Жұлдыздық динамика: Жұлдыздардың өзара тартылысы мен қозғалысы

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.

There are three related approximations made in the Newtonian EOM and Poisson Equation above.

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 .

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.

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, , 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, , 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 to equal , 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 objects is approximately

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. A better approximation of the ratio of timescales is , 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 have much smoother potential, typically takes weak encounters to build a strong deflection to change orbital energy significantly.

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. 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. 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. 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.

Insert in the cylindrical Poisson eq. which drops with radius, and is zero beyond and uniform along the z direction within the boundary.