Part 1: Phase Space & Statistical Abstraction

When Stars Become Particles | Statistical Thinking Module 2 | COMP 536

Author

Anna Rosen

Learning Objectives

By the end of Part 1, you will be able to:

Part 1: The Conceptual Bridge - Phase Space for Stars

Warning📏 Unit Conventions

We mix unit systems for pedagogical clarity: - Distances: pc (parsecs) or kpc for galactic scales - Velocities: km/s (matches observations) - Masses: Solar masses (\(M_\odot\)) - Time: Myr (megayears) or Gyr (gigayears)

In these units: \(G \approx 0.00449\) pc³/(\(M_\odot\)·Myr²)

For N-body codes, use one consistent system throughout!

1.1 Changing Perspective: Stars as Particles

Priority: 🟡 Standard Path We’ve just seen how statistical mechanics reduces \(10^{57}\) atoms in a star to four differential equations. Now prepare yourself for an even more audacious leap: what if the stars themselves are just “atoms” in a larger system? What if a star cluster is just a “gas” made of stellar “particles”?

This isn’t poetic license – it’s mathematical truth.

Now we’re about to make a profound shift in perspective that will seem absurd at first but reveals the deep universality of statistical mechanics. Instead of thinking about atoms as particles moving inside a star, we’re going to think about stars themselves as particles moving inside a cluster. This isn’t just a cute analogy – the mathematics is literally identical. The same Boltzmann equation, the same moment-taking procedure, the same conservation laws emerge, just with different labels on the particles.

To understand this shift, let’s first clarify what we mean by a “particle” in statistical mechanics. A particle isn’t defined by its size or mass – it’s defined by its role in the system. A statistical “particle” is any entity that:

  1. Can be characterized by position and velocity
  2. Interacts with other similar entities through forces
  3. Exists in large enough numbers for statistics to apply \((N \gg 1)\)
  4. Evolves according to deterministic or stochastic rules

By this definition, atoms in a gas are particles. But so are stars in a cluster. The scale is irrelevant – what matters is the statistical behavior.

System “Particle” Mass Typical \(N\) Forces What You’ll Model
Gas Atoms \(10^{-24}\) g \(10^{23}\) Electromagnetic Not in this course
Star Atoms \(10^{-24}\) g \(10^{57}\) EM + Gravity Module 2 (theory only)
Open Cluster Stars \(2 \times 10^{33}\) g \(10^{2}\)-\(10^{4}\) Gravity only Project 2 (N-body)
Globular Cluster Stars \(2 \times 10^{33}\) g \(10^{5}\)-\(10^{6}\) Gravity only Project 2 (advanced)
Stellar Association Stars \(2 \times 10^{33}\) g \(10^{2}\)-\(10^{3}\) Gravity only Often unbound

Note: Systems with more particles (galaxies with \(10^{11}\) stars, galaxy clusters) require additional physics beyond pure N-body that we won’t cover.

Despite mass differences of \(10^{57}\), the statistical framework is identical for what we CAN model!

Phase Space Phase Space: The space of all possible states. For particles: 6D space \((x,y,z,v_x,v_y,v_z)\). Each point represents one complete state of the system.

Each star in a cluster occupies a point in 6-dimensional phase space:

  • Position: \((x, y, z)\) - location in the cluster
  • Velocity: \((v_x, v_y, v_z)\) - motion through space

Consider a star in a globular cluster, located 10 pc from the cluster center, orbiting at about 15 km/s.

In phase space, this star is a single point with coordinates:

  • Position: \((x, y, z) = (10 \text{ pc}, 0, 0) = (3.1 \times 10^{19} \text{ cm}, 0, 0)\) (if we put it on the x-axis)
  • Velocity: \((v_x, v_y, v_z) = (0, 15 \text{ km/s}, 0) = (0, 1.5 \times 10^6 \text{ cm/s}, 0)\) (circular orbit)

Every second, this point moves through phase space as the star orbits. Multiply by a million stars, and you have the cluster’s phase space distribution – a swirling cloud of points that maintains its overall structure for billions of years, just like atoms in a gas maintain pressure despite individual chaos.

