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. 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.[1]

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.[1]

The motions of stars in a galaxy or in a globular cluster are principally determined by the average distribution of the other, distant 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 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,

which is simply a formulation of the N-body problem. For an N-body system, any individual member, is influenced by the gravitational potentials of the remaining 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, , of a system is related to the gravitational field, by:

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

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.[3] 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, , 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 relaxation time for a stellar system of objects is approximately equal to:

where is known as the crossing time, the time it takes for a star to travel across the galaxy 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.[2] 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.[3]

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, , is defined in a way such that

represents the probability of finding a given star with position around a differential volume and velocity around a differential volume . 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.

See also

Further reading


  1. ^ a b c Murdin, Paul (2001). "Stellar Dynamics". Encyclopedia of Astronomy and Astrophysics. Nature Publishing Group. p. 1. ISBN 978-0750304405.
  2. ^ a b Binney, James; Tremaine, Scott (2008). Galactic Dynamics. Princeton: Princeton University Press. pp. 35, 63, 65, 698. ISBN 978-0-691-13027-9.
  3. ^ a b Sparke, Linda; Gallagher, John (2007). Galaxies in the Universe. New York: Cambridge. p. 131. ISBN 978-0521855938.
  4. ^ Henon, M (June 21, 1982). "Vlasov Equation?". Astronomy and Astrophysics. 114: 211–212.
  5. ^ Lynden-Bell, Donald (1962). "The stability and vibrations of a gas of stars". Monthly Notices of the Royal Astronomical Society. 124: 279–296.
  6. ^ Einstin, Albert (2002). "A Simple Application of the Newtonian Law of Gravitation to Star Clusters" (PDF). The Collected Papers of Albert Einstein. 7: 230–233 – via Princeton University Press.
  7. ^ Zwicky, Fritz (2009). "Republication of: The redshift of extragalactic nebulae". General Relativity and Gravitation. 41: 207–224.
  8. ^ Choudhuri, Arnab Rai (2010). Astrophysics for Physicists. New York: Cambridge University Press. pp. 213–214. ISBN 978-0-521-81553-6.

Astrophysics is the branch of astronomy that employs the principles of physics and chemistry "to ascertain the nature of the astronomical objects, rather than their positions or motions in space". Among the objects studied are the Sun, other stars, galaxies, extrasolar planets, the interstellar medium and the cosmic microwave background. Emissions from these objects are examined across all parts of the electromagnetic spectrum, and the properties examined include luminosity, density, temperature, and chemical composition. Because astrophysics is a very broad subject, astrophysicists apply concepts and methods from many disciplines of physics, including classical mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics.

In practice, modern astronomical research often involves a substantial amount of work in the realms of theoretical and observational physics. Some areas of study for astrophysicists include their attempts to determine the properties of dark matter, dark energy, black holes, and other celestial bodies; whether or not time travel is possible, wormholes can form, or the multiverse exists; and the origin and ultimate fate of the universe. Topics also studied by theoretical astrophysicists include Solar System formation and evolution; stellar dynamics and evolution; galaxy formation and evolution; magnetohydrodynamics; large-scale structure of matter in the universe; origin of cosmic rays; general relativity and physical cosmology, including string cosmology and astroparticle physics.

Box orbit

In stellar dynamics, a box orbit refers to a particular type of orbit that can be seen in triaxial systems, i.e. systems that do not possess a symmetry around any of its axes. They contrast with the loop orbits that are observed in spherically symmetric or axisymmetric systems.

In a box orbit, a star oscillates independently along the three different axes as it moves through the system. As a result of this motion, it fills in a (roughly) box-shaped region of space. Unlike loop orbits, the stars on box orbits can come arbitrarily close to the center of the system. As a special case, if the frequencies of oscillation in different directions are commensurate, the orbit will lie on a one- or two-dimensional manifold and can avoid the center. Such orbits are sometimes called "boxlets".

Cartwheel Galaxy

The Cartwheel Galaxy (also known as ESO 350-40 or PGC 2248) is a lenticular galaxy and ring galaxy about 500 million light-years away in the constellation Sculptor. It is an estimated 150,000 light-years diameter, and has a mass of about 2.9–4.8 × 109 solar masses; its outer ring has a circular velocity of 217 km/s.It was discovered by Fritz Zwicky in 1941. Zwicky considered his discovery to be "one of the most complicated structures awaiting its explanation on the basis of stellar dynamics."An estimation of the galaxy's span resulted in a conclusion of 150,000 light years, which is slightly larger than the Milky Way.

