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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 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.

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 is usually concerned with the more global, statistical properties of several orbits rather than with the specific data on the positions and velocities of individual orbits.

The motions of stars in a galaxy or in a globular cluster are principally determined by the average distribution of the other, distant stars, and little influenced by the nearest stars. Stellar encounters involve processes such as relaxation, mass segregation, tidal forces, and dynamical friction that influence the trajectories of the system's members.

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.

Key Concepts
Stellar dynamics involves determining the gravitational potential of a substantial amount 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 describing the interactions of an isolated stellar system can be written down as,

$$m_i\frac{d \mathbf{r_i}}{dt} = \sum_{i=1 \atop i \ne j}^N \frac{G m_i m_j \left(\mathbf{r}_i - \mathbf{r}_j\right)}{\left\| \mathbf{r}_i - \mathbf{r}_j\right\|^3} $$

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, it is not feasible to calculate 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. The gravitational potential, $$\Phi$$, of a system is related to the gravitational field, $$\mathbf{\vec{g}}$$ by:

$$\mathbf{\vec{g}} = -\nabla \Phi$$

whereas the mass density, $$\rho $$, is related to the potential via Poisson's equation:

$$\nabla^2\Phi = 4\pi G \rho $$

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 if the change in potential energy between the two is greater than or equal to their initial kinetic energy. Strong encounters are rare, and they are typically only considered important in dense stellar systems, such as the cores of globular clusters. Weak encounters have a more profound effect on the evolution of a stellar system over the course of many orbits. 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:

$$t_{\text{relax}} \backsimeq \frac{0.1N}{\ln N}t_\text{cross}$$

where $$t_{\text{cross}}$$ is known as the crossing time, the time it takes for a star to travel across the stellar system once.

The relaxation time identifies collisionless vs. collisional stellar systems. Dynamics on timescales less than the relaxation time are defined to be collisionless. They are also identified as systems where subject stars interact with a smooth gravitational potential as opposed to the sum of point-mass potentials. 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.

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

$$f(\mathbf{x},\mathbf{v},t)\,\text{d}\mathbf{x}\,\text{d}\mathbf{v}$$

represents the probability of finding a given star with position $$\mathbf{x}$$ around a differential volume $$\text{d}\mathbf{x}$$ and velocity $$\text{v}$$ around a differential volume $$\text{d}\mathbf{v}$$. The distribution is function is normalized such that integrating it over all positions and velocities will equal unity. 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 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. 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. 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.

Applications
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.