Part 2: Stellar Dynamics as Collisionless Statistics
When Stars Become Particles | Statistical Thinking Module 2 | COMP 536
Learning Objectives
By the end of Part 2, you will be able to:
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:
\[\frac{df}{dt} = \frac{\partial f}{\partial t} + \vec{v} \cdot \vec{\nabla}_r f + \vec{a} \cdot \vec{\nabla}_v f = \left(\frac{\partial f}{\partial t}\right)_{coll}\]
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:
\[\left(\frac{\partial f}{\partial t}\right)_{coll} = 0\]
This gives us the collisionless Boltzmann equation (also called the Vlasov equation):
\[\boxed{\frac{\partial f}{\partial t} + \vec{v} \cdot \vec{\nabla}_r f - \vec{\nabla}\Phi \cdot \vec{\nabla}_v f = 0}\]
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:
\[b_{90} = \frac{2GM_*}{v_{rel}^2}\]
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 = \(10^6\) 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: \[b_{90} \sim \frac{2 \times 6.67 \times 10^{-8} \times 2 \times 10^{33}}{(1.4 \times 10^{6})^2} \sim 10^{14} \text{ cm} \sim 7 \text{ AU}\]
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: \(10^3\)-\(10^6\) stars/pc³ (47 Tuc core: ~\(3 \times 10^4\) stars/pc³) - Nuclear star cluster: \(10^6\)-\(10^7\) 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:
\[t_{\text{relax}} = \frac{0.1 N}{\ln N} t_{\text{cross}}\]
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 (\(N \sim 10^5\)-\(10^6\)): \(t_\text{relax} \sim 10\) 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:
Velocity Dispersion (\(\sigma\))
The RMS spread of stellar velocities around the mean. Typical values: ~1 km/s for open clusters, ~10 km/s for globular clusters, ~200 km/s for galaxy bulges. Directly observable from Doppler broadening of spectral lines.
\[\boxed{\sigma^2 = \langle v^2 \rangle - \langle v \rangle^2 = \text{Var}(v)}\]
The key insight is that velocity dispersion directly gives us the kinetic energy per unit mass:
\[\boxed{\text{Kinetic energy per star} = \frac{1}{2}M_\star \sigma^2}\]
For a cluster with \(N\) stars, the total kinetic energy is: \[K = \frac{1}{2}N M_\star \sigma^2_{3D} = \frac{3}{2}N M_\star \sigma^2_{1D}\]
But here’s the crucial difference:
- Gas: Particles collide \(\to\) energy redistributes \(\to\) Maxwell-Boltzmann distribution \(\to\) temperature has meaning
- Star cluster: Stars don’t collide \(\to\) no thermalization \(\to\) velocity distribution set by gravity \(\to\) no temperature
::: {.callout-note info} ## 🔭 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: \(M \sim R\sigma^2/G \approx 5 \times 10^5 M_\odot\) - Luminous mass: ~\(2 \times 10^5 M_\odot\) - 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 (~\(4000 M_\odot\)), 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! :::
In your N-body simulations, you’ll calculate velocity dispersion as: \[\sigma^2 = \frac{1}{N}\sum_{i=1}^N |\vec{v}_i - \vec{v}_{cm,0}|^2\]
Important: Use \(\vec{v}_{cm,0}\), the INITIAL center of mass velocity, not the current COM velocity. Recalculating COM at each timestep can hide numerical drift!
Watch how \(\sigma\) evolves: it should remain roughly constant for a virialized cluster but increase during collapse (virial theorem: as \(R\) decreases, \(\sigma\) increases).
What really matters: The Virial Theorem For any self-gravitating system in equilibrium: \[\boxed{2K + W = 0}\]
This gives us: \[\sigma^2_{3D} = \frac{|W|}{NM_\star} = \frac{3GM_{\text{total}}}{2R}\]
Velocity dispersion is completely determined by the gravitational potential - not by any “temperature” or thermal process.
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.
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: ~\(10^5\) 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: ~\(10^{10}\) 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:
\[\boxed{\frac{\partial \nu}{\partial t} + \vec{\nabla} \cdot (\nu \vec{u}) = 0}\]
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:
\[\boxed{\frac{1}{\nu}\frac{d(\nu \sigma_r^2)}{dr} + \frac{2\beta\sigma_r^2}{r} = -\frac{d\Phi}{dr} = -\frac{GM_r}{r^2}}\]
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.
Anisotropy Parameter \(\beta\)
Measures orbital shape distribution in stellar systems. Defined in spherical coordinates as \(\beta = 1 - (\sigma_\theta^2 + \sigma_\phi^2)/(2\sigma_r^2)\). Values: \(\beta = 0\) for isotropic orbits, \(\beta > 0\) for radially-biased orbits, \(\beta < 0\) for tangentially-biased orbits.
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:
\[\beta = 1 - \frac{\sigma_\theta^2 + \sigma_\phi^2}{2\sigma_r^2}\]
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).
Besides two-body relaxation, there’s “violent relaxation” (Lynden-Bell 1967): - Occurs when gravitational potential changes rapidly (galaxy mergers!) - ALL stars feel time-varying forces simultaneously - Timescale: ~1 crossing time (not \(N\)-dependent like two-body) - Creates elliptical galaxies from spiral mergers - Explains why ellipticals look “relaxed” despite \(t_{relax} >> t_{universe}\)
This is how galaxies can appear virialized even though star-star collisions never occur!
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 (\(10^{10}\) yr) |
Both equations come from the same source – the first moment of the Boltzmann equation! The only differences are:
- Collision timescale: Atoms collide constantly \(\to\) isotropic pressure. Stars rarely “collide” \(\to\) 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
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 \(\to\) continuity equation
- Take the 1st moment \(\to\) force balance equation
- Take the 2nd moment \(\to\) 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 \(10^{57}\) atoms in a star or \(10^{5}\) 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.
Where we’ve been: You’ve seen how the collisionless Boltzmann equation leads to the Jeans equations – the stellar dynamics analog of fluid dynamics. The absence of collisions creates fundamental differences: no thermalization, persistent anisotropy, and rich phase space structure.
Where we’re going: Part 3 will introduce the virial theorem, the universal diagnostic that revealed dark matter. You’ll learn how this simple relationship (\(2K + W = 0\)) applies to every gravitating system from molecular clouds to galaxy clusters, and how it becomes your primary tool for N-body simulations.
The key insight to carry forward: Stellar systems are statistical but not thermal. This distinction creates the diverse structures we observe in the universe.