Chandrasekhar's variational principle

In astrophysics, Chandrasekhar's variational principle provides the stability criterion for a static barotropic star, subjected to radial perturbation, named after the Indian American astrophysicist Subrahmanyan Chandrasekhar.

Chandrasekhar's white dwarf equation

In astrophysics, Chandrasekhar's white dwarf equation is an initial value ordinary differential equation introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar, in his study of the gravitational potential of completely degenerate white dwarf stars. The equation reads as

with initial conditions

where measures the density of white dwarf, is the non-dimensional radial distance from the center and is a constant which is related to the density of the white dwarf at the center. When , this equation reduces to Lane–Emden equation with polytropic index . In the Lane–Emden equation, the density at the centre can be scaled out of the equation, but for white-dwarfs, the central density is directly tied to the equation.

Chandrasekhar limit

The Chandrasekhar limit () is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about 1.4 M☉ (2.765×1030 kg).White dwarfs resist gravitational collapse primarily through electron degeneracy pressure (compare main sequence stars, which resist collapse through thermal pressure). The Chandrasekhar limit is the mass above which electron degeneracy pressure in the star's core is insufficient to balance the star's own gravitational self-attraction. Consequently, a white dwarf with a mass greater than the limit is subject to further gravitational collapse, evolving into a different type of stellar remnant, such as a neutron star or black hole. Those with masses under the limit remain stable as white dwarfs.The limit was named after Subrahmanyan Chandrasekhar, an Indian astrophysicist who improved upon the accuracy of the calculation in 1930, at the age of 20, in India by calculating the limit for a polytrope model of a star in hydrostatic equilibrium, and comparing his limit to the earlier limit found by E. C. Stoner for a uniform density star. Importantly, the existence of a limit, based on the conceptual breakthrough of combining relativity with Fermi degeneracy, was indeed first established in separate papers published by Wilhelm Anderson and E. C. Stoner in 1929. The limit was initially ignored by the community of scientists because such a limit would logically require the existence of black holes, which were considered a scientific impossibility at the time. That the roles of Stoner and Anderson are often forgotten in the astronomy community has been noted.

Chandrasekhar potential energy tensor

In astrophysics, Chandrasekhar potential energy tensor provides the gravitational potential of a body due to its own gravity created by the distribution of matter across the body, named after the Indian American astrophysicist Subrahmanyan Chandrasekhar. The Chandrasekhar tensor is a generalization of potential energy in other words, the trace of the Chandrasekhar tensor provides the potential energy of the body.

Division on Dynamical Astronomy

The Division on Dynamical Astronomy (DDA) is a branch of the American Astronomical Society that focuses on the advancement of all aspects of dynamical astronomy, including celestial mechanics, solar system dynamics, stellar dynamics, as well as the dynamics of the interstellar medium and galactic dynamics, and coordination of such research with other branches of science. It awards the Brouwer Award every year, which was established to recognize outstanding contributions to the field of Dynamical Astronomy, including celestial mechanics, astrometry, geophysics, stellar systems, galactic and extra galactic dynamics. The Division also awards the Vera Rubin Early Career Prize for promise of continued excellence for an astronomer no more than 10 years beyond receipt of their doctorate.

Dynamical friction

In astrophysics, dynamical friction or Chandrasekhar friction, sometimes called gravitational drag, is loss of momentum and kinetic energy of moving bodies through gravitational interactions with surrounding matter in space. It was first discussed in detail by Subrahmanyan Chandrasekhar in 1943.

Emden–Chandrasekhar equation

In astrophysics, the Emden–Chandrasekhar equation is a dimensionless form of the Poisson equation for the density distribution of a spherically symmetric isothermal gas sphere subjected to its own gravitational force, named after Robert Emden and Subrahmanyan Chandrasekhar. The equation was first introduced by Robert Emden in 1907. The equation reads

where is the dimensionless radius and is the related to the density of the gas sphere as , where is the density of the gas at the centre. The equation has no known explicit solution. If a polytropic fluid is used instead of an isothermal fluid, one obtains the Lane–Emden equation. The isothermal assumption is usually modeled to describe the core of a star. The equation is solved with the initial conditions,

The equation appears in other branches of physics as well, for example the same equation appears in the Frank-Kamenetskii explosion theory for a spherical vessel. The relativistic version of this spherically symmetric isothermal model was studied by Subrahmanyan Chandrasekhar in 1972.

