Stellar dynamics

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. From Newton's second law an equation describing the interactions of an isolated stellar system can be written down as,

${\displaystyle m_{i}{\frac {d^{2}\mathbf {r_{i}} }{dt^{2}}}=\sum _{i=1 \atop i\neq j}^{N}{\frac {Gm_{i}m_{j}\left(\mathbf {r} _{i}-\mathbf {r} _{j}\right)}{\left\|\mathbf {r} _{i}-\mathbf {r} _{j}\right\|^{3}}}}$

which is simply a formulation of the N-body problem. For an N-body system, any individual member, ${\displaystyle m_{i}}$ is influenced by the gravitational potentials of the remaining ${\displaystyle m_{j}}$ 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.[2] The gravitational potential, ${\displaystyle \Phi }$, of a system is related to the gravitational field, ${\displaystyle \mathbf {\vec {g}} }$ by:

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

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

${\displaystyle \nabla ^{2}\Phi =4\pi G\rho }$

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, ${\displaystyle f(\mathbf {x} ,\mathbf {v} ,t)}$, is defined in a way such that

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

represents the probability of finding a given star with position ${\displaystyle \mathbf {x} }$ around a differential volume ${\displaystyle {\text{d}}\mathbf {x} }$ and velocity ${\displaystyle {\text{v}}}$ around a differential volume ${\displaystyle {\text{d}}\mathbf {v} }$. 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.[4] 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.[5]

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.[6][7] 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.[8]

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.[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.

• Stellar classification
• Boltzmann equation
• Dynamical friction
• Jeans equations
• Mass segregation (astronomy)
• N-body problem
• Virial theorem

• Dynamics and Evolution of Galactic Nuclei, D. Merritt (2013). Princeton University Press.
• Galactic Dynamics, J. Binney and S. Tremaine (2008). Princeton University Press.
• Gravitational N-Body Simulations: Tools and Algorithms, S. Aarseth (2003). Cambridge University Press.
• Principles of Stellar Dynamics, S. Chandrasekhar (1960). Dover.