Part 2: Stellar Dynamics as Collisionless Statistics

Learning Objectives¶

By the end of Part 2, you will be able to:

Write the collisionless Boltzmann equation for stellar systems
Derive the Jeans equations as moments of the distribution function
Explain why stellar systems don’t thermalize like gases
Calculate velocity dispersions and understand their role as “temperature”
Apply these equations to real stellar systems from clusters to galaxies

Part 2: Application - Star Clusters (Stars as Particles)¶

Having established that stars can be treated as particles in phase space, let’s now apply our statistical mechanics machinery to derive the fundamental equations of stellar dynamics.

2.1 The Collisionless Boltzmann Equation (Again)¶

Priority: 🟡 Standard Path

Let’s start from the full Boltzmann equation that governs any system of particles:

\frac{df}{dt} = \frac{\partial f}{\partial t} + \vec{v} \cdot \vec{\nabla}_r f + \vec{a} \cdot \vec{\nabla}_v f = \left(\frac{\partial f}{\partial t}\right)_{coll}

(1)

where the right-hand side represents changes due to “collisions” (close encounters). For star clusters, gravitational “collisions” (close encounters between stars) are extremely rare. When the relaxation time vastly exceeds the age of the system, we can set the collision term to zero:

\left(\frac{\partial f}{\partial t}\right)_{coll} = 0

(2)

This gives us the collisionless Boltzmann equation (also called the Vlasov equation):

\boxed{\frac{\partial f}{\partial t} + \vec{v} \cdot \vec{\nabla}_r f - \vec{\nabla}\Phi \cdot \vec{\nabla}_v f = 0}

(3)

Note that gravity enters through the gravitational potential $\Phi$ gradient $\vec{\nabla}\Phi$ rather than individual forces. This represents the mean field approximation - each star moves in the smooth potential created by all other stars.

Why can we ignore collisions? Let’s do an order-of-magnitude estimate to see when two stars significantly deflect each other.

The Setup: Two stars strongly interact when they pass close enough that their gravitational interaction significantly changes their velocities. The characteristic distance for a 90° deflection is:

b_{90} = \frac{2GM_*}{v_{rel}^2}

(4)

Order-of-Magnitude Estimate (typical globular cluster parameters):

Average stellar mass: $M_* \sim M_{\odot} = 2 \times 10^{33}$ g
Cluster velocity dispersion: $\sigma \sim 10$ km/s = 10⁶ cm/s
Relative velocity between stars: $v_{rel} \sim \sqrt{2}\sigma \sim 14$ km/s = $1.4 \times 10^6$ cm/s

This gives:

b_{90} \sim \frac{2 \times 6.67 \times 10^{-8} \times 2 \times 10^{33}}{(1.4 \times 10^{6})^2} \sim 10^{14} \text{ cm} \sim 7 \text{ AU}

(5)

Typical stellar densities in different environments:

Solar neighborhood: 0.1 stars/pc³
Open cluster core: 1-10 stars/pc³
Globular cluster halo: 0.1-1 stars/pc³
Globular cluster core: 10³-10⁶ stars/pc³ (47 Tuc core: ~ $3 \times 10^4$ stars/pc³)
Nuclear star cluster: 10⁶-10⁷ stars/pc³

With stellar density $n_* \sim 10$ stars/pc³ (in a typical cluster), the collision rate is extremely low. The relaxation time (time for gravitational encounters to redistribute energy) is:

t_{\text{relax}} = \frac{0.1 N}{\ln N} t_{\text{cross}}

(6)

where $t_\text{cross} = R_\text{cl}/\sigma$ is the crossing time.

Why the 0.1 factor? This comes from detailed N-body experiments by Spitzer (1987). It represents the typical fraction of stars that must undergo strong encounters to redistribute energy throughout the system. The $\ln N$ term is the Coulomb logarithm, accounting for the cumulative effect of many weak deflections being more important than rare close encounters.

For concrete examples:

Open cluster ( $N \sim 10^3$ ): $t_\text{relax} \sim 100$ Myr (will evaporate)
Globular cluster ( $N \sim 10^5$ -10⁶): $t_\text{relax} \sim 10$ Gyr (quasi-stable)
Galaxy ( $N \sim 10^{11}$ ): $t_\text{relax} \sim 10^{18}$ yr (truly collisionless)

This exceeds the age of many clusters, so they never reach “thermodynamic” equilibrium!

