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
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,
at stellar surfaces, around Sun-like stars or km-sized stellar black holes, 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
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 for M13 Star Cluster, for M31 Disk Galaxy, 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,
Here in the N-body system, any individual member, is influenced by the gravitational potentials of the remaining 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, , of a system is related to the acceleration and the gravitational field, by:
whereas the potential is related to a (smoothened) mass density, , via the Poisson's equation in the integral form
or the more common differential form
An example of the Poisson Equation and escape speed in a uniform sphere
Consider an analytically smooth spherical potential
where takes the meaning of the speed to "escape to the edge" , and 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 by computing the corresponding density using the spherical Poisson Equation
where the enclosed mass
Hence the potential model corresponds to a uniform sphere of radius , total mass with
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 as typical stellar 3-dimensional speed, 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,
hence radiation force is negligible in general, except perhaps around a luminous O-type star of mass , or around a black hole accreting gas at the Eddington limit so that its luminosity-to-mass ratio is defined by .
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
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 have small angular momentum per unit mass Their small angular momentum (due to ) does not make a high enough barrier near to force the particle to turn around.
The effective potential is always positive infinity in Newtonian gravity. However, in GR, it nosedives to minus infinity near if
Sparing a rigorous GR treatment, one can verify this by computing the last stable circular orbit, where the effective potential is at an inflection point using an approximate classical potential of a Schwarzschild black hole
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.,
For typical black holes of the destruction radius where 0.001pc is the stellar spacing in the densest stellar systems (e.g., the nuclear star cluster in the Milky Way centre). Hence (main sequence) stars are generally too compact internally and too far apart spaced to be disrupted by even the strongest black hole tides in galaxy or cluster environment.
Radius of sphere of influence A particle of mass with a relative speed V will be deflected when entering the (much larger) cross section of a black hole. This so-called sphere of influence is loosely defined by, up to a Q-like fudge factor ,
hence for a Sun-like star we have,
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 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 inside all star clusters and in galaxies.
Connections between star loss cone and gravitational gas accretion physics First consider a heavy black hole of mass is moving through a dissipational gas of (rescaled) thermal sound speed and density , then every gas particle of mass m will likely transfer its relative momentum to the BH when coming within a cross-section of radius In a time scale 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 , dissipate kinetic energy by gas collisions and fall in the black hole. The gas capture rate is
where the polytropic index is the sound speed in units of velocity dispersion squared, and the rescaled sound speed allows us to match the Bondi spherical accretion rate, for the adiabatic gas , compared to of the isothermal case .
Coming back to star tidal disruption and star capture by a (moving) black hole, setting , we could summarise the BH's growth rate from gas and stars, with,
because the black hole consumes a fractional/most part of star/gas particles passing its sphere of influence.
Gravitational dynamical friction Consider the case that a heavy black hole of mass moves relative to a background of stars in random motion in a cluster of total mass with a mean number density within a typical size .
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
where is called a dynamical friction time.
Dynamical friction time vs Crossing time in a virialised system Consider a Mach-1 BH, which travels initially at the sound speed , hence its Bondi radius satisfies
where the sound speed is with the prefactor fixed by the fact that for a uniform spherical cluster of the mass density , half of a circular period is the time for "sound" to make a oneway crossing in its longest dimension, i.e.,
It is customary to call the "half-diameter" crossing time the dynamical time scale.
Assume the BH stops after traveling a length of with its momentum deposited to stars in its path over crossings, then the number of stars deflected by the BH's Bondi cross section per "diameter" crossing time is
More generally, the Equation of Motion of the BH at a general velocity in the potential of a sea of stars can be written as
and the Coulomb logarithm modifying factor discounts friction on a supersonic moving BH with mass . As a rule of thumb, it takes about a sound crossing time to "sink" subsonic BHs, from the edge to the centre without overshooting, if they weigh more than 1/8th of the total cluster mass. Lighter and faster holes can stay afloat much longer.
More rigorous formulation of dynamical friction
The full Chandrasekhar dynamical friction formula for the change in velocity of the object involves integrating over the phase space density of the field of matter and is far from transparent.
It reads as
where
is the number of particles in an infinitesimal cylindrical volume of length and a cross-section within the black hole's sphere of influence.
Like the "Couloumb logarithm" factors in the contribution of distant background particles, here the factor 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 . Also the bigger the system, the greater is .