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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 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.[1]
Connection with fluid dynamics
Stellar dynamics also has connections to the field of plasma physics.[2] 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.
In accretion disks and stellar surfaces, the dense plasma or gas particles collide very frequently, and collisions result in equipartition and perhaps viscosity under magnetic field. We see various sizes for accretion disks and stellar atmosphere, both made of enormous number of microscopic particle mass,
- 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.
As a rule of thumb, the typical scales concerned (see the Upper Portion of P.C.Budassi's Logarithmic Map of the Universe) are
- 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, where the equation of motion (EOM) for internal interactions of an isolated stellar system of N members can be written down as,
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.[1]
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.[3] The gravitational potential, , of a system is related to the acceleration and the gravitational field, by:
An example of the Poisson Equation and escape speed in a uniform sphere
Consider an analytically smooth spherical potential
We can fix the normalisation by computing the corresponding density using the spherical Poisson Equation
Hence the potential model corresponds to a uniform sphere of radius , total mass with
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.[4]
Relativistic Approximations
There are three related approximations made in the Newtonian EOM and Poisson Equation above.
SR and GR
Firstly above equations neglect relativistic corrections, which are of order of
Eddington Limit
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,
Loss cone
Thirdly a star can be swallowed if coming within a few Schwarzschild radii of the black hole. This radius of Loss is given by
The loss cone can be visualised by considering infalling particles aiming to the black hole within a small solid angle (a cone in velocity). These particle with small have small angular momentum per unit mass
The effective potential
Sparing a rigorous GR treatment, one can verify this by computing the last stable circular orbit, where the effective potential is at an inflection point using an approximate classical potential of a Schwarzschild black hole
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,[5] i.e.,
For typical black holes of the destruction radius
Radius of sphere of influence
A particle of mass with a relative speed V will be deflected when entering the (much larger) cross section of a black hole. This so-called sphere of influence is loosely defined by, up to a Q-like fudge factor ,
Connections between star loss cone and gravitational gas accretion physics
First consider a heavy black hole of mass is moving through a dissipational gas of (rescaled) thermal sound speed and density , then every gas particle of mass m will likely transfer its relative momentum to the BH when coming within a cross-section of radius
Coming back to star tidal disruption and star capture by a (moving) black hole, setting , we could summarise the BH's growth rate from gas and stars, with,
Gravitational dynamical friction
Consider the case that a heavy black hole of mass moves relative to a background of stars in random motion in a cluster of total mass with a mean number density
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.
After certain time of relaxations the heavy black hole's kinetic energy should be in equal partition with the less-massive background objects. The slow-down of the black hole can be described as
Dynamical friction time vs Crossing time in a virialised system
Consider a Mach-1 BH, which travels initially at the sound speed , hence its Bondi radius satisfies
Assume the BH stops after traveling a length of with its momentum deposited to stars in its path over crossings, then the number of stars deflected by the BH's Bondi cross section per "diameter" crossing time is
More generally, the Equation of Motion of the BH at a general velocity in the potential of a sea of stars can be written as
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.
It reads as
Like the "Couloumb logarithm" factors in the contribution of distant background particles, here the factor also factors in the probability of finding a background slower-than-BH particle to contribute to the drag. The more particles are overtaken by the BH, the more particles drag the BH, and the greater is . Also the bigger the system, the greater is .
A background of elementary (gas or dark) particles can also induce dynamical friction, which scales with the mass density of the surrounding medium, ; 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.
Summing up the gravitational drag of both collisional gas and collisionless stars, we have
Interestingly, the 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.
We see the force is also proportional to the inverse square of the velocity at the high end, hence the fractional rate of energy loss drops rapidly at high velocities. Dynamical friction is, therefore, unimportant for objects that move relativistically, such as photons. This can be rationalized by realizing that the faster the object moves through the media, the less time there is for a wake to build up behind it. Friction tends to be the highest at the sound barrier, where .
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.[7] This means that two stars need to come within a separation,
Mean free path
The mean free path of strong encounters in a typically stellar system is then
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.
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
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.
Relation between friction and relaxation
Clearly that the dynamical friction of a black hole is much faster than the relaxation time by roughly a factor , but these two are very similar for a cluster of black holes,
For a star cluster or galaxy cluster with, say, , we have . Hence encounters of members in these stellar or galaxy clusters are significant during the typical 10 Gyr lifetime.
On the other hand, typical galaxy with, say, stars, would have a crossing time and their relaxation time is much longer than the age of the Universe. This justifies modelling galaxy potentials with mathematically smooth functions, neglecting two-body encounters throughout the lifetime of typical galaxies. And inside such a typical galaxy the dynamical friction and accretion on stellar black holes over a 10-Gyr Hubble time change the black hole's velocity and mass by only an insignificant fraction
if the black hole makes up less than 0.1% of the total galaxy mass . Especially when , we see that a typical star never experiences an encounter, hence stays on its orbit in a smooth galaxy potential.
