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 $N\gg 10;$ 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]

Slingshot of a test body in a two-body potential

The study of many particles in the phase space of a static potential

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

$\sim (10^{-8}{\text{pc}}/100{\text{km/s}},1M_{\odot }/10^{55}=m_{p})$ for the Sun,
$\sim (\mathrm {10pc/10km/s} ,1000M_{\odot }/1000)$ for M13 Star Cluster,
$\sim (\mathrm {100kpc/100km/s} ,10^{11}M_{\odot }/10^{11})$ for M31 Disk Galaxy,
$\sim (\mathrm {10Mpc/1000km/s} ,10^{14}M_{\odot }/10^{77}=m_{\nu })$ , where we counted neutrinos in the Bullet Clusters of N = 1000 galaxies. Note 1000pc/1km/s = 1Gyr = HubbleTime/14.

Galaxies collide occasionally in galaxy clusters, and stars collide occasionally in star clusters, too. , resulting in dynamical friction and relaxation. Unlike stars in galaxy disks, gas particles collide very frequently in accretion disks, which have $(L/V,M/N)$

$\sim (10^{-4}{\text{pc}}/10{\text{km/s}},0.1M_{\odot }/10^{54}\sim m_{p})$ around stars or stellar black holes,
$\sim (10^{-1}{\text{pc}}/100{\text{km/s}},10M_{\odot }/10^{56}\sim m_{p})$ around million solar mass black holes in centres of galaxies. The plasma or gas collisions result in equipartition and perhaps viscosity under magnetic field.

Historically, the methods utilized 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]

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.^[2]

Stellar dynamics also has connections to the field of plasma physics.^[3] 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.

Key concepts[]

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. From Newton's second law an equation of motion describing the interactions of an isolated stellar system can be written down as,

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} _{i}-\mathbf {r} _{j}\right)}{\left\|\mathbf {r} _{i}-\mathbf {r} _{j}\right\|^{3}}}

which is simply a formulation of the N-body problem. For an N-body system, any individual member,

m_{i}

is influenced by the gravitational potentials of the remaining

m_{j}

members. In practice, except for in the highest performance computer simulations, it is not feasible to calculate rigorously the system's gravitational potential by adding all of the point-mass potentials in the system, so stellar dynamicists develop potential models that can accurately model the system while remaining computationally inexpensive.^[4] The gravitational potential,

\Phi

, of a system is related to the acceleration and the gravitational field,

\mathbf {g}

by:

{\frac {d^{2}\mathbf {r_{i}} }{dt^{2}}}=\mathbf {g} =-\nabla \Phi

whereas the potential is related to a (smoothened) mass density,

\rho

, via Poisson's equation:

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

Both the equations of motion and Poisson Equation can take on other forms, e.g., given here for a spherical system.

Note three approximations here. Firstly above equations neglect relativistic corrections, which are of order of $(v/c)^{2}\ll 10^{-4}$ as typical stellar 3-dimensional speed, $v\sim 3-3000$ km/s, is much below the speed of light.

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,

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},

hence radiation force is negligible in general, except perhaps around a luminous O-type star of mass

30M_{\odot }

, or around a black hole accreting gas at the Eddington limit so that its luminosity-to-mass ratio

L_{\bullet }/M_{\bullet }

is defined by

Q^{\text{Eddington}}=1

.

Thirdly a star is 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.,

1\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}]}\approx \left[{s_{\text{Hill}} \over R_{\odot }\left({M_{\bullet } \over 0.5M_{\odot }}\right)^{1/3}}\right]^{3}.

A star can also swallowed if coming within a few Schwarzschild radii of the black hole. This radius of Loss is given by ${\textstyle s\leq s_{\text{Loss}}={\frac {6GM_{\bullet }}{c^{2}}}}$ and the minimum angular momentum per unit mass ${\textstyle J\leq J_{\text{loss}}={\frac {4GM_{\bullet }}{c}}}$ . This is found by computing the position of the angular momentum barrier of an effective potential

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

where

\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]

is an approximate classical potential of a Schwarzschild black hole.

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

\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} ,

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.

Stars will be deflected when entering the (much larger) cross section $\pi s_{\bullet }^{2}$ of a black hole. This so-called sphere of influence is defined by,

{\frac {GM_{\bullet }}{s_{\bullet }}}m={\frac {V^{2}}{2}}m,

hence

s_{\bullet }={GM_{\bullet } \over V^{2}/2}={M_{\bullet } \over M_{\odot }}{V_{\odot }^{2} \over V^{2}}R_{\odot }>\max[s_{\text{Hill}},s_{\text{Loss}}]=(1-4000)R_{\odot },

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 ${\textstyle V_{\odot }={\sqrt {2GM_{\odot }/R_{\odot }}}=615\mathrm {km/s} }$ 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 ${\textstyle V\sim {\sqrt {2G(NM_{\odot })/R}}\ll \mathrm {300km/s} }$ inside all star clusters and in galaxies.

Connections to gravitational dynamical friction and gravitational gas accretion physics[]

First consider a heavy black hole of mass $M_{\bullet }$ is moving through a dissipational gas of thermal sound speed $c_{s}$ and density $\rho _{\text{gas}}$ , then every gas particle of mass m will likely transfer its momentum $mV$ to the black hole where coming within a cross-section of radius $s_{\bullet }$ . In a time scale $t_{\text{fric}}$ 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 $s_{\bullet }$ , dissipate kinetic energy by gas collisions and fall in the black hole. The rate of change is given by, within an order of magnitude,

{\dot {M}}_{\bullet }\approx -{M_{\bullet }{\dot {V}} \over V}\equiv {M_{\bullet } \over t_{\text{fric}}^{\text{gas}}}\sim (\pi s_{\bullet }^{2})\rho _{\text{gas}}{\sqrt {V^{2}+c_{s}^{2}}}.

