln \left[((\sqrt {(u^{3}-1)^{2}+\lambda ^{3))}+u^{3}-1 \over {\sqrt {1+\lambda ^{3))}-1}\right]^{1 \over 3},~~u\equiv {|V_{\bullet }|t \over {\text{ς'))t},~~\lambda \equiv ({s_{\bullet } \over {\text{ς'))t})\\{\ln \Lambda _{\text{lag))^{*} \over \ln \Lambda }&\equiv \int _{0}^{|mV_{\bullet }|}\!\!\!\!{(4\pi p^{2}dp)e^{-{p^{2} \over 2(m\sigma )^{2))} \over ({\sqrt {2\pi ))m\sigma )^{3))\left.\right|_{p=m|v|}\approx {|\mathbf {V} _{\bullet }|^{3} \over |\mathbf {V} _{\bullet }|^{3}+3.45\sigma ^{3)),\\\ln \Lambda &=\int {d\mathbf {x_{1)) ^{3}~2Heaviside[{n(\mathbf {x_{1)) ) \over n(\mathbf {x} )}-1-{M_{\bullet } \over NM_{\odot ))] \over (s_{\bullet }^{2}+|\mathbf {x_{1)) -\mathbf {x} |^{2})^{3 \over 2))\approx \ln {\sqrt {1+\left({0.123NM_{\odot } \over M_{\bullet ))\right)^{2))},\end{aligned))}$$

${\displaystyle t=0}$

${\displaystyle {\text{ς))}$

${\displaystyle mn(\mathbf {x} )}$

${\displaystyle p=mv}$

${\displaystyle \sigma }$

${\displaystyle \sigma \leq {\text{ς))}$

Interestingly, the ${\displaystyle G^{2}(m+M_{\bullet })(mn(\mathbf {x} ))}$ 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 ${\displaystyle \ln \Lambda _{\text{lag))^{gas}\left.\right|_{u=1}=\ln ((\text{ς'))t \over s_{\bullet ))}$.

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.^[7] This means that two stars need to come within a separation,

$${\displaystyle s_{*}={GM_{\odot }+GM_{\odot } \over V^{2}/2}={2 \over 1.5}{GM_{\odot } \over {\text{ς))^{2))={3.29R \over N-1},}$$

$${\displaystyle 1\sim Q^{\text{virial))\equiv {\overbrace {2K} ^{(NM_{\odot })V^{2)) \over |W|}={NM_{\odot }{\text{ς))^{2}+NM_{\odot }{\text{ς))^{2}+NM_{\odot }{\text{ς))^{2} \over {N(N-1) \over 2}{GM_{\odot }^{2} \over R_{pair))},}$$

${\displaystyle N(N-1)/2}$

${\displaystyle R_{\text{pair))={\pi ^{2} \over 24}R\approx 0.411234R}$

${\displaystyle Q^{\text{virial))\leftarrow \rightarrow {\sqrt {\ln \Lambda )).}$

Mean free path

The mean free path of strong encounters in a typically ${\displaystyle (N-1)=4.19nR^{3}\gg 100}$ stellar system is then

$${\displaystyle l_{\text{strong))={1 \over (\pi s_{*}^{2})n}\approx {(N-1) \over 8.117}R\gg R,}$$

${\displaystyle 0.123N}$

${\displaystyle \pi s_{*}^{2))$

${\displaystyle R/V}$

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, ${\displaystyle \mathbf {v} }$, 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, ${\displaystyle \delta \mathbf {v} }$, 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 ${\displaystyle \delta \mathbf {v} }$ to equal ${\displaystyle \mathbf {v} }$, 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 ${\displaystyle N}$ objects is approximately

$${\displaystyle {t_{\text{relax)) \over t_{\text{ς))}=N^{\text{relax))\backsimeq {\frac {0.123(N-1)}{\ln(N-1)))\gg 1}$$

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 ${\displaystyle \left.{\frac {N'}{\ln {\sqrt {1+N'^{2))))}\right|_{N'=0.123(N-2)))$, 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 ${\displaystyle N\sim 10^{2}-10^{10))$ have much smoother potential, typically takes ${\displaystyle \sim \ln N'\approx (2-20)}$ 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 ${\displaystyle M_{\odot }/M_{\bullet ))$, but these two are very similar for a cluster of black holes,

$${\displaystyle N^{\text{fric))={t_{\text{fric)) \over t_{\text{ς))}\rightarrow {t_{\text{relax)) \over t_{\text{ς))}=N^{\text{relax))\sim {(N-1) \over 10-100},~{\text{when))~{M_{\bullet }\rightarrow m\leftarrow M_{\odot )).}$$