The dynamical friction or relaxation time identifies collisionless vs. collisional particle systems. Dynamics on timescales much less than the relaxation time is effectively collisionless because typical star will deviate from its initial orbit size by a tiny fraction . 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.
First important concept is "gravity balancing motion" near the perturber and for the background as a whole
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 of the system:
E.g., in case Q is the perturber's mass , then we can estimate the Dynamical friction time via the (gas/star) Accretion rate
In the limit the perturber is just 1 of the N background particle, , 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, by neglecting gravitational friction and relaxation of the perturber, working in the limit 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.[7]
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 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
Convention and notation in case of a thermal distribution
In most of stellar dynamics literature, it is convenient to adopt the convention that the particle mass is unity in solar mass unit , hence a particle's momentum and velocity are identical, i.e.,
For example, the thermal velocity distribution of air molecules (of typically 15 times the proton mass per molecule) in a room of constant temperature would have a Maxwell distribution
where the energy per unit mass
and is the width of the velocity Maxwell distribution, identical in each direction and everywhere in the room, and the normalisation constant (assume the chemical potential such that the Fermi-Dirac distribution reduces to a Maxwell velocity distribution) is fixed by the constant gas number density at the floor level, where
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 as
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.[8] 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.[9]
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
Jeans computed the weighted velocity of the Boltzmann Equation after integrating over velocity space
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, epecially 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.[10][11] 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.[12]
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.
A unified thick disk potential
Consider an oblate potential in cylindrical coordinates
First we can see the total mass of the system is because
We can also show that some special cases of this unified potential become the potential of the Kuzmin razor-thin disk, that of the Point mass , and that of a uniform-Needle mass distribution:
A worked example of gravity vector field in a thick disk
First consider the vertical gravity at the boundary,
Note that both the potential and the vertical gravity are continuous across the boundaries, hence no razor disk at the boundaries. Thanks to the fact that at the boundary, is continuous. Apply Gauss's theorem by integrating the vertical force over the entire disk upper and lower boundaries, we have
The vertical gravity drops with
Density of a thick disk from Poisson Equation
Insert in the cylindrical Poisson eq.
Surface density and mass of a thick disk
Integrating over the entire thick disc of uniform thickness , we find the surface density and the total mass as
This confirms that the absence of extra razor thin discs at the boundaries. In the limit, , this thick disc potential reduces to that of a razor-thin Kuzmin disk, for which we can verify .
Oscillation frequencies in a thick disk
To find the vertical and radial oscillation frequencies, we do a Taylor expansion of potential around the midplane.
At large radii three frequencies satisfy . E.g., in the case that and , the oscillations forms a resonance.
In the case that , the density is zero everywhere except uniform needle between along the z-axis.
If we further require , then we recover a well-known property for closed ellipse orbits in point mass potential,
A worked example for neutrinos in galaxies
For example, the phase space distribution function of non-relativistic neutrinos of mass m anywhere will not exceed the maximum value set by
Let's approximate the distribution is at maximum, i.e.,
Take the simple case , and estimate the density at the centre with an escape speed , we have
Clearly eV-scale neutrinos with is too light to make up the 100–10000 over-density in galaxies with escape velocity , while neutrinos in clusters with could make up times cosmic background density.
By the way the freeze-out cosmic neutrinos in your room have a non-thermal random momentum , 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
Consider building a steady state model of the fore-mentioned uniform sphere of density and potential
First a recap on motion "inside" the uniform sphere potential. Inside this constant density core region, individual stars go on resonant harmonic oscillations of angular frequency with
Example on Jeans theorem and CBE on Uniform Sphere Potential
Generally for a time-independent system, Jeans theorem predicts that is an implicit function of the position and velocity through a functional dependence on "constants of motion".
For the uniform sphere, a solution for the Boltzmann Equation, written in spherical coordinates and its velocity components is
It is easy to see in spherical coordinates that
Insert the potential and these definitions of the orbital energy E and angular momentum J and its z-component Jz along every stellar orbit, we have
To verify the above being constants of motion in our spherical potential, we note
Likewise the x and y components of the angular momentum are also conserved for a spherical potential. Hence .