In case that a heavy black hole of mass $M_{\bullet }$ moves relative to a background of stars in random motion in a cluster of total mass $(NM_{\odot })$ with a mean number density $n\sim (N-1)/(4\pi R^{3}/3)$ within a typical size $R$ . 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

-{M_{\bullet }{\dot {V}}}={M_{\bullet }V \over t_{\text{fric}}^{\text{star}}},

where

t_{\text{fric}}^{\text{star}}

is called a dynamical friction time, the time that takes to lose the heavy body's initial streaming motion, satisfying

{{\dot {V}} \over V}=t_{\text{fric}}^{-1}\equiv {1 \over N_{\bullet }^{\text{fric}}t_{\text{cross}}},

where we assume that for the traveling black hole to lose most of its momentum

M_{\bullet }V

to sling-shot stars, it needs to make

N_{\bullet }^{\text{fric}}

crossings of the system with

N_{\bullet }^{\text{fric}}={(M_{\bullet }V) \over N^{\text{defl}}(M_{\odot }V)}={0.0625M_{\odot } \over M_{\bullet }}N,

where the number of such deflected stars per crossing is

N^{\text{defl}}={\pi s_{\bullet }^{2} \over \pi R^{2}}N={M_{\bullet }^{2} \over (0.25M_{\odot })^{2}}N.

and we used the balance of centrifugal and gravitational force in a virtualised system,

1\sim Q^{\text{viral}}\approx {{V^{2} \over R} \over {G(NM_{\odot }/2) \over R^{2}}}\approx {(R/V)^{-2} \over (G\rho )}

where

t_{\text{cross}}\equiv {R \over V}\sim (G\rho )^{-1/2},~\rho =nM_{\odot },

is the so-called crossing time or dynamical time scale, the time it takes for a star to travel across the galaxy once.

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

{d(M_{\bullet }\mathbf {V} _{\bullet })}=-M_{\bullet }\mathbf {V} _{\bullet }{dt \over t_{\text{fric}}^{\text{star}}}=-\mathbf {V} _{\bullet }\left[mn(\mathbf {x} )dx^{3}\right]\ln \Lambda ,

~~dx^{3}=dtV_{\bullet }(\pi s_{\bullet }^{2})=dt|V_{\bullet }|\pi \left[{G(m+M_{\bullet }) \over |V_{\bullet }|^{2}/2}\right]^{2}\int _{0}^{|mV_{\bullet }|}f_{p}(4\pi p^{2}dp)

where

\ln(\Lambda )\sim \ln(N)

is the so-called "Coulomb logarithm" depending on the size of the system,

dx^{3}

is a infinitesimal cylindrical volume of length

|V_{\bullet }dt|

and a weighted cross-section

\pi s_{\bullet }^{2}

within the black hole's sphere of influence, and

f_{p}(4\pi p^{2}dp)={(4\pi p^{2}dp) \over ({\sqrt {2\pi }}m\sigma )^{3}}e^{-{p^{2} \over 2(m\sigma )^{2}}},

is the probability of finding a background star at momentum range p+dp in a Gaussian distribution or Maxwell distribution of velocity spread (called velocity dispersion

\sigma \sim V

) of the background star of (mass) density

mn(\mathbf {x} )

. Note the

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 of the massive body in its two-body encounters with background objects. A background of elementary (gas or dark) particles can also induce dynamical friction, which scales with the mass density of the surrounding medium,

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. The force is also proportional to the inverse square of the velocity, 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.

Coming back to the gas accretion, star tidal disruption and star capture by a (moving) black hole, we note that

{\dot {M}}_{\bullet }\sim {\sqrt {\sigma ^{2}+V_{\bullet }^{2}}}\pi (s_{\text{Hill}}^{2}+s_{\text{Loss}}^{2})\left[mn_{\text{star}}(\mathbf {x} )\right]\ln {M_{\text{stars}} \over M_{\odot }}+{\sqrt {c_{s}^{2}+V^{2}}}(\pi s_{\bullet }^{2})\rho _{\text{gas}}\ln {M_{\text{gas}} \over M_{\bullet }}

because the black hole consumes the majority of gas but only a minority of stars passing its sphere of influence.

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/miniscule 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.^[6] This means that two stars need to come within a separation,

s={GM_{\odot }M_{\odot } \over M_{\odot }V^{2}/2}={4R \over N-1},

where we used the Virial Theorem, i.e., "mutual potential energy balances twice kinetic energy on average",

1\sim Q^{\text{viral}}\equiv {2K \over |W|}={(NM_{\odot })V^{2} \over {N(N-1) \over 2}{GM_{\odot }^{2} \over R}},

where the factor

N(N-1)/2

is the number of handshakes between a pair of stars without double-counting.

The mean free path of strong encounters in a typically $N\gg 100$ stellar system is then

l={1 \over n(\pi s^{2})}\sim [0.0833(N-1)]R\gg R,

i.e., it takes about

0.1N

radius crossings for a typical star to come within a cross-section

\pi s^{2}

to be deflected from its path completely. Hence the mean free time of a strong encounter is much longer than the crossing time

R/V

.

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, $\mathbf {v}$ , 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, $\delta \mathbf {v}$ , 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 $\delta \mathbf {v}$ to equal $\mathbf {v}$ , or the time it takes for the small deviations in velocity to equal the star's initial velocity. The relaxation time for a stellar system of $N$ objects is approximately equal to:

t_{\text{relax}}\backsimeq N^{\text{relax}}t_{\text{cross}},~N^{\text{relax}}={\frac {0.1(N-1)}{\ln(N-1)}}\gg 1

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. And it takes many weak encounters to build a final strong deflection for a system with

N\geq 3

.