The distribution function becomes: \[f(\vec{r}, \vec{v}, t) \, d^3r \, d^3v = \text{number of stars in phase space volume } d^3r \, d^3v\]

This is exactly the same \(f\) we used for atoms — only now each point represents an entire star rather than a single atom.

::: {.callout-note info} ## 🛰️ Gaia: Mapping Stellar Phase Space

The Gaia space telescope is measuring phase space coordinates for over 1 billion stars in our Milky Way:

Precision varies with stellar brightness: - Bright stars (G < 15 mag): 10-25 microarcsecond position precision - Faint stars (G = 20 mag): ~700 microarcsecond precision - Proper motions: Similar precision scaled by observation baseline (~5 years) - Radial velocities: 1-15 km/s precision depending on stellar type

Discoveries from phase space mapping: - Stellar streams from disrupted star clusters and dwarf galaxies
- Substructure in the Milky Way’s halo - Evidence for past galaxy mergers (the “Gaia-Enceladus” merger ~10 Gyr ago)

While we focus on star clusters in this module, the same phase space principles apply to galactic scales! :::

::: {.callout-note hint} ## 🤔 Quick Check: When is a Star Not a Particle?

We’re treating stars as point particles. When would this approximation break down?

Think about: What happens when two stars get very close?

Answer: The approximation fails when:

  1. Binary star formation: When stars get close enough to form bound pairs
  2. Tidal disruption: When a star passes close to a massive object and gets torn apart
  3. Stellar collisions: In extremely dense environments, stars can actually collide
  4. Close encounters: When stars pass within ~100 AU (\(1.5 \times 10^{15}\) cm), they exchange energy in complex ways

For most star clusters, stars are separated by ~0.1-1 pc (20,000-200,000 AU), so point particles work beautifully! :::

1.2 A Star’s Journey Through Phase Space

Let’s make this concrete by following a single star orbiting in a globular cluster. Consider a star located 5 pc from the cluster center, orbiting at about 10 km/s.

In phase space, this star is a single point with coordinates:

  • Position: \((x, y, z) = (5 \text{ pc}, 0, 0) = (1.5 \times 10^{19} \text{ cm}, 0, 0)\) (if we put it on the x-axis)
  • Velocity: \((v_x, v_y, v_z) = (0, 10 \text{ km/s}, 0) = (0, 10^6 \text{ cm/s}, 0)\) (circular orbit)

As time evolves, this point traces a path through phase space. In position space alone, the star traces an orbit. But in the full 6D phase space, the trajectory is more complex — the velocity vector continuously changes as the star feels the collective gravity of all other stars.

Now imagine not one star but thousands of stars in a tidal stream — debris from a disrupting globular cluster. In position space, they form a narrow ribbon extending from the cluster. But in phase space, they form a thin sheet — stars at the same position have small velocity dispersions, while stars with identical velocities are spread along the stream. This phase space structure encodes the entire disruption history!

Phase mixing: Even without collisions, initially clumped distributions spread out in phase space, like cream stirred into coffee. Stars with slightly different energies orbit at different periods, causing “spiral winding” that eventually smooths out any initial structure. This is why old stellar populations look smooth while young streams show complex patterns.

Note💻 Connection to Your N-body Code (Project 2)

In your N-body simulation, you’re literally watching points move through phase space! Each timestep updates:

  • Positions: \(\vec{r}_{new} = \vec{r}_{old} + \vec{v} \Delta t\)
  • Velocities: \(\vec{v}_{new} = \vec{v}_{old} + \vec{a} \Delta t\)

Your integrator must preserve phase space volume (Liouville’s theorem). This is why symplectic integrators like Leapfrog are superior — they exactly preserve this volume!

You’ll simulate star clusters with \(N = 100\) to \(10,000\) particles — real systems you could observe with a telescope!

