Stellar dynamics on arvau

Wikipedia / 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

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.

Connection with fluid dynamics

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.

at stellar surfaces,
around Sun-like stars or km-sized stellar black holes,
around million solar mass black holes (about AU-sized) in centres of galaxies.

The system crossing time scale is long in stellar dynamics, where it is handy to note that

The long timescale means that, unlike gas particles in accretion disks, stars in galaxy disks very rarely see a collision in their stellar lifetime. However, galaxies collide occasionally in galaxy clusters, and stars have close encounters occasionally in star clusters.

for M13 Star Cluster,
for M31 Disk Galaxy,
for neutrinos in the Bullet Clusters, which is a merging system of N = 1000 galaxies.

Connection with Kepler problem and 3-body problem

At a superficial level, all of stellar dynamics might be formulated as an N-body problem by Newton's second law members.

In practice, except for in the highest performance computer simulations, it is not feasible to calculate rigorously the future of a large N system this way. Also this EOM gives very little intuition. Historically, the methods utilised 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 can address both the global, statistical properties of many orbits as well as the specific data on the positions and velocities of individual orbits.

Concept of a gravitational potential field

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. Without getting a system's gravitational potential by adding all of the point-mass potentials in the system at every second, stellar dynamicists develop potential models that can accurately model the system while remaining computationally inexpensive., via the Poisson's equation

An example of the Poisson Equation and escape speed in a uniform sphere

is the speed to "escape from the edge to infinity". The gravity is like the restoring force of harmonic oscillator inside the sphere, and Keplerian outside as described by the Heaviside functions.

Key concepts

While both the equations of motion and Poisson Equation can also take on non-spherical forms, depending on the coordinate system and the symmetry of the physical system, the essence is the same: The motions of stars in a galaxy or in a globular cluster are principally determined by the average distribution of the other, distant stars. 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.

Relativistic Approximations

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

SR and GR

km/s, is much below the speed of light.

Eddington Limit

Loss cone

Thirdly a star can be swallowed if coming within a few Schwarzschild radii

to force the particle to turn around.

Tidal disruption radius

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.,

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.

Radius of sphere of influence

inside all star clusters and in galaxies.

Connections between star loss cone and gravitational gas accretion physics

because the black hole consumes a fractional/most part of star/gas particles passing its sphere of influence.

Gravitational dynamical friction

Intuition says that gravity causes the light bodies to accelerate and gain momentum and kinetic energy (see slingshot effect). By conservation of energy and momentum, we may conclude that the heavier body will be slowed by an amount to compensate. Since there is a loss of momentum and kinetic energy for the body under consideration, the effect is called dynamical friction.

is called a dynamical friction time.

Dynamical friction time vs Crossing time in a virialised system

the dynamical time scale.

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. Lighter and faster holes can stay afloat much longer.

More rigorous formulation of dynamical friction

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.

within the black hole's sphere of influence.

; 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.

Here the "lagging-behind" fraction for gas in a Maxwell distribution with a Gaussian distribution).

dependence suggests that dynamical friction is from the gravitational pull of by the wake, which is induced by the gravitational focusing of the massive body in its two-body encounters with background objects.

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/weak if their mutual potential energy at the closest passage is comparable/minuscule to their initial kinetic energy. Strong encounters are rare, and they are typically only considered important in dense stellar systems, e.g., a passing star can be sling-shot out by binary stars in the core of a globular cluster.

Mean free path

Weak encounters

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.

, that is perpendicular to the impact parameter, is approximately equal to the acceleration at the impact parameter, multiplied by the time duration of the acceleration.

from a more rigorous calculation than the above mean free time estimates for strong deflection.

weak encounters to build a strong deflection to change orbital energy significantly.

Relation between friction and relaxation

. Hence encounters of members in these stellar or galaxy clusters are significant during the typical 10 Gyr lifetime.

, we see that a typical star never experiences an encounter, hence stays on its orbit in a smooth galaxy potential.

. 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.

A Spherical-Cow Summary of Continuity Eq. in Collisional and Collisionless Processes

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.

by consistently omitting etc for clarity, approximating 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 in one or three directions, or the (non)-uniform isotropic Sound speed to emphasize of the logic behind the order of magnitude of the friction time scale.