Clearly that the dynamical friction timescale of a black hole is much shorter than the relaxation time by roughly a factor $M_{\odot }/M_{\bullet }$ , but

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

For a star cluster or galaxy cluster with, say, $N=10^{3},~R=\mathrm {1pc-10^{5}pc} ,~V=\mathrm {1km/s-10^{3}km/s}$ , we have $t_{\text{relax}}\sim 100T_{\rm {cross}}\approx 100\mathrm {Myr} -10\mathrm {Gyr}$ . Hence we expect stellar or galaxy encounters to be important during the typical 10 Gyr lifetime of these clusters.

On the other hand, typical galaxy with, say, $N=10^{6}-10^{11}$ stars, would have a crossing time $t_{\text{cross}}\sim {1\mathrm {kpc} -100\mathrm {kpc} \over 1\mathrm {km/s} -100\mathrm {km/s} }\sim 100\mathrm {Myr}$ 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

\Delta \sim {M_{\bullet } \over 0.1NM_{\odot }}{t \over t_{\text{cross}}}\leq {M_{\bullet } \over 0.1\%NM_{\odot }}

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

The relaxation time identifies collisionless vs. collisional stellar 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 $t/t_{\text{relax}}\ll 1$ . They are also identified as systems where subject stars interact with a smooth gravitational potential as opposed to the sum of point-mass potentials.^[4] 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.^[6]

A concise 1-page summary of some main equations in stellar dynamics and accretion discs physics are shown here.

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, $f(\mathbf {x} ,\mathbf {v} ,t)$ , is defined in a way such that

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

where

dN/N

represents the probability of finding a given star with position

\mathbf {x}

around a differential volume

d\mathbf {x}

and velocity

{\text{v}}

around a differential velocity space volume

d\mathbf {v}

. The distribution function is normalized (sometimes) such that integrating it over all positions and velocities will equal N, the total number of bodies of the system. For collisional systems, Liouville's theorem is applied to study the microstate of a stellar system, and is also commonly used to study the different statistical ensembles of statistical mechanics.

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

\mathbf {p} =m\mathbf {v} =\mathbf {v} ,~m=1,~N_{\text{total}}=M_{\text{total}},~{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 $T_{0}\sim \mathrm {300K}$ would have a Maxwell distribution

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}\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

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

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 $e^{\mu \over kT_{0}}$ (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 at the floor level, where $\Phi (x,y,z)\equiv g_{0}z=0$ ,

n_{0}=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})\approx {(2\pi )^{3/2}(m\sigma _{1})^{3} \over (2\pi \hbar )^{3}}e^{\mu  \over m\sigma _{1}^{2}}.

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 ${\mathcal {L}}$ as

{\mathcal {L}}f(t,\mathbf {x} ,\mathbf {p} )={f_{\text{fit}}^{\text{Max}}-f(t,\mathbf {x} ,\mathbf {p} ) \over t_{\text{relax}}},\quad {\mathcal {L}}\equiv {\frac {\partial }{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla +\mathbf {F} \cdot {\frac {\partial }{\partial \mathbf {p} }}\,.

where

\mathbf {F} =-m\nabla \Phi

is the gravitational force and

f_{\text{fit}}^{\text{Max}}

is the Maxwell (equipartition) distribution (to fit the same density, same mean and rms velocity as

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

). The equation means the non-Gaussianity will decay on a (relaxation) time scale of

t_{\text{relax}}

, and the system will ultimately relaxes to a Maxwell (equipartition) distribution.

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.^[7] 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.^[8]

Jeans computed the weighted velocity of the Boltzmann Equation after integrating over velocity space

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

and obtain the Momentum (Jeans) Eqs. of a

^{p}

opulation:

\overbrace {\left({\partial  \over \partial t}+\sum _{j=1}^{3}{\bar {v_{j}^{p}}}{\partial  \over \partial x_{j}}\right){\bar {v_{i}^{p}}}} ^{{\dot {\bar {v}}}_{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}-{\bar {\mathbf {v} }}_{j}^{p})(\mathbf {v} _{i}-{\bar {\mathbf {v} }}_{i}^{p})m_{p}f_{p}}-{\underbrace {{\bar {v_{i}^{p}}}\overbrace {[{\dot {m}}_{p}/m_{p}]} ^{1/t|_{{\text{visc}}~m_{p}=M_{\text{gas}}}^{\text{fric}}}} _{\text{snow.plough}}},

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.

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.^[9]^[10] 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.^[11]

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 worked example of Poisson Equation and oscillation frequencies in a thick disk[]

Consider an oblate potential in cylindrical coordinates

\Phi (R,z)={GM_{0} \over 2z_{0}}\log {({\sqrt {1+u^{2}q^{2}}}+uq)^{2} \over \left[{\sqrt {1+u^{2}q_{+}^{2}}}+uq_{+}\right]\left[{\sqrt {1+u^{2}q_{-}^{2}}}+uq_{-}\right]},~~u\equiv {R_{0} \over R},~~q\equiv 1+{\max[0,z_{0}-|z|] \over R_{0}},~~q_{\pm }\equiv {R_{\pm } \over R_{0}}\equiv 1+{\left|z_{0}\pm |z|\right| \over R_{0}},

where

z_{0},R_{0}

are (positive) vertical and radial length scales. We will show how some special cases of the potential become that of the Kuzmin razor-thin disk, that of the point mass

M_{0}

, and that of a uniform-rod mass distribution:

\Phi (R,z)=-{GM_{0} \over {\sqrt {R^{2}+(|z|+R_{0})^{2}}}},~~z_{0}=0,

\Phi (R,z)=-{GM_{0} \over {\sqrt {R^{2}+z^{2}}}},~~z_{0}=R_{0}=0,

\Phi (R,z)={GM_{0} \over 2z_{0}}\left[2\arcsin {h}{\max[0,z_{0}-|z|] \over R}-\arcsin {h}{z_{0}+|z| \over R}-\arcsin {h}{\left|z_{0}-|z|\right| \over R}\right],~~R_{0}=0.