Fesenkov Astrophysical Institute

Fesenkov Astrophysical Institute (Астрофизический институт имени В. Г. Фесенкова, АФИФ), or FAPHI, is a research institute in Almaty, Kazakhstan. The institute was founded in 1941 as the Institute for Astronomy and Physics of the Kazakh branch of the USSR Academy of Sciences, when a group of Soviet astronomers was evacuated during World War II from the European parts of the USSR to Almaty. In 1948 G.A. Tikhov had organized an independent sector of astrobotany, and in 1950 astronomers established the Astrophysical Institute of the Kazakh SSR. In 1989 the Institute was renamed after Vasily Fesenkov, one of its founders.

FAPHI conducts both observational and theoretical research. The prime objects of observations are the Sun, outer planets, comets, Herbig Ae/Be stars, and active galaxies. The topics of theoretical research include stellar dynamics and computational astrophysics, active galactic nuclei, cosmology, physics of comets and interstellar medium. The institute runs three observational bases in mountains near Almaty: Kamenskoe Plateau Observatory, Assy-Turgen Observatory and Tien Shan Astronomical Observatory.

FAPHI is a member of the International Astronomical Union.

Jack G. Hills

Jack G. Hills is a theorist of stellar dynamics. He worked on the Oort cloud; the inner part of it, the Hills cloud, was named after him. He spent much of his professional career at Los Alamos National Laboratory, which named him a Laboratory Fellow in 1998.

Jeans equations

The Jeans equations describe the motion of a collection of stars in a gravitational field.

If n = n(x,t) is the density of stars in space, as a function of position x = (x1x2x3) and time t, v = (v1v2v3) is the velocity, and Φ = Φ(x,t) is the gravitational potential, the Jeans equations may be written as

Here, the <…> notation means an average at a given point and time (x,t), so that, for example, is the average of component 1 of the velocity of the stars at a given point and time. The second set of equations may alternately be written as

where measures the velocity dispersion in components i and j at a given point.

The Jeans equations are analogous to the Euler equations for fluid flow and may be derived from the collisionless Boltzmann equation. They were originally derived by James Clerk Maxwell but were first applied to stellar dynamics by James Jeans.

Lindblad resonance

A Lindblad resonance, named for the Swedish galactic astronomer Bertil Lindblad, is an orbital resonance in which an object's epicyclic frequency (the rate at which one periapse follows another) is a simple multiple of some forcing frequency. Resonances of this kind tend to increase the object's orbital eccentricity and to cause its longitude of periapse to line up in phase with the forcing. Lindblad resonances drive spiral density waves both in galaxies (where stars are subject to forcing by the spiral arms themselves) and in Saturn's rings (where ring particles are subject to forcing by Saturn's moons).

Lindblad resonances affect stars at such distances from a disc galaxy's centre where the natural frequency of the radial component of a star's orbital velocity is close to the frequency of the gravitational potential maxima encountered during its course through the spiral arms. If a star's orbital speed around the galactic centre is greater than that of the part of the spiral arm through which it is passing, then an inner Lindblad resonance occurs - if smaller, then an outer Lindblad resonance. At an inner resonance, a star's orbital speed is increased, moving the star outwards, and decreased for an outer resonance causing inward movement.

Michel Hénon

Michel Hénon (French: [enɔ̃]; 1931 in Paris – 7 April 2013 in Nice) was a French mathematician and astronomer. He worked for a long time at the Nice Observatory.

In astronomy, Hénon is well known for his contributions to stellar dynamics. In the late 1960s and early 1970s he made important contributions on the dynamical evolution of star clusters, in particular globular clusters. He developed a numerical technique using Monte Carlo methods to follow the dynamical evolution of a spherical star cluster much faster than the so-called n-body methods.

In mathematics, he is well known for the Hénon map, a simple discrete dynamical system that exhibits chaotic behavior.

He published a two-volume work on the restricted three-body problem.