For a star cluster or galaxy cluster with, say, ${\displaystyle N=10^{3},~R=\mathrm {1pc-10^{5}pc} ,~V=\mathrm {1km/s-10^{3}km/s} }$, we have ${\displaystyle t_{\text{relax))\sim 100t_{\text{ς))\approx 100\mathrm {Myr} -10\mathrm {Gyr} }$. 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, ${\displaystyle N=10^{6}-10^{11))$ stars, would have a crossing time ${\displaystyle t_{\text{ς))\sim {1\mathrm {kpc} -100\mathrm {kpc} \over 1\mathrm {km/s} -100\mathrm {km/s} }\sim 100\mathrm {Myr} }$ 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

$${\displaystyle \Delta \sim {M_{\bullet } \over 0.1NM_{\odot )){t \over t_{\text{ς))}\leq {M_{\bullet } \over 0.1\%NM_{\odot ))}$$

if the black hole makes up less than 0.1% of the total galaxy mass ${\displaystyle NM_{\odot }\sim 10^{6-11}M_{\odot ))$. Especially when ${\displaystyle M_{\bullet }\sim M_{\odot ))$, 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 ${\displaystyle t/t_{\text{relax))\ll 1}$. 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

$${\displaystyle {\text{Perturber Virial))\approx {GM_{\bullet } \over s_{\bullet ))\approx V_{\text{cir))^{2}\approx \langle V\rangle ^{2}\approx {\overline {\langle V^{2}\rangle ))\approx \sigma ^{2}\approx \left({R \over t_{\text{ς))}\right)^{2}\approx c_{\text{ς))^{2}\approx {G(Nm) \over R}\approx {\text{Background Virial)),}$$

${\displaystyle 4\pi }$

${\displaystyle \pi }$

${\displaystyle \ln {\text{Λ))}$

${\displaystyle M_{\bullet }+m\approx M_{\bullet ))$

${\displaystyle V_{\text{cir))}$

${\displaystyle \langle V\rangle }$

${\displaystyle \sigma }$

${\displaystyle c_{\text{ς))}$

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:

$${\displaystyle {dQ \over dt}\approx {\pm Q \over ({l \over c_{\text{ς))})},~{\text{Q being mass M, energy E, momentum (M V), Phase density f, size R, density)){Nm \over {4\pi \over 3}R^{3))...,}$$

${\displaystyle \pm }$

${\displaystyle l=c_{\text{ς))t_{\text{fric))}$

${\displaystyle t_{\text{fric))}$

$${\displaystyle s^{2}\approx \max \left[{\text{Bondi radius))~s_{\bullet },{\text{Tidal radius))~s_{\text{Hill)),{\text{physical size))~s_{\text{Loss cone))\right]^{2}.}$$

E.g., in case Q is the perturber's mass ${\displaystyle Q=M_{\bullet ))$, then we can estimate the Dynamical friction time via the (gas/star) Accretion rate

$${\displaystyle {\begin{aligned}{\dot {M))_{\bullet }=&{M_{\bullet } \over t_{\text{fric))}\approx \int _{0}^{s^{2))d({\text{area)))~({\text{background mean flux)))\approx s^{2}(\rho c_{\text{ς)))\\\approx &{\frac ((\text{Perturber influenced cross section))~(s^{2})}((\text{background system cross section))~(R^{2})))\times {\frac ((\text{background mass))~(Nm)}((\text{crossing time))~t_{\text{ς))\approx {R \over c_{\text{ς))}\approx {1 \over {\sqrt {G(Nm) \over R^{3))}\sim {\sqrt {G\rho ))\sim \kappa ))}\\\approx &{GM_{\bullet } \over Gt_{\text{ς))}{GM_{\bullet } \over G(Nm)}\approx (\rho c_{\text{ς)))\left({GM_{\bullet } \over c_{\text{ς))^{2))\right)^{2},~~{\text{if consider only gravitationally focusing,))\\\approx &{M_{\bullet } \over Nt_{\text{ς))},~~{\text{if for a light perturber))M_{\bullet }\rightarrow m=M_{\odot }\\\rightarrow &0,~~{\text{if practically collisionless))~~N\rightarrow \infty ,\end{aligned))}$$

In the limit the perturber is just 1 of the N background particle, ${\displaystyle M_{\bullet }\rightarrow m}$, 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 ${\displaystyle N\rightarrow \infty }$ 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.^[7]

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.