Note the somewhat pointed end of the equal potential in the (R,z) meridional plane of this R0=5z0=1 model

First consider the vertical gravity at the boundary,

g_{z}(R,z)=-\partial _{z}\Phi (R,z)=-{GM_{0}z \over 2z_{0}^{2}}\left[{1 \over {\sqrt {R_{0}^{2}+R^{2}}}}-{1 \over {\sqrt {(R_{0}+2z_{0})^{2}+R^{2}}}}\right],~~z=\pm z_{0},

Note that both the potential and the vertical gravity are continuous across the boundaries, hence no razor disk at the boundaries. Thanks to the fact that at the boundary, $q=q_{-}=1=q_{+}-{2z_{0} \over R_{0}},~~\partial _{|z|}(2q)-\partial _{|z|}q_{-}=\partial _{|z|}\left(q_{+}-{\frac {2z_{0}}{R_{0}}}\right)={1 \over R_{0}}$ is continuous. Apply Gauss's theorem by integrating the vertical force over the entire disk upper and lower boundaries, we have

2\int _{0}^{\infty }(2\pi RdR)|g_{z}(R,z_{0})|=4\pi GM_{0},

confirming that

M_{0}

takes the meaning of the total disk mass.

The vertical gravity drops with $-g_{z}\rightarrow GM_{0}z(1+R_{0}/z_{0})/R^{3}$ at large radii, which is enhanced over the vertical gravity of a point mass $GM_{0}z/R^{3}$ due to the self-gravity of the thick disk.

Insert in the cylindrical Poisson eq.

\rho (R,z)={\partial _{z}\partial _{z}\Phi  \over 4\pi G}+{\partial _{R}(R\partial _{R}\Phi ) \over 4\pi GR}={M_{0}R_{0}/z_{0} \over 4\pi (R^{2}+R_{0}^{2})^{3/2}}H(z_{0}-|z|),

which drops with radius, and is zero beyond

|z|>z_{0}

and uniform along the z-direction within the boundary.

Note the vertically uniform thick disk density contour in this R0=5z0=1 model

Integrating over the entire thick disc of uniform thickness $2z_{0}$ , we find the surface density and the total mass as

\Sigma (R)=(2z_{0})\rho (R,0),~~M_{0}=\int _{0}^{\infty }(2\pi RdR)\Sigma (R).

This confirms that the absence of extra razor thin discs at the boundaries. In the limit, $z_{0}\rightarrow 0$ , this thick disc potential reduces to that of a razor-thin Kuzmin disk, for which we can verify ${|g_{z}(R,0+)| \over 2\pi G}\rightarrow \Sigma (R)\rightarrow {M_{0}R_{0} \over 2\pi (R^{2}+R_{0}^{2})^{3/2}}$ .

To find the vertical and radial oscillation frequencies, we do a Taylor expansion of potential around the midplane.

\Phi (R_{1},z)\approx \Phi (R,0)+{\omega ^{2}R}(R_{1}-R)+{\kappa ^{2} \over 2}(R_{1}-R)^{2}+{\nu ^{2} \over 2}z^{2}

and we find the circular speed

V_{\text{cir}}

and the vertical and radial epicycle frequencies to be given by

V_{\text{cir}}^{2}=(R\omega )^{2}=\left[{(1+R_{0}/z_{0})GM_{0} \over {\sqrt {R^{2}+(R_{0}+z_{0})^{2}}}}-{(R_{0}/z_{0})GM_{0} \over {\sqrt {R^{2}+R_{0}^{2}}}}\right],~~\nu ^{2}={GM_{0}(R_{0}/z_{0}+1) \over (R^{2}+(R_{0}+z_{0})^{2})^{3/2}},~~\kappa ^{2}+\nu ^{2}-2\omega ^{2}=4\pi G\rho (R,0)={GM_{0}R_{0}/z_{0} \over (R^{2}+R_{0}^{2})^{3/2}}.

Interestingly the rotation curve $V_{\text{cir}}$ is solid-body-like near the centre $R\ll R_{0}$ , and is Keplerian far away.

At large radii three frequencies satisfy ${\textstyle \left.\left[\omega ,\nu ,\kappa ,{\sqrt {4\pi G\rho }}\right]\right|_{R\to \infty }\to [1,1+R_{0}/z_{0},1,R_{0}/z_{0}]^{1 \over 2}{\sqrt {GM_{0} \over R^{3}}}}$ . E.g., in the case that $R\to \infty$ and $R_{0}/z_{0}=3$ , the oscillations $\omega :\nu :\kappa =1:2:1$ forms a resonance. In the case that $R_{0}=0$ , the density is zero everywhere except uniform on the z-axis between $|z|\leq z_{0}$ . If we further require $z_{0}=0$ , then we recover a property $\omega :\nu :\kappa =1:1:1$ everywhere, as expected for closed ellipse orbits in point mass potential.

A worked example for neutrinos in galaxies[]

For example, the phase space distribution function of non-relativistic neutrinos of mass m anywhere will not exceed the maximum value set by

f(\mathbf {x} ,\mathbf {v} ,t)={dN \over dx^{3}dv^{3}}\leq {6 \over (2\pi \hbar /m)^{3}},

where the Fermi-Dirac statistics says there are at most 6 flavours of neutrinos within a volume

dx^{3}

and a velocity volume

dv^{3}=(dp/m)^{3}=[(2\pi \hbar /dx)/m]^{3}

. Let's approximate the distribution as

f(x,y,z,V_{x},V_{y},V_{z})={6 \over (2\pi \hbar /m)^{3}}q^{\alpha  \over 2},~~0\leq q(E)={\Phi _{\max }-E \over V_{0}^{2}/2}\leq 1,

where

0\geq \Phi _{\max }\geq E=\Phi (x,y,z)+{V_{x}^{2}+V_{y}^{2}+V_{z}^{2} \over 2}\geq \Phi _{\min }\equiv \Phi _{\max }-{V_{0}^{2} \over 2}

such that

E_{\min },E_{\max }

, respectively, is the potential energy of at the centre or the edge of the gravitational bound system. The corresponding neutrino mass density, assume spherical, would be