N-body units

N-body units are a completely self-contained system of units used for N-body simulations of self-gravitating systems in astrophysics. In this system, the base physical units are chosen so that the total mass, M, the gravitational constant, G, and the virial radius, R, are normalized. The underlying assumption is that the system of N objects (stars) satisfies the virial theorem. The consequence of standard N-body units is that the velocity dispersion of the system, v, is and that the dynamical or crossing time, t, is . The use of standard N-body units was advocated by Michel Hénon in 1971. Early adopters of this system of units included H. Cohn in 1979 and D. Heggie and R. Mathieu in 1986. At the conference MODEST14 in 2014, D. Heggie proposed that the community abandon the name "N-body units" and replace it with the name "Hénon units" to commemorate the originator.

NEMO (Stellar Dynamics Toolbox)

NEMO (Not Everybody Must Observe) is a toolkit for stellar dynamics. At its core it manipulates an n-body system (snapshot), but can also derive or compute orbits, derive images and extract tables to take to other analysis systems.

Schönberg–Chandrasekhar limit

In stellar astrophysics, the Schönberg–Chandrasekhar limit is the maximum mass of a non-fusing, isothermal core that can support an enclosing envelope. It is expressed as the ratio of the core mass to the total mass of the core and envelope. Estimates of the limit depend on the models used and the assumed chemical compositions of the core and envelope; typical values given are from 0.10 to 0.15 (10% to 15% of the total stellar mass). This is the maximum to which a helium-filled core can grow, and if this limit is exceeded, as can only happen in massive stars, the core collapses, releasing energy that causes the outer layers of the star to expand to become a red giant. It is named after the astrophysicists Subrahmanyan Chandrasekhar and Mario Schönberg, who estimated its value in a 1942 paper. They estimated it to be

The Schönberg–Chandrasekhar limit comes into play when fusion in a main-sequence star exhausts the hydrogen at the center of the star. The star then contracts until hydrogen fuses in a shell surrounding a helium-rich core, both of which are surrounded by an envelope consisting primarily of hydrogen. The core increases in mass as the shell burns its way outwards through the star. If the star's mass is less than approximately 1.5 solar masses, the core will become degenerate before the Schönberg–Chandrasekhar limit is reached, and, on the other hand, if the mass is greater than approximately 6 solar masses, the star leaves the main sequence with a core mass already greater than the Schönberg–Chandrasekhar limit so its core is never isothermal before helium fusion. In the remaining case, where the mass is between 1.5 and 6 solar masses, the core will grow until the limit is reached, at which point it will contract rapidly until helium starts to fuse in the core.

Subrahmanyan Chandrasekhar

Subrahmanyan Chandrasekhar ; 19 October 1910 – 21 August 1995) was an Indian American astrophysicist who spent his professional life in the United States. He was awarded the 1983 Nobel Prize for Physics with William A. Fowler for "...theoretical studies of the physical processes of importance to the structure and evolution of the stars". His mathematical treatment of stellar evolution yielded many of the current theoretical models of the later evolutionary stages of massive stars and black holes. The Chandrasekhar limit is named after him.

Chandrasekhar worked on a wide variety of physical problems in his lifetime, contributing to the contemporary understanding of stellar structure, white dwarfs, stellar dynamics, stochastic process, radiative transfer, the quantum theory of the hydrogen anion, hydrodynamic and hydromagnetic stability, turbulence, equilibrium and the stability of ellipsoidal figures of equilibrium, general relativity, mathematical theory of black holes and theory of colliding gravitational waves. At the University of Cambridge, he developed a theoretical model explaining the structure of white dwarf stars that took into account the relativistic variation of mass with the velocities of electrons that comprise their degenerate matter. He showed that the mass of a white dwarf could not exceed 1.44 times that of the Sun – the Chandrasekhar limit. Chandrasekhar revised the models of stellar dynamics first outlined by Jan Oort and others by considering the effects of fluctuating gravitational fields within the Milky Way on stars rotating about the galactic centre. His solution to this complex dynamical problem involved a set of twenty partial differential equations, describing a new quantity he termed 'dynamical friction', which has the dual effects of decelerating the star and helping to stabilize clusters of stars. Chandrasekhar extended this analysis to the interstellar medium, showing that clouds of galactic gas and dust are distributed very unevenly.

Chandrasekhar studied at Presidency College, Madras (now Chennai) and the University of Cambridge. A long-time professor at the University of Chicago, he did some of his studies at the Yerkes Observatory, and served as editor of The Astrophysical Journal from 1952 to 1971. He was on the faculty at Chicago from 1937 until his death in 1995 at the age of 84, and was the Morton D. Hull Distinguished Service Professor of Theoretical Astrophysics.

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