2.2 Velocity Dispersion and Kinetic Energy¶

Priority: 🔴 Essential Just as temperature measures the kinetic energy per particle ( $\sim k_B T$ ) in a gas, velocity dispersion $\sigma$ measures the kinetic energy per star in a cluster:

\boxed{\sigma^2 = \langle v^2 \rangle - \langle v \rangle^2 = \text{Var}(v)}

(7)

The key insight is that velocity dispersion directly gives us the kinetic energy per unit mass:

\boxed{\text{Kinetic energy per star} = \frac{1}{2}M_\star \sigma^2}

(8)

For a cluster with $N$ stars, the total kinetic energy is:

K = \frac{1}{2}N M_\star \sigma^2_{3D} = \frac{3}{2}N M_\star \sigma^2_{1D}

(9)

But here’s the crucial difference:

Gas: Particles collide → energy redistributes → Maxwell-Boltzmann distribution → temperature has meaning
Star cluster: Stars don’t collide → no thermalization → velocity distribution set by gravity → no temperature

🔭 How We Actually Measure Velocity Dispersions

Astronomers measure $\sigma$ through Doppler broadening of absorption lines:

Point spectrograph at cluster center: Collect light from many stars simultaneously
Measure line profiles: Each star’s radial velocity shifts its spectral lines
Fit Gaussian to broadened line: Width gives line-of-sight $\sigma_{los}$
Convert to 3D: $\sigma_{3D} = \sqrt{3} \times \sigma_{los}$ (assumes isotropy - see warning below)

Real Example: Globular cluster M15

Observed line width: ~11 km/s (line-of-sight velocity dispersion)
If isotropic: $\sigma_{3D} \approx 19$ km/s
Using virial theorem with $R_{eff} \sim 1$ pc:
Estimated mass: $M \sim R\sigma^2/G \approx 5 \times 10^5 M_☉$
Luminous mass: ~ $2 \times 10^5 M_☉$
Mass discrepancy: Extra mass likely from stellar remnants (white dwarfs, neutron stars, stellar-mass black holes). Some studies suggest a possible intermediate-mass black hole (~ $4000 M_☉$ ), but this remains controversial!

WARNING about isotropy assumption: The conversion $\sigma_{3D} = \sqrt{3} \times \sigma_{los}$ assumes the velocity dispersion is the same in all directions (isotropic orbits, $\beta = 0$ ). Real clusters often have:

Radially-biased orbits ( $\beta > 0$ ) in outer regions: True mass is HIGHER than isotropic estimate
Tangentially-biased orbits ( $\beta < 0$ ) near center: True mass is LOWER than isotropic estimate

These anisotropies can cause 30-50% errors in mass estimates!

What really matters: The Virial Theorem For any self-gravitating system in equilibrium:

\boxed{2K + W = 0}

(11)

This gives us:

\sigma^2_{3D} = \frac{|W|}{NM_\star} = \frac{3GM_{\text{total}}}{2R}

(12)

Velocity dispersion is completely determined by the gravitational potential - not by any “temperature” or thermal process.

📊 Statistical Insight: Energy is Universal, Temperature is Not

Energy is always well-defined - every moving object has kinetic energy. But temperature only makes sense when:

Particles exchange energy through collisions
The system reaches thermodynamic equilibrium
Velocities follow a Maxwell-Boltzmann distribution

Star clusters violate all three conditions! The velocity dispersion $\sigma$ measures kinetic energy, not temperature. This distinction matters because:

Energy conservation always applies: Total $E = K + W$ is conserved
Virial equilibrium replaces thermal equilibrium: Systems settle into $2K + W = 0$
Jeans equations replace fluid equations: We use moments of the collisionless Boltzmann

The mathematics looks similar (both involve velocity moments), but the physics is fundamentally different.

🔗 Connection to Project 3 (Monte Carlo Radiative Transfer)

Just as photons random walk through stellar atmospheres in your Monte Carlo RT simulations, stars undergo a gravitational “random walk” in phase space through distant encounters. Both are statistical processes, but with vastly different physics:

Photon Random Walk (Project 3):

Mean free path: ~1 cm in stellar interior
Timescale: ~10⁵ years from core to surface
Process: Absorption and re-emission
Result: Energy diffusion outward

Stellar Phase Space Walk:

“Mean free path”: >> cluster size (essentially infinite)
Timescale: ~10¹⁰ years for energy exchange
Process: Gravitational deflections
Result: Slow evolution toward equilibrium

Both require statistical treatment, but photons thermalize quickly while stellar systems never truly thermalize!

2.3 The Jeans Equations: Stellar Fluid Dynamics¶

Priority: 🔴 Essential Taking moments of the collisionless Boltzmann equation gives the Jeans equations – the stellar dynamics equivalent of fluid equations.

Zeroth Moment: Continuity¶

Multiply by 1 and integrate over velocity space:

\boxed{\frac{\partial \nu}{\partial t} + \vec{\nabla} \cdot (\nu \vec{u}) = 0}

(13)

where $\nu(\vec{r},t)$ is the stellar number density and $\vec{u} = \langle \vec{v} \rangle$ is the mean stellar velocity. This is mass conservation for a “fluid” of stars.