\rho (r)=n(x,y,z)m=\int dV_{x}\int dV_{y}\int dV_{z}~m~f(x,y,z,V_{x},V_{y},V_{z})={6\pi 2^{5/2}B\left(1+{\alpha  \over 2},{3 \over 2}\right)m(\Phi _{\max }-\Phi (r))^{3+\alpha  \over 2} \over (2\pi \hbar /m)^{3}(\Phi _{\max }-\Phi _{\min })^{\alpha  \over 2}}

Take the simple case $\alpha \to 0$ , and estimate the density at the centre $r=0$ with an escape speed $V_{0}$ , we have

\rho (r)\leq \rho (0)\rightarrow {m^{4}V_{0}^{3} \over \pi ^{2}\hbar ^{3}}\approx m_{\mathrm {eV} }^{4}V_{200}^{3}\times {\text{[Cosmic Critical Density]}}.

Clearly eV-scale neutrinos with $m_{eV}\sim 0.1-1$ is too light to make up the 100–10000 over-density in galaxies with escape velocity $V_{200}\equiv V/(\mathrm {200km/s} )\sim 0.1-3.4$ . Neutrinos in clusters with $V\sim \mathrm {2000km/s}$ could make up $100-1000$ times cosmic background density. By the way the freeze-out cosmic neutrinos in your room have a non-thermal random momentum ${\textstyle \sim {(\mathrm {2.7K} )k \over c}\sim (1~\mathrm {eV} /c^{2})(\mathrm {70km/s} )}$ , and do not follow a Maxwell distribution, and are not in thermal equilibrium with the air molecules because of the extremely low cross-section of neutrino-baryon interactions.

A worked example of the Poisson Equation and escape speed in a uniform sphere[]

Consider a spherical potential

\Phi (r)=-V_{e}(0)^{2}-{V_{e}^{2}(r) \over 2},

where

{V_{e}^{2} \over 2}\equiv \left[1-{r^{2} \over r_{0}^{2}},~~{2r_{0} \over r}-2\right]_{\max }{V_{0}^{2} \over 2},

takes the meaning of the speed to "escape to the edge"

r_{0}

. We can fix the normalisation

V_{0}

by computing the corresponding density using the spherical Poisson Equation

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),

where the enclosed mass

GM(r)={r^{2}d\Phi  \over dr}=\int _{0}^{r}dr\int _{0}^{\pi }(rd\theta )\int _{0}^{2\pi }(r\sin \theta d\varphi )\rho _{0}H(r_{0}-r).

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

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

A worked example on Jeans theorem and CBE[]

Generally for a time-independent system, Jeans theorem predicts that $f(\mathbf {x} ,\mathbf {v} )$ is an implicit function of the position and velocity through a functional dependence on constants of motion. E.g., in a spherical system the orbital energy E and angular momentum J and its z-component Jz along every stellar orbit satisfy

dE[\mathbf {x} ,\mathbf {v} ]/dt=dJ[\mathbf {x} ,\mathbf {v} ]/dt=dJ_{z}[\mathbf {x} ,\mathbf {v} ]/dt=0,

hence

f(\mathbf {x} ,\mathbf {v} )=f(E[\mathbf {x} ,\mathbf {v} ],J[\mathbf {x} ,\mathbf {v} ],J_{z}[\mathbf {x} ,\mathbf {v} ])

is a solution to the Collisionless Boltzmann Equation. To show conservation of energy and z-angular momentum, we find

dE/dt={\partial E \over \partial t}+\mathbf {v} {\partial E \over \partial \mathbf {x} }+(\mathbf {-\nabla \Phi } ){\partial E \over \partial \mathbf {v} }={\partial \Phi  \over \partial t}+\mathbf {v} {\partial \Phi  \over \partial \mathbf {x} }+(\mathbf {-\nabla \Phi } )\mathbf {v} ={\partial \Phi  \over \partial t}=0

for any steady state potential.

dJ_{z}/dt={\partial J_{z} \over \partial t}+{\partial J_{z} \over \partial \mathbf {x} }\cdot \mathbf {v} -(\mathbf {\nabla \Phi } )\cdot {\partial J_{z} \over \partial \mathbf {v} }=0+[(V_{y})V_{x}+(-V_{x})V_{y}]-\left[(-y){x \over R}{\partial \Phi (R,z) \over \partial R}+(x){y \over R}{\partial \Phi (R,z) \over \partial R}\right]=0

around the z-axis of any axisymmetric potential, where

{\textstyle R={\sqrt {x^{2}+y^{2}}}}

.

Likewise the x and y components of the angular momentum are also conserved for a spherical potential. Hence $dJ/dt=0$ .

Using the chain rule, we have

{d \over dt}f(E[\mathbf {x} ,\mathbf {v} ],J[\mathbf {x} ,\mathbf {v} ],J_{z}[\mathbf {x} ,\mathbf {v} ])={\partial f \over \partial E}{dE \over dt}+{\partial f \over \partial J_{z}}{dJ_{z} \over dt}+{\partial f \over \partial J}{dJ \over dt}=0,

i.e.,

{\textstyle {d \over dt}f=0}

, so that CBE is satisfied.

