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.

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.

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,

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.

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.

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

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

After certain time of relaxations the heavy black hole's kinetic energy should be in equal partition with the less-massive background objects.

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.

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.

A background of elementary (gas or dark) particles can also induce dynamical friction, which scales with the mass density of the surrounding medium,

; 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.

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

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.

and being ambiguous whether the geometry of the system is a thin/thick gas/stellar disk or a (non)-uniform stellar/dark sphere with or without a boundary, and about the subtle distinctions among the kinetic energies from the local Circular rotation speed

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:

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.

[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.

[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.

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.

[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.

Apply Gauss's theorem by integrating the vertical force over the entire disk upper and lower boundaries, we have

, 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.

Inside this constant density core region, individual stars go on resonant harmonic oscillations of angular frequency

where the global average (indicated by the overline bar) of flow implies uniform pattern of flat azimuthal rotation, but zero net streaming everywhere in the meridional

Note while there is no Dark Matter in producing the previous flat rotation curve, the price is shown by the reduction factor

, which are useful in modelling ocean currents on the rotating earth surface or angular momentum transfer in accretion disks, where the frictional term

and Phase space density and especially the Jeans equation, we can extract a general theme, again using the Spherical cow approach.

One can project the phase space into these moments, which is easily if in a highly spherical system, which admits conservations of energy