So for any time-independent spherical potential (including our uniform sphere model), the orbital energy E and angular momentum J and its z-component Jz along every stellar orbit satisfy
Hence using the chain rule, we have
A worked example on moments of distribution functions in a uniform spherical cluster
We can find out various moments of the above distribution function, reformatted as with the help of three Heaviside functions,
In fact, the positivity carves the ( ) left-half of an ellipsoid in the velocity space ("velocity ellipsoid"),
The velocity ellipsoid (in this case) has rotational symmetry around the r axis or axis. It is more squashed (in this case) away from the radial direction, hence more tangentially anisotropic because everywhere , except at the origin, where the ellipsoid looks isotropic. Now we compute the moments of the phase space.
E.g., the resulting density (moment) is
The streaming velocity is computed as the weighted mean of the velocity vector
Incidentally, the angular momentum global average of this flat-rotation sphere is
Likewise thanks to the symmetry of , we have , , everywhere}.
Likewise the rms velocity in the rotation direction is computed by a weighted mean as follows, E.g.,
Here
Likewise
So the pressure tensor or dispersion tensor is
The larger tangential kinetic energy than that of radial motion seen in the diagonal dispersions is often phrased by an anisotropy parameter
A worked example of Virial Theorem
Twice kinetic energy per unit mass of the above uniform sphere is
The average Virial per unit mass can be computed from averaging its local value , which yields
A worked example of Jeans Equation in a uniform sphere
Jeans Equation is a relation on how the pressure gradient of a system should be balancing the potential gradient for an equilibrium galaxy. In our uniform sphere, the potential gradient or gravity is
The radial pressure gradient
The reason for the discrepancy is partly due to centrifugal force
Now we can verify that
A worked example of Jeans equation in a thick disk
Consider again the thick disk potential in the above example. If the density is that of a gas fluid, then the pressure would be zero at the boundary . To find the peak of the pressure, we note that
So the fluid temperature per unit mass, i.e., the 1-dimensional velocity dispersion squared would be
Along the rotational z-axis,
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.
Jeans equation links gravity with pressure gradient, it is a generalisation of the Eq. of Motion for single particles. While Jeans equation can be solved in disk systems, the most user-friendly version of the Jeans eq. is the spherical anisotropic version for a static frictionless system , hence the local velocity speed everywhere for each of the three directions . One can project the phase space into these moments, which is easily if in a highly spherical system, which admits conservations of energy and angular momentum J. The boundary of the system sets the integration range of the velocity bound in the system.
In summary, in the spherical Jeans eq.,
See also
Further reading
- 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.
References
- ^ a b c Murdin, Paul (2001). "Stellar Dynamics". Encyclopedia of Astronomy and Astrophysics. Nature Publishing Group. p. 1. ISBN 978-0750304405.
- ^ https://cds.cern.ch/record/1053485/files/p37.pdf [bare URL PDF]
- ^ Binney, James; Tremaine, Scott (2008). Galactic Dynamics. Princeton: Princeton University Press. pp. 35, 63, 65, 698. ISBN 978-0-691-13027-9.
- ^ de Vita, Ruggero; Trenti, Michele; MacLeod, Morgan (2019-06-01). "Correlation between mass segregation and structural concentration in relaxed stellar clusters". Monthly Notices of the Royal Astronomical Society. 485 (4): 5752–5760. arXiv:1903.07619. doi:10.1093/mnras/stz815. ISSN 0035-8711.
- ^ Binney, James. "Galaxy Dynamics" (PDF). Princeton University Press. Retrieved 4 January 2022.
- ^ Ostriker, Eva (1999). "Dynamical Friction in a Gaseous Medium". The Astrophysical Journal. 513 (1): 252. arXiv:astro-ph/9810324. Bibcode:1999ApJ...513..252O. doi:10.1086/306858. S2CID 16138105.
- ^ a b Sparke, Linda; Gallagher, John (2007). Galaxies in the Universe. New York: Cambridge. p. 131. ISBN 978-0521855938.
- ^ Henon, M (June 21, 1982). "Vlasov Equation?". Astronomy and Astrophysics. 114 (1): 211–212. Bibcode:1982A&A...114..211H.
- ^ Lynden-Bell, Donald (1962). "The stability and vibrations of a gas of stars". Monthly Notices of the Royal Astronomical Society. 124 (4): 279–296. Bibcode:1962MNRAS.124..279L. doi:10.1093/mnras/124.4.279.
- ^ Einstein, 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.
- ^ Zwicky, Fritz (2009). "Republication of: The redshift of extragalactic nebulae". General Relativity and Gravitation. 41 (1): 207–224. Bibcode:2009GReGr..41..207Z. doi:10.1007/s10714-008-0707-4. S2CID 119979381.
- ^ Choudhuri, Arnab Rai (2010). Astrophysics for Physicists. New York: Cambridge University Press. pp. 213–214. ISBN 978-0-521-81553-6.