E.g., consider a steady state model of the above uniform sphere. A solution for the Boltzmann Equation, written in spherical coordinates $(r,\theta ,\phi )$ and its velocity components $(V_{r},V_{\theta },V_{\phi })$ is

f(r,\theta ,\varphi ,V_{r},V_{\theta },V_{\varphi })={C_{0} \over V_{0}^{2}[0,2Q]_{\max }^{1/2}},

where we define

Q[\mathbf {x} ,\mathbf {v} ]\equiv \left(-V_{0}^{2}-E\right)+{J^{2} \over 2r_{0}^{2}}\left[{J_{z} \over |J_{z}|},0\right]_{\max }=\left({V_{e}^{2} \over 2}-{V_{r}^{2}+V_{\theta }^{2}+V_{\varphi }^{2} \over 2}\right)+{r^{2} \over r_{0}^{2}}{V_{\theta }^{2}+V_{\varphi }^{2} \over 2}\left[{V_{\varphi } \over |V_{\varphi }|},0\right]_{\max },

where

C_{0}=2\pi ^{-2}\rho _{0}

is a normalisation constant, which has the dimension of (mass) density. It is easy to see that

{\frac {d}{dt}}f={\frac {d}{dt}}Q(E,J,J_{z})={dE \over dt}={dJ_{z} \over dt}={dJ \over dt}=0.

A worked example on moments of distribution functions in a uniform spherical cluster[]

We can find out various moments of the above distribution function. E.g., the resulting density is

{\begin{aligned}\rho (r,\theta ,\varphi )&=\int _{-V_{e}(r)}^{V_{e}(r)}dV_{r}\int _{-V_{0}}^{V_{0}}dV_{\theta }\int _{0}^{V_{0}}dV_{\varphi }{C_{0} \over \max[0,2V_{0}^{4}Q]^{1/2}}\\[4pt]&=\int _{-1}^{1}\int _{-1}^{1}\int _{0}^{1}{(V_{e}du_{r})(V_{0}du_{\theta })(V_{0}du_{\varphi })C_{0} \over [0,(2V_{0}^{4})(V_{0}^{2}/2)(1-r^{2}/r_{0}^{2})(1-u_{r}^{2}-u_{\theta }^{2}-u_{\varphi }^{2})]_{\max }^{1/2}}\\[4pt]&=C_{0}{\int _{0}^{1}(1-u^{2})^{-1/2}(2\pi u^{2}du)}=\rho _{0}\end{aligned}}

is indeed a spherical (angle-independent) and uniform (radius-independent) density inside the edge, where the normalisation constant

C_{0}=2\pi ^{-2}

, and

(u_{r},u_{\theta },u_{\varphi })

is

(V_{r},V_{\theta },V_{\varphi })

rescaled by the function

V_{e}(r)

or

{\textstyle V_{0}=V_{e}(r)/{\sqrt {1-r^{2}/r_{0}^{2}}}}

respectively.

The streaming velocity is computed as the weighted mean of the velocity vector

{\begin{aligned}{\bar {\mathbf {V} }}&\equiv {\int fd\mathbf {V} ^{3}\mathbf {V}  \over \int fd\mathbf {V} ^{3}}\\&=\rho ^{-1}[V_{r},V_{\theta },V_{\varphi }]{C_{0} \over (2V_{0}^{4}Q)^{1/2}}\\&=\left[0,0,{\int _{0}^{1}(2du_{r})\int _{0}^{\sqrt {1-u_{r}^{2}}}(2du_{\theta })\int _{0}^{\sqrt {1-u_{r}^{2}-u_{\theta }^{2}}}du_{\varphi }{u_{\varphi }V_{0}(1-u_{r}^{2}-u_{\theta }^{2}-u_{\varphi }^{2})^{-1/2}} \over \int _{0}^{1}(2\pi UdU)\int _{0}^{\sqrt {1-U^{2}}}{du_{\varphi }(1-U^{2}-u_{\varphi }^{2})^{-1/2}}}\right]\\&=\left[0,0,{4V_{0} \over 3\pi }\right]\end{aligned}}

which has no net streaming in the meridional

(r,\theta )

plane, and has a flat azimuthal rotation.

Likewise thanks to the symmetry of $f(r,\theta ,\varphi ,V_{r},V_{\theta },V_{\varphi })=f(r,\theta ,\pm \varphi ,\pm V_{r},\pm V_{\theta },V_{\varphi })$ , we

${\overline {(\pm V_{r})V_{\varphi }}}=0,~{\overline {(\pm V_{r})V_{\theta }}}=0,~{\overline {(\pm V_{\theta })V_{\varphi }}}=0.$

Likewise the rms velocity in the rotation direction is computed by a weighted mean as follows, E.g.,

{\begin{aligned}{\overline {V_{\varphi }^{2}}}&\equiv {\int fd\mathbf {V} ^{3}V_{\varphi }^{2} \over \rho }\\&={\int _{0}^{1}(2du_{r})\int _{0}^{\sqrt {1-u_{r}^{2}}}(2du_{\theta })\int _{0}^{\sqrt {1-u_{r}^{2}-u_{\theta }^{2}}}du_{\varphi }{(u_{\varphi }V_{0})^{2} \over (1-u_{r}^{2}-u_{\theta }^{2}-u_{\varphi }^{2})^{1/2}} \over \int _{0}^{1}{(2\pi u^{2}du)(1-u^{2})^{-1/2}}}\\&=\left({\frac {V_{0}}{2}}\right)^{2}\\&={\int _{0}^{1}(2du_{r})\int _{0}^{\sqrt {1-u_{r}^{2}}}(2du_{\varphi })\int _{0}^{\sqrt {1-u_{r}^{2}-u_{\varphi }^{2}}}du_{\theta }{(u_{\theta }V_{0})^{2} \over (1-u_{r}^{2}-u_{\theta }^{2}-u_{\varphi }^{2})^{1/2}} \over \int _{0}^{1}{(2\pi u^{2}du)(1-u^{2})^{-1/2}}}\\&={\overline {V_{\theta }^{2}}}.\end{aligned}}