Connections to statistical mechanics and plasma physics

The statistical nature of stellar dynamics originates from the application of the kinetic theory of gases to stellar systems by physicists such as James Jeans in the early 20th century. The Jeans equations, which describe the time evolution of a system of stars in a gravitational field, are analogous to Euler's equations for an ideal fluid, and were derived from the collisionless Boltzmann equation. This was originally developed by Ludwig Boltzmann to describe the non-equilibrium behavior of a thermodynamic system. Similarly to statistical mechanics, stellar dynamics make use of distribution functions that encapsulate the information of a stellar system in a probabilistic manner. The single particle phase-space distribution function, ${\displaystyle f(\mathbf {x} ,\mathbf {v} ,t)}$, is defined in a way such that

$${\displaystyle f(\mathbf {x} ,\mathbf {v} ,t)\,d\mathbf {x} \,d\mathbf {v} =dN}$$

${\displaystyle dN/N}$

${\displaystyle \mathbf {x} }$

${\displaystyle d\mathbf {x} }$

${\displaystyle {\text{v))}$

${\displaystyle d\mathbf {v} }$

Convention and notation in case of a thermal distribution

In most of stellar dynamics literature, it is convenient to adopt the convention that the particle mass is unity in solar mass unit ${\displaystyle M_{\odot ))$, hence a particle's momentum and velocity are identical, i.e.,

$${\displaystyle \mathbf {p} =m\mathbf {v} =\mathbf {v} ,~m=1,~N_{\text{total))=M_{\text{total)),}$$

$${\displaystyle {dM \over dx^{3}dv^{3))=f(\mathbf {x} ,\mathbf {v} ,t)=f(\mathbf {x} ,\mathbf {p} ,t)\equiv {dN \over dx^{3}dp^{3))}$$

For example, the thermal velocity distribution of air molecules (of typically 15 times the proton mass per molecule) in a room of constant temperature ${\displaystyle T_{0}\sim \mathrm {300K} }$ would have a Maxwell distribution

$${\displaystyle f^{\text{Max))(x,y,z,mV_{x},mV_{y},mV_{z})={1 \over (2\pi \hbar )^{3)){1 \over \exp \left({E(x,y,z,p_{x},p_{y},p_{z})-\mu \over kT_{0))\right)+1))$$

$${\displaystyle f^{\text{Max))\sim {1 \over (2\pi \hbar /m)^{3))e^{\mu \over kT_{0))e^{-E \over m\sigma _{1}^{2)),}$$

where the energy per unit mass

$${\displaystyle E/m=\Phi (x,y,z)+(V_{x}^{2}+V_{y}^{2}+V_{z}^{2})/2,}$$

${\displaystyle \Phi (x,y,z)\equiv g_{0}z=0}$

and ${\textstyle \sigma _{1}={\sqrt {kT_{0}/m))\sim \mathrm {0.3km/s} }$ is the width of the velocity Maxwell distribution, identical in each direction and everywhere in the room, and the normalisation constant ${\displaystyle e^{\mu \over kT_{0))}$ (assume the chemical potential ${\textstyle \mu \sim (m\sigma _{1}^{2})\ln \left[n_{0}\left(((\sqrt {2\pi ))\hbar \over m\sigma _{1))\right)^{3}\right]\ll 0}$ such that the Fermi-Dirac distribution reduces to a Maxwell velocity distribution) is fixed by the constant gas number density ${\displaystyle n_{0}=n(x,y,0)}$ at the floor level, where

$${\displaystyle n(x,y,0)=\!\!\int _{-\infty }^{\infty }mdV_{x}\!\!\int _{-\infty }^{\infty }mdV_{y}\!\!\int _{-\infty }^{\infty }mdV_{z}f(x,y,0,mV_{x},mV_{y},mV_{z})}$$

$${\displaystyle n\approx {(2\pi )^{3/2}(m\sigma _{1})^{3} \over (2\pi \hbar )^{3))e^{\mu \over m\sigma _{1}^{2)).}$$

The CBE

In plasma physics, the collisionless Boltzmann equation is referred to as the Vlasov equation, which is used to study the time evolution of a plasma's distribution function.