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 sign is generally negative except for the (accreting) mass M, and the Mean free path can be due to direct molecular viscosity from a physical collision Cross section, or due to gravitational scattering (bending/focusing/Sling shot) of particles; generally the influenced area is the greatest of the competing processes of Bondi accretion, Tidal disruption, and Loss cone

, then we can estimate the Dynamical friction where we have applied the relations motion-balancing-gravity.

, this friction time is identified with the (gravitational) Relaxation time. Again all Coulomb logarithm etc are suppressed without changing the estimations from these qualitative equations.

For the rest of Stellar dynamics, we will consistently work on precise calculations through primarily Worked Examples as approximated true in most galaxies on the 14Gyrs Hubble time scale, even though this is sometimes violated for some clusters of stars or clusters of galaxies.of the cluster.

A concise 1-page summary of some main equations in Stellar dynamics and Accretion disc physics are shown here, where one attempts to be more rigorous on the qualitative equations above.

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. 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. 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.

Convention and notation in case of a thermal distribution

would have a Maxwell distribution

The CBE

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.

The Boltzmann equation is often written more generally with the Liouville operator, and the system will ultimately relaxes to a Maxwell (equipartition) distribution.

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.

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. Applying the collisionless Boltzmann equation, these moments are then related by various forms of continuity equations, of which most notable are the Jeans equations and Virial theorem.

Probability-weighted moments and hydrostatic equilibrium

opulation (e.g., gas, stars, dark matter):

The general version of Jeans equation, involving (3 x 3) velocity moments is cumbersome. It only becomes useful or solvable if we could drop some of these moments, especially drop the off-diagonal cross terms for systems of high symmetry, and also drop net rotation or net inflow speed everywhere.

The isotropic version is also called Hydrostatic equilibrium equation where balancing pressure gradient with gravity; the isotropic version works for axisymmetric disks as well, after replacing the derivative dr with vertical coordinate dz. 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.

Applications and examples

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.

A unified thick disk potential

are (positive) vertical and radial length scales. Despite its complexity, we can easily see some limiting properties of the model.

A worked example of gravity vector field in a thick disk

takes the meaning of the total disk mass.

due to the self-gravity of the thick disk.

Density of a thick disk from Poisson Equation

and uniform along the z-direction within the boundary.

Surface density and mass of a thick disk

Oscillation frequencies in a thick disk

, and is Keplerian far away.

forms a resonance.

along the z-axis.

A worked example for neutrinos in galaxies

times cosmic background density.

, 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.

A Recap on Harmonic Motions in Uniform Sphere Potential

, i.e., the phase space density or distribution function, such that their overall stellar number density reproduces the constant core, hence their collective "steady-state" potential. Once this is reached, we call the system is a self-consistent equilibrium.

Example on Jeans theorem and CBE on Uniform Sphere Potential

is an implicit function of the position and velocity through a functional dependence on "constants of motion".

are excluded.

It is easy to see in spherical coordinates that

for any "steady state" potential.

is a solution to the Collisionless Boltzmann Equation for our static spherical potential.

A worked example on moments of distribution functions in a uniform spherical cluster

, from the distribution function (DF, i.e., phase space density).

respectively.

, except at the origin, where the ellipsoid looks isotropic. Now we compute the moments of the phase space.

plane.

, and this does not contradict the global non-zero rotation.

everywhere}.

a positive anisotropy would have meant that radial motion dominated, and a negative anisotropy means that tangential motion dominates (as in this uniform sphere).

A worked example of Virial Theorem

Twice kinetic energy per unit mass of the above uniform sphere is

Hence we have verified the validity of Virial Theorem for a uniform sphere under self-gravity, i.e., the gravity due to the mass density of the stars is also the gravity that stars move in self-consistently; no additional dark matter halo contributes to its potential, for example.

A worked example of Jeans Equation in a uniform sphere

at the very edge.

everywhere at all time.

A worked example of Jeans equation in a thick disk

A recap on worked examples on Jeans Eq., Virial and Phase space density

Having looking at the a few applications of Poisson Eq. and Phase space density and especially the Jeans equation, we can extract a general theme, again using the Spherical cow approach.

and angular momentum J. The boundary of the system sets the integration range of the velocity bound in the system.

kinetic energy of an equilibrium equals the average kinetic energy on circular orbits with purely transverse motion.