Likewise

{\overline {V_{r}^{2}}}={\int _{0}^{1}(du_{\varphi })\int _{0}^{\sqrt {1-u_{\varphi }^{2}}}(2du_{\theta })\int _{0}^{\sqrt {1-u_{\varphi }^{2}-u_{\theta }^{2}}}(2du_{r}){(u_{r}V_{e}(r))^{2} \over (1-u_{r}^{2}-u_{\theta }^{2}-u_{\varphi }^{2})^{1/2}} \over \int _{0}^{1}{(2\pi u^{2}du)(1-u^{2})^{-1/2}}}={V_{0}^{2}(1-r^{2}/r_{0}^{2}) \over 4}

So the pressure tensor or dispersion tensor is

{P \over \rho }=\sigma _{ij}^{2}={\overline {\mathbf {V} _{i}\mathbf {V} _{j}}}-{\bar {\mathbf {V} _{i}}}{\bar {\mathbf {V} _{j}}}=\left[\left(1-{r^{2} \over r_{0}^{2}}\right)\left({V_{0} \over 2}\right)^{2},0,0\right],\left[0,\left({V_{0} \over 2}\right)^{2},0\right],\left[0,0,\left(1-{64 \over 9\pi ^{2}}\right)\left({V_{0} \over 2}\right)^{2}\right]

with zero off-diagonal terms because of the symmetric velocity distribution. The larger tangential kinetic energy than that of radial motion seen in the diagonal dispersions is often phrased by an anisotropy parameter

\beta (r)\equiv 1-{{\bar {V_{\theta }^{2}}} \over {\bar {V_{r}^{2}}}}=-{r^{2} \over r_{0}^{2}-r^{2}};

a positive anisotropy would have meant that radial motion dominated, and a negative anisotropy means that tangential motion dominates.

A worked example of Virial Theorem[]

Twice kinetic energy per unit mass of the above uniform sphere is

{\begin{aligned}{2K \over M_{0}}&=M_{0}^{-1}\int _{0}^{M_{0}}{\overline {V_{r}^{2}+V_{\theta }^{2}+V_{\varphi }^{2}}}dM\\&=M_{0}^{-1}\int _{0}^{1}\left({V_{0}^{2} \over 4}+{V_{0}^{2} \over 4}+{(1-x^{2})V_{0}^{2} \over 4}\right)d(x^{3}M_{0})=0.6V_{0}^{2}\end{aligned}}

which balances the potential energy per unit mass of the uniform sphere

{\begin{aligned}{W \over M_{0}}&=M_{0}^{-1}\int _{0}^{r_{0}}{-GM(r)dM \over r}\\&=M_{0}^{-1}\int _{0}^{1}{x^{3}d(x^{3}) \over x}{-GM_{0}^{2} \over r_{0}}=-{3GM_{0} \over 5r_{0}}=-0.6V_{0}^{2}\end{aligned}}

as required by the Virial Theorem.

A worked example of Jeans Equation in a uniform sphere[]

Jeans Equation is a relation on how the pressure gradient of a system should be balancing the potential gradient for an equilibrium galaxy. In our uniform sphere, the potential gradient or gravity is

\nabla \Phi ={d\Phi  \over dr}={\Omega ^{2}r},~~\Omega ={V_{0} \over r_{0}}.

The radial pressure gradient

-{d(\rho \sigma _{r}^{2}) \over \rho dr}={\Omega ^{2}r \over 2}.

The reason for the discrepancy is partly due to centrifugal force

{{\bar {V}}_{\varphi }^{2} \over r}={64 \over 9\pi ^{2}}{V_{0}^{2} \over r},

and partly due to anisotropic pressure

{(\sigma _{\theta }^{2}-\sigma _{r}^{2}) \over r}+{(\sigma _{\varphi }^{2}-\sigma _{r}^{2}) \over r}={\Omega ^{2}r \over 2}+\left[{\Omega ^{2}r \over 2}-{64 \over 9\pi ^{2}}{V_{0}^{2} \over r}\right].

Now we can verify that

{\begin{aligned}{\Omega ^{2}r}&={\partial \Phi  \over \partial r}\\&=-{d(\rho \sigma _{r}^{2}) \over \rho dr}+{{\bar {V}}_{\varphi }^{2}+{\bar {V}}_{\theta }^{2}-2{\bar {V}}_{r}^{2} \over r}+{(\sigma _{\varphi }^{2}+\sigma _{\theta }^{2}-2\sigma _{r}^{2}) \over r}-{\partial V_{r} \over \partial t}\\&={-d(\rho \sigma _{r}^{2}) \over \rho dr}+{-2\beta {\overline {V_{r}^{2}}} \over r}-{\partial V_{r} \over \partial t}\end{aligned}}

This is essentially the Jeans equation in an anisotropic rotating spherical system.

A worked example of Jeans equation in a thick disk[]

Consider again the thick disk potential in the above example. If the density is that of a gas fluid, then the pressure would be zero at the boundary $z=\pm z_{0}$ . To find the peak of the pressure, we note that

P(R,z)=\int _{z}^{z_{0}}\partial _{z}\Phi \rho (R)dz=\rho (R)[\Phi (R,z_{0})-\Phi (R,z)].

So the fluid temperature per unit mass, i.e., the 1-dimensional velocity dispersion squared would be

\sigma ^{2}(R,z)={P(R,z) \over \rho (R)}={GM_{0} \over 2z_{0}}\log {(R_{0}+z_{0}+z+{\sqrt {R^{2}+(R_{0}+z0+z)^{2}}})(R_{0}+z_{0}-z+{\sqrt {R^{2}+(R_{0}+z_{0}-z)^{2}}}) \over (R_{0}+{\sqrt {R_{0}^{2}+R^{2}}})(R_{0}+2z_{0}+{\sqrt {R^{2}+(R_{0}+2z_{0})^{2}}})}.~~|z|\leq z_{0}

Along the rotational z-axis,

\sigma ^{2}(0,z)={GM_{0} \over 2z_{0}}\log {4(R_{0}+z0+z)(R_{0}+z_{0}-z) \over 4R_{0}(R_{0}+2z_{0})}

\sigma (0,z)={\sqrt {GM_{0} \over 2z_{0}}}{\sqrt {\log {(R_{0}+z_{0})^{2}-z^{2} \over (R_{0}+z_{0})^{2}-z_{0}^{2}}}},

which is clearly the highest at the centre and zero at the boundaries

z=\pm z_{0}

. Both the pressure and the dispersion peak at the midplane

z=0

. In fact the hottest and densest point is the centre, where

P(0,0)={M_{0} \over 4\pi R_{0}^{2}z_{0}}{-GM_{0}\log[1-(1+R_{0}/z_{0})^{-2}] \over 2z_{0}}.

