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 ≫ 10. {\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.
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,
( L / V , M / N ) {\displaystyle (L/V,M/N)}
∼ ( 10 − 8 pc / 500 km/s , 1 M ⊙ / 10 55 = m p ) {\displaystyle \sim (10^{-8}{\text{pc}}/500{\text{km/s}},1M_{\odot }/10^{55}=m_{p})} at stellar surfaces,
∼ ( 10 − 4 pc / 10 km/s , 0.1 M ⊙ / 10 54 ∼ m p ) {\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,
∼ ( 10 − 1 pc / 100 km/s , 10 M ⊙ / 10 56 ∼ m p ) {\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
1000 pc / 1 km/s = 1000 Myr = HubbleTime / 14. {\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 ( L / V , M / N ) {\displaystyle (L/V,M/N)}
∼ ( 10 p c / 10 k m / s , 1000 M ⊙ / 1000 ) {\displaystyle \sim (\mathrm {10pc/10km/s} ,1000M_{\odot }/1000)} for M13 Star Cluster,
∼ ( 100 k p c / 100 k m / s , 10 11 M ⊙ / 10 11 ) {\displaystyle \sim (\mathrm {100kpc/100km/s} ,10^{11}M_{\odot }/10^{11})} for M31 Disk Galaxy,
∼ ( 10 M p c / 1000 k m / s , 10 14 M ⊙ / 10 77 = m ν ) {\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,
m i d 2 r i d t 2 = ∑ i = 1 i ≠ j N G m i m j ( r j − r i ) ‖ r j − r i ‖ 3 . {\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}}}.}
Here in the N-body system, any individual member, m i {\displaystyle m_{i}} is influenced by the gravitational potentials of the remaining m j {\displaystyle m_{j}} 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.
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. The gravitational potential, Φ {\displaystyle \Phi } , of a system is related to the acceleration and the gravitational field, g {\displaystyle \mathbf {g} } by:
d 2 r i d t 2 = g → = − ∇ r i Φ ( r i ) , Φ ( r i ) = − ∑ k = 1 k ≠ i N G m k ‖ r i − r k ‖ , {\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\|}},}
whereas the potential is related to a (smoothened) mass density, ρ {\displaystyle \rho } , via the Poisson's equation in the integral form
Φ ( r ) = − ∫ G ρ ( R ) d 3 R ‖ r − R ‖ {\displaystyle \Phi (\mathbf {r} )=-\int {G\rho (\mathbf {R} )d^{3}\mathbf {R} \over \left\|\mathbf {r} -\mathbf {R} \right\|}}
or the more common differential form
∇ 2 Φ = 4 π G ρ . {\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
Φ ( r ) ≡ ( − V 0 2 ) + [ r 2 − r 0 2 2 r 0 2 , 1 − r 0 r ] max V 0 2 ≡ Φ ( r 0 ) − V e 2 ( r ) 2 , Φ ( r 0 ) = − V 0 2 , g = − ∇ Φ ( r ) = − Ω 2 r H ( r 0 − r ) − G M 0 r 2 H ( r − r 0 ) , Ω = V 0 r 0 , M 0 = V 0 2 r 0 G , {\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}}}
where V e ( r ) {\displaystyle V_{e}(r)} takes the meaning of the speed to "escape to the edge" r 0 {\displaystyle r_{0}} , and 2 V 0 {\displaystyle {\sqrt {2}}V_{0}} 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 V 0 {\displaystyle V_{0}} by computing the corresponding density using the spherical Poisson Equation
G ρ = d 4 π r 2 d r r 2 d Φ d r = d ( G M ) 4 π r 2 d r = 3 V 0 2 4 π r 0 2 H ( r 0 − r ) , {\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),}
where the enclosed mass
M ( r ) = r 2 d Φ G d r = ∫ 0 r d r ∫ 0 π ( r d θ ) ∫ 0 2 π ( r sin θ d φ ) ρ 0 H ( r 0 − r ) = M 0 x 3 | x = r r 0 . {\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 r 0 {\displaystyle r_{0}} , total mass M 0 {\displaystyle M_{0}} with
V 0 r 0 ≡ 4 π G ρ 0 3 = G M 0 r 0 3 . {\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.
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
( v / c ) 2 ≪ 10 − 4 {\displaystyle (v/c)^{2}\ll 10^{-4}} as typical stellar 3-dimensional speed, v ∼ 3 − 3000 {\displaystyle v\sim 3-3000} 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,
Q Eddington = σ e 4 π m H c L ⊙ r 2 G M ⊙ r 2 = 1 30 , 000 , {\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},}
hence radiation force is negligible in general, except perhaps around a luminous O-type star of mass 30 M ⊙ {\displaystyle 30M_{\odot }} , or around a black hole accreting gas at the Eddington limit so that its luminosity-to-mass ratio L ∙ / M ∙ {\displaystyle L_{\bullet }/M_{\bullet }} is defined by Q Eddington = 1 {\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 s ≤ s Loss = 6 G M ∙ c 2 {\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 θ ≪ 1 {\displaystyle \theta \ll 1} have small angular momentum per unit mass J ≡ r v sin θ ≤ J loss = 4 G M ∙ c . {\displaystyle J\equiv rv\sin \theta \leq J_{\text{loss}}={\frac {4GM_{\bullet }}{c}}.} Their small angular momentum (due to ) does not make a high enough barrier near s Loss {\displaystyle s_{\text{Loss}}} to force the particle to turn around. The effective potential
Φ eff ( r ) ≡ E − r ˙ 2 2 = J 2 2 r 2 + Φ ( r ) , {\displaystyle \Phi _{\text{eff}}(r)\equiv E-{{\dot {r}}^{2} \over 2}={J^{2} \over 2r^{2}}+\Phi (r),} is always positive infinity in Newtonian gravity. However, in GR, it nosedives to minus infinity near 6 G M ∙ c 2 {\displaystyle {\frac {6GM_{\bullet }}{c^{2}}}} if J ≤ 4 G M ∙ c . {\displaystyle J\leq {\frac {4GM_{\bullet }}{c}}.}
Sparing a rigorous GR treatment, one can verify this s loss , J loss {\displaystyle s_{\text{loss}},J_{\text{loss}}} by computing the last stable circular orbit, where the effective potential is at an inflection point Φ eff ″ ( s loss ) = Φ eff ′ ( s loss ) = 0 {\displaystyle \Phi ''_{\text{eff}}(s_{\text{loss}})=\Phi '_{\text{eff}}(s_{\text{loss}})=0} using an approximate classical potential of a Schwarzschild black hole
Φ ( r ) = − ( 4 G M ∙ / c ) 2 2 r 2 [ 1 + 3 ( 6 G M ∙ / c 2 ) 2 8 r 2 ] − G M ∙ r [ 1 − ( 6 G M ∙ / c 2 ) 2 r 2 ] . {\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, i.e.,
( 1 − 1.5 ) ≥ Q tide ≡ G M ⊙ / R ⊙ 2 [ G M ∙ / s Hill 2 − G M ∙ / ( s Hill + R ⊙ ) 2 ] , s Hill → R ⊙ ( ( 2 − 3 ) G M ∙ G M ⊙