Liouville’s Theorem Liouville’s Theorem: Phase space volume is conserved along dynamical trajectories. Systems can trade position spread for velocity spread, but total phase space volume remains constant.

::: {.callout-note hint} ## 🤔 Quick Check: Liouville’s Theorem in Action

Consider two star clusters with different configurations:

Part A: Two clusters:

  • Cluster A: Tight (1 pc radius) but hot (\(\sigma = 20\) km/s)
  • Cluster B: Diffuse (10 pc radius) but cool (\(\sigma = 2\) km/s)

Which occupies more phase space volume?

Part B: A star cluster contracts gravitationally. What must happen to its velocity dispersion?

Answer:

Part A: Phase space volume ~ (position spread)³ \(\times\) (velocity spread)³

  • Cluster A: Volume ~ \((1 \text{ pc})^3 \times (20 \text{ km/s})^3\) = proportional to \(1 \times 8000 = 8000\) units
  • Cluster B: Volume ~ \((10 \text{ pc})^3 \times (2 \text{ km/s})^3\) = proportional to \(1000 \times 8 = 8000\) units

They occupy the same phase space volume! This is the virial theorem in action.

Part B: Velocity dispersion must increase! Liouville says volume is conserved:

  • If position spread decreases, velocity spread must increase
  • This is why contracting clusters “heat up”
  • Real example: Core collapse in globular clusters

Note: The conversion between position and velocity spread isn’t arbitrary - it’s set by the virial theorem! :::

1.3 The Power of Abstraction: Same Math, Different Scales

This change of perspective reveals the scale invariance of statistical mechanics:

Property Stellar Interior Star Cluster Scale Ratio
Particles atoms/ions stars –
Mass \(\sim 2 \times 10^{-24}\) g \(\sim 2 \times 10^{33}\) g \(10^{57}\)
Number \(\sim 10^{57}\) \(\sim 10^{3}\)-\(10^{6}\) \(10^{-54}\) to \(10^{-51}\)
Collision time \(\sim 10^{-9}\) s \(\sim 10^9\) yr (\(3 \times 10^{16}\) s) \(3 \times 10^{25}\)
Size \(\sim 7 \times 10^{10}\) cm \(\sim 10\) pc (\(3 \times 10^{19}\) cm) \(4 \times 10^{8}\)
Mean free path \(\sim 10^{-5}\) cm >> cluster size –
Temperature from kinetic energy NO temperature (no collisions!) –
Pressure from momentum flux NO pressure (no thermalization!) –
Equilibrium via Collisions (instant) Virial theorem –

Despite mass ratios of \(10^{57}\) and timescale ratios of \(10^{25}\), the mathematical framework is the same!

Important📊 Statistical Insight: Scale-Free Methods in Data Science

The scale invariance you’re seeing in physics appears throughout machine learning:

Scale-Free Networks: Many real networks follow power laws – the same statistical patterns regardless of scale.

Transfer Learning: Neural networks trained on one scale often work at another – features learned on everyday objects transfer to microscopy or astronomy.

Renormalization Group and Deep Learning: Deep neural networks perform a kind of renormalization – each layer extracts features at a different scale.

The profound connection: The same statistical universality that lets us use one framework for atoms and stars lets neural networks recognize patterns across vastly different domains!

1.4 Why This Abstraction Works

The reason we can treat stars as particles isn’t just mathematical convenience – it reflects a deep truth about statistical systems. When you have enough of anything, individual identity becomes irrelevant and statistical properties dominate.

For star clusters, the “collision” time (close encounters between stars) is typically much longer than the age of the universe. This means stars effectively move through a smooth gravitational potential created by all the other stars. The discreteness of individual stars matters as little as the discreteness of individual atoms matters for gas dynamics.

This universality means the techniques you learn for one scale apply to similar systems:

  • The virial theorem works for all self-gravitating systems
  • Jeans equations describe star clusters just as hydrostatic equilibrium describes stars
  • Relaxation processes govern both atomic and stellar systems (with vastly different