References[]

^ ^a ^b ^c Murdin, Paul (2001). "Stellar Dynamics". Encyclopedia of Astronomy and Astrophysics. Nature Publishing Group. p. 1. ISBN 978-0750304405.
^ de Vita, Ruggero; Trenti, Michele; MacLeod, Morgan (2019-06-01). "Correlation between mass segregation and structural concentration in relaxed stellar clusters". Monthly Notices of the Royal Astronomical Society. 485 (4): 5752–5760. arXiv:1903.07619. doi:10.1093/mnras/stz815. ISSN 0035-8711.
^ https://cds.cern.ch/record/1053485/files/p37.pdf
^ ^a ^b Binney, James; Tremaine, Scott (2008). Galactic Dynamics. Princeton: Princeton University Press. pp. 35, 63, 65, 698. ISBN 978-0-691-13027-9.
^ Binney, James. "Galaxy Dynamics" (PDF). Princeton University Press. Retrieved 4 January 2022.
^ ^a ^b Sparke, Linda; Gallagher, John (2007). Galaxies in the Universe. New York: Cambridge. p. 131. ISBN 978-0521855938.
^ Henon, M (June 21, 1982). "Vlasov Equation?". Astronomy and Astrophysics. 114 (1): 211–212. Bibcode:1982A&A...114..211H.
^ Lynden-Bell, Donald (1962). "The stability and vibrations of a gas of stars". Monthly Notices of the Royal Astronomical Society. 124 (4): 279–296. Bibcode:1962MNRAS.124..279L. doi:10.1093/mnras/124.4.279.
^ Einstein, Albert (2002). "A Simple Application of the Newtonian Law of Gravitation to Star Clusters" (PDF). The Collected Papers of Albert Einstein. 7: 230–233 – via Princeton University Press.
^ Zwicky, Fritz (2009). "Republication of: The redshift of extragalactic nebulae". General Relativity and Gravitation. 41 (1): 207–224. Bibcode:2009GReGr..41..207Z. doi:10.1007/s10714-008-0707-4. S2CID 119979381.
^ Choudhuri, Arnab Rai (2010). Astrophysics for Physicists. New York: Cambridge University Press. pp. 213–214. ISBN 978-0-521-81553-6.

[:0-1] Murdin, Paul (2001). "Stellar Dynamics". Encyclopedia of Astronomy and Astrophysics. Nature Publishing Group. p. 1. ISBN 978-0750304405.

[2] Vita, Ruggero; Trenti, Michele; MacLeod, Morgan (2019-06-01). "Correlation between mass segregation and structural concentration in relaxed stellar clusters". Monthly Notices of the Royal Astronomical Society. 485 (4): 5752–5760. arXiv:1903.07619. doi:10.1093/mnras/stz815. ISSN 0035-8711.

[3] ttps://cds.cern.ch/record/1053485/files/p37.pdf

[:12-4] Binney, James; Tremaine, Scott (2008). Galactic Dynamics. Princeton: Princeton University Press. pp. 35, 63, 65, 698. ISBN 978-0-691-13027-9.

[Galaxy_Dynamics-5] Binney, James. "Galaxy Dynamics" (PDF). Princeton University Press. Retrieved 4 January 2022.

[:2-6] Sparke, Linda; Gallagher, John (2007). Galaxies in the Universe. New York: Cambridge. p. 131. ISBN 978-0521855938.

[7] Henon, M (June 21, 1982). "Vlasov Equation?". Astronomy and Astrophysics. 114 (1): 211–212. Bibcode:1982A&A...114..211H.

[8] Lynden-Bell, Donald (1962). "The stability and vibrations of a gas of stars". Monthly Notices of the Royal Astronomical Society. 124 (4): 279–296. Bibcode:1962MNRAS.124..279L. doi:10.1093/mnras/124.4.279.

[9] Einstein, Albert (2002). "A Simple Application of the Newtonian Law of Gravitation to Star Clusters" (PDF). The Collected Papers of Albert Einstein. 7: 230–233 – via Princeton University Press.

[10] Zwicky, Fritz (2009). "Republication of: The redshift of extragalactic nebulae". General Relativity and Gravitation. 41 (1): 207–224. Bibcode:2009GReGr..41..207Z. doi:10.1007/s10714-008-0707-4. S2CID 119979381.

[11] Choudhuri, Arnab Rai (2010). Astrophysics for Physicists. New York: Cambridge University Press. pp. 213–214. ISBN 978-0-521-81553-6.

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Stellar dynamics

Contents

Key concepts[]

Connections to gravitational dynamical friction and gravitational gas accretion physics[]

Gravitational encounters and relaxation[]

Connections to statistical mechanics and plasma physics[]

Applications and examples[]

A worked example of Poisson Equation and oscillation frequencies in a thick disk[]

A worked example for neutrinos in galaxies[]

A worked example of the Poisson Equation and escape speed in a uniform sphere[]

A worked example on Jeans theorem and CBE[]

A worked example on moments of distribution functions in a uniform spherical cluster[]

A worked example of Virial Theorem[]

A worked example of Jeans Equation in a uniform sphere[]

A worked example of Jeans equation in a thick disk[]

See also[]

Further reading[]

References[]