The Boltzmann equation is often written more generally with the Liouville operator ${\displaystyle {\mathcal {L))}$ as

$${\displaystyle {\mathcal {L))f(t,\mathbf {x} ,\mathbf {p} )={f_{\text{fit))^{\text{Max))-f(t,\mathbf {x} ,\mathbf {p} ) \over t_{\text{relax))},}$$

$${\displaystyle {\mathcal {L))\equiv {\frac {\partial }{\partial t))+{\frac {\mathbf {p} }{m))\cdot \nabla +\mathbf {F} \cdot {\frac {\partial }{\partial \mathbf {p} ))\,.}$$

${\displaystyle \mathbf {F} \equiv \mathbf {\dot {p)) =-m\nabla \Phi }$

${\displaystyle f_{\text{fit))^{\text{Max))}$

${\displaystyle f(t,\mathbf {x} ,\mathbf {p} )}$

${\displaystyle t_{\text{relax))}$

Whereas Jeans applied the collisionless Boltzmann equation, along with Poisson's equation, to a system of stars interacting via the long range force of gravity, Anatoly Vlasov applied Boltzmann's equation with Maxwell's equations to a system of particles interacting via the Coulomb Force.^[8] 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.^[9]

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

$${\displaystyle {1 \over \rho _{p))\int \!\left\{\mathbf {v} _{p}{d[f_{p}m_{p}] \over dt}-\langle {\mathbf {v} }\rangle _{p}{d[f_{p}m_{p}] \over dt}\right\}d^{3}\mathbf {v} =0,}$$

${\displaystyle ^{p))$

$${\displaystyle {\begin{aligned}\overbrace {\left({\partial \over \partial t}+\sum _{j=1}^{3}\langle {v_{j}^{p))\rangle {\partial \over \partial x_{j))\right)\langle {v_{i}^{p))\rangle } ^((\dot {\langle {v}\rangle ))_{i}^{p))&\underbrace {=} _{EoM}\overbrace {-\partial \Phi (t,\mathbf {x} ) \over \partial x_{i)) ^{g_{i}\sim O(-GM/R^{2})}~~\underbrace {-} _{\text{balance))^{\text{pressure))~~\sum _{j=1}^{3}{\partial \over \rho ^{p}\partial x_{j))\overbrace {[\underbrace {\rho ^{p}(t,\mathbf {x} )} _{\int _{\infty }\!\!\!\!m_{p}f_{p}d^{3}\mathbf {v} }\underbrace {\sigma _{ji}^{p}(t,\mathbf {x} )} _{O(c_{s}^{2})}]} ^{\int \limits _{\infty }\!\!d\mathbf {v} ^{3}(\mathbf {v} _{j}-\langle {v}\rangle _{j}^{p})(\mathbf {v} _{i}-\langle {v}\rangle _{i}^{p})m_{p}f_{p))-{\underbrace {\langle {v_{i}^{p))\rangle \overbrace {[{\dot {m))_{p}/m_{p}]} ^{1/t|_((\text{visc))~m_{p}=M_{\text{gas))}^{\text{fric)))) _{\text{snow.plough))},\\0&=-{\partial \Phi (t,\mathbf {x} ) \over \partial x_{i))-{\partial (n\sigma ^{2}) \over n\partial x_{i)),~~{\text{hydrostatic isotropic velocity, no flow and friction )).\end{aligned))}$$

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, epecially 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.^[10][11] 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.^[12]

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.^[1] Stellar dynamical models are also used to study the evolution of active galactic nuclei and their black holes, as well as to estimate the mass distribution of dark matter in galaxies.

A unified thick disk potential

Consider an oblate potential in cylindrical coordinates

$${\displaystyle {\begin{aligned}\Phi (R,z)&={GM_{0} \over 2z_{0))\left[2\sinh ^{-1}\!\!Q-\sinh ^{-1}\!\!Q_{+}-\sinh ^{-1}\!\!Q_{-}\right]\\&={GM_{0} \over 2z_{0))\log {({\sqrt {1+Q^{2))}+Q)^{2} \over \left[{\sqrt {1+Q_{+}^{2))}+Q_{+}\right]\left[{\sqrt {1+Q_{-}^{2))}+Q_{-}\right]},\\Q_{\pm }&\equiv {R_{0}+\left|~|z|\pm z_{0}~\right| \over R},\\Q&\equiv {R_{0}+[0,|z|-z_{0}]_{\max } \over R},\\\end{aligned))}$$

${\displaystyle z_{0},R_{0))$

First we can see the total mass of the system is ${\displaystyle M_{0))$ because

$${\displaystyle \Phi (R,z)\rightarrow {GM_{0} \over 2z_{0))(2Q_{-}-Q_{-}-Q_{+})=-{GM_{0} \over R},}$$

${\displaystyle R\rightarrow \infty ,~|z|\geq z_{0},}$

${\displaystyle Q=Q_{-}=Q_{+}-{2z_{0} \over R}={|z|+(R_{0}-z_{0}) \over R}\rightarrow 0.}$

We can also show that some special cases of this unified potential become the potential of the Kuzmin razor-thin disk, that of the Point mass ${\displaystyle M_{0))$, and that of a uniform-Needle mass distribution: