Stellar dynamics explained

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

N \gg 10 .

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,

(L/V,M/N) \sim ( 10 ^{-8} pc / 500 km/s , 1 M ^{55} _{\odot/ 10} = m _{p} ) at stellar surfaces,\sim ( 10 ^{-4} { pc }/10,0.1M_/10^\sim m_) around Sun-like stars or km-sized stellar black holes, \sim ( 10 ^{-1} { pc }/100,10M_/10^\sim m_) 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

1000 pc / 1 km/s = 1000 Myr = HubbleTime / 14 .

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

(L/V,M/N) \sim ( 10pc/10km/s , 1000 M _{\odot/1000)} for M13 Star Cluster, \sim ( 100kpc/100km/s , 10 ^{11} M _{\odot} / 10 ^{11} ) for M31 Disk Galaxy, \sim ( 10Mpc/1000km/s , 10 ^{14} M _{\odot} / 10 ^{77} = m _{\nu)} 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 problemby 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,$$m\_i\backslash frac\; =\; \backslash sum\_^N\; \backslash frac.$$Here in the N-body system, any individual member,

m _{i} is influenced by the gravitational potentials of the remaining m _{j} 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.^[3] The gravitational potential,

\Phi , of a system is related to the acceleration and the gravitational field, g by:$$\backslash frac\; \}=\backslash mathbf\; =-\backslash nabla\; \_\backslash Phi\; (\backslash mathbf),~~\backslash Phi\; (\backslash mathbf\; \_)=-\backslash sum\; \_^.$$

Likewise the rms velocity in the rotation direction is computed by a weighted mean as follows, E.g.,

Here

\langle V ^{2} _{t} \rangle = \langle V ^{2} _{\theta} + V ^{2 \rangle} _{\varphi} =0 . 5 V ^{2 .} _{0}

Likewise$$\backslash langle\backslash mathbf\_r^2\backslash rangle(\backslash mathbf)\; =\; =\; \backslash left(\backslash sqrt\; \backslash right)^2.$$

So the pressure tensor or dispersion tensor iswith zero off-diagonal terms because of the symmetric velocity distribution. Note while there is no Dark Matter in producing the previous flat rotation curve, the price is shown by the reduction factor

{ 8 \over 3 \pi} = 0 . 8488 in the random velocity spread in the azimuthal direction. Among the diagonal dispersion tensor moments, \sigma _{\theta} \equiv \sqrt{\sigma ^{2} _{\theta\theta} } = 0.5V_0 is the biggest among the three at all radii, and \sigma _{\varphi} \equiv \sqrt{\sigma ^{2} _{\varphi\varphi} } \ge \sigma_r \equiv \sqrt only near the edge between 0 . 8488 r _{0} \le r \le r _{0} .

The larger tangential kinetic energy than that of radial motion seen in the diagonal dispersions is often phrased by an anisotropy parameter$$\backslash beta(r)\; \backslash equiv\; 1\; -\; =\; 1\; -\; =\; -\; \backslash le\; 0;$$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

which balances the potential energy per unit mass of the uniform sphere, inside which

M \propto r ^{3} \propto x ^{3} .

The average Virial per unit mass can be computed from averaging its local value

r ⋅ ( - \nabla \Phi) , which yieldsas required by the Virial Theorem. For this self-gravitating sphere, we can also verify that the virial per unit mass equals the averages of half of the potential 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

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$$\backslash nabla\; \backslash Phi\; =\; =\; \backslash ge\; 0,\; ~~\backslash Omega\; =\; .$$

The radial pressure gradient$$-\; =\; -\; -\; =\; +\; 0\; \backslash ge\; 0.$$

The reason for the discrepancy is partly due to centrifugal force$$=\; >\; 0,$$and partly due to anisotropic pressureso

0 . 2643 V _{0} = \sigma _{\varphi} < \sigma _{r} =0 . 5 V _{0} at the very centre, but the two balance at radius r =0 . 8488 r _{0} , and then reverse to 0 . 2643 V _{0} = \sigma _{\varphi} > \sigma _{r} =0 at the very edge.

Now we can verify thatHere the 1st line above is essentially the Jeans equation in the r-direction, which reduces to the 2nd line, the Jeans equation in an anisotropic (aka

\beta \ne 0 ) rotational (aka \langle V _{\varphi\rangle} \ne 0 ) axisymmetric (\partial _{\varphi} \Phi( x ,t) =0 ) sphere (aka \partial _{\theta} n( x ,t) =0 ) after much coordinate manipulations of the dispersion tensor; similar equation of motion can be obtained for the two tangential direction, e.g., {\partial \langle V _{\varphi\rangle} \over \partial t} , which are useful in modelling ocean currents on the rotating earth surface or angular momentum transfer in accretion disks, where the frictional term - { \langle V _{\varphi\rangle} \over t _{fric} } is important. The fact that the l.h.s. {\partial V _{r} \over \partial t} =0 means that the force is balanced on the r.h.s. for this uniform (aka \nabla _{x} m n( x ,t) =0 ) spherical model of a galaxy (cluster) to stay in a steady state (aka time-independent equilibrium {\partial n( x ,t) \over \partial t} = 0 everywhere) statically (aka with zero flow \langle V ( x ,t) \rangle =0 everywhere). Note systems like accretion disk can have a steady net radial inflow \langle V ( x ) \rangle < 0 everywhere at all time.

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

z = \pm z _{0} . To find the peak of the pressure, we note that $$P(R,z)\; =\; \backslash int^\_z\; \backslash partial\_z\backslash Phi\; \backslash rho(R)\; dz\; =\; \backslash rho(R)\; [\backslash Phi(R,z\_0)\; -\; \backslash Phi(R,z)].$$

So the fluid temperature per unit mass, i.e., the 1-dimensional velocity dispersion squared would be $$\backslash sigma^2(R,z)\; =,\; ~~\; |z|\; \backslash le\; z\_0$$

$$\backslash sigma^2=\; \backslash log,\; ~~\; Q(z)\; \backslash equiv\; R\_0\; +\; z\_0\; +\; z\; +\backslash sqrt.$$

Along the rotational z-axis, $$\backslash sigma^2(0,z)\; =\; \backslash log$$$$\backslash sigma(0,z)\; =\; \backslash sqrt\; \backslash sqrt,$$which is clearly the highest at the centre and zero at the boundaries

z = \pm z _{0} . Both the pressure and the dispersion peak at the midplane z =0 . In fact the hottest and densest point is the centre, where P( 0 , 0 ) = { M _{0} \over 4 \pi R ^{2} _{0} z _{0} } { - G M _{0} log [ 1- ( 1+ R _{0/z} ^{-2} ^{0 )} ] \over 2 z _{0}} .

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

\langle{v _{j}\rangle} =0 frictionless system t _{fric} → inf ty , hence the local velocity speed \sigma ^{2} _{j} (r) = \langle{v ^{2 }\rangle(r)} _{j} - \underbrace{\langle{v ^{2 (r)}} _{=0} = { \int\limits _{infty} dv _{r} dv _{\theta} dv _{\varphi} ({v} ^{=0} _{j}} ) ^{2} f _{p} \over \int\limits _{infty} dv _{r} dv _{\theta} dv _{\varphi} f _{p},} everywhere for each of the three directions ~ _{j} = ~ _{r,} ~ _{\theta,} ~ _{\varphi} .One can project the phase space into these moments, which is easily if in a highly spherical system, which admits conservations of energy E = 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., which matches the expectation from the Virial theorem

\overline{r \partial _{r} \Phi} = \overline{v ^{2 }} _{cir} = \overline{G M \over r} = \overline{\langle v ^{2} _{t} \rangle} , or in other words, the \overline{ globalaverage } kinetic energy of an equilibrium equals the average kinetic energy on circular orbits with purely transverse motion.

See also

• Stellar classification
• Boltzmann equation
• Dynamical friction
• Jeans equations
• Mass segregation (astronomy)
• N-body problem
• Virial theorem
• Stellar kinematics
• Poisson's equation
• Vector calculus
• Accretion disk
• Relaxation (physics)

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.

Notes and References

1. Book: Murdin, Paul . Encyclopedia of Astronomy and Astrophysics . Nature Publishing Group. 2001 . 978-0750304405 . 1. Stellar Dynamics.
2. https://cds.cern.ch/record/1053485/files/p37.pdf
3. Book: Galactic Dynamics. Binney. James. Tremaine. Scott . Princeton University Press . 2008 . 978-0-691-13027-9 . Princeton. 35, 63, 65, 698.