First Moment: Momentum (Jeans Equation)¶

Multiply by $\vec{v}$ and integrate. For a spherically symmetric system in steady state:

\boxed{\frac{1}{\nu}\frac{d(\nu \sigma_r^2)}{dr} + \frac{2\beta\sigma_r^2}{r} = -\frac{d\Phi}{dr} = -\frac{GM_r}{r^2}}

(14)

This is the stellar dynamics equivalent of hydrostatic equilibrium! The left side represents “pressure” support from stellar random motions, while the right side is gravity.

The anisotropy parameter $\beta$ (in spherical coordinates $r, \theta, \phi$ ) captures something gas doesn’t have – stars can have different velocity dispersions in different directions:

\beta = 1 - \frac{\sigma_\theta^2 + \sigma_\phi^2}{2\sigma_r^2}

(15)

where $\sigma_r$ is radial dispersion and for spherical symmetry $\sigma_\theta = \sigma_\phi$ .

$\beta = 0$ : isotropic orbits ( $\sigma_r = \sigma_\theta = \sigma_\phi$ )
$\beta \to 1$ : radial orbits dominate (like comets)
$\beta < 0$ : tangential orbits dominate (like planets)

Physical meaning: The anisotropy parameter tells us about orbital shapes. Radial orbits plunge through the center, tangential orbits avoid it. Real clusters often have radial orbits in their halos (stars falling in) and tangential orbits near the center (survivors of tidal stripping).

2.4 The Beautiful Parallel: From Atoms to Stars¶

Priority: 🔴 Essential The profound insight is that the same mathematical framework describes both stellar interiors (atoms as particles) and star clusters (stars as particles):

Quantity	Stellar Interior (atoms)	Star Cluster (stars)
“Particles”	Atoms with mass $m$	Stars with mass $M_\star$
Number density	$n(\vec{r})$ atoms/cm³	$\nu(\vec{r})$ stars/pc³
Velocity spread	Temperature $T$	Dispersion $\sigma^2$
“Pressure”	$P = nkT$ (thermal)	$\Pi_{ij} = \nu\sigma_{ij}^2$ (kinetic)
Equilibrium	$\frac{dP}{dr} = -\rho g$	Jeans equation
Energy exchange	Collisions (nanoseconds)	Relaxation (10¹⁰ yr)

Both equations come from the same source – the first moment of the Boltzmann equation! The only differences are:

Collision timescale: Atoms collide constantly → isotropic pressure. Stars rarely “collide” → anisotropic “pressure”
Thermalization: Atoms reach Maxwell-Boltzmann quickly. Stars never thermalize
Extra term: The $2\beta\sigma_r^2/r$ term appears because stellar systems can have anisotropic orbits

🎯 The Profound Unity: One Framework, All Scales

We’ve now derived fundamental equations at vastly different scales using the SAME mathematical machinery:

The Moment Method Applied to Different “Particles”:

System	“Particle”	Particle Mass	0th Moment	1st Moment
Stellar interior	Atoms	$\sim 2 \times 10^{-24}$ g	Mass continuity	Hydrostatic equilibrium
Star cluster	Stars	$\sim 2 \times 10^{33}$ g	Stellar continuity	Jeans equation
Galaxy	Star clusters	$\sim 10^{38}$ g	Cluster continuity	Cluster Jeans equation
Universe	Galaxies	$\sim 10^{44}$ g	Galaxy continuity	Cosmic Jeans equation

That’s a range of 68 orders of magnitude in mass! Yet the mathematics is identical:

Write down the Boltzmann equation for your “particles”
Take the 0th moment → continuity equation
Take the 1st moment → force balance equation
Take the 2nd moment → energy equation (if needed)

Why this works: Statistical mechanics doesn’t care what you call a “particle” – only that you have many of them. Whether it’s 10⁵⁷ atoms in a star or 10⁵ stars in a cluster, the statistical framework applies.

Part 2 Synthesis: Collisionless Statistics Creates Structure¶

You’ve discovered that stellar dynamics is just statistical mechanics without thermalization. The key differences from gases:

No collisions = No thermalization: Stars maintain distinct orbits for billions of years. This creates rich structures (spiral arms, bars, streams) impossible in gases.
Anisotropy matters: Without collisions to isotropize velocities, radial and tangential dispersions differ. The $\beta$ parameter becomes essential.
Phase space structure persists: Stellar streams maintain coherent phase space structure for gigayears. This “memory” lets us reconstruct galactic history.
Same math, different physics: The Jeans equations are mathematically identical to fluid equations but describe fundamentally different physics – orbits rather than pressure.

The profound realization: The universe recycles the same statistical framework at every scale. Master it once with atoms, apply it to stars, extend it to galaxies. The labels change but the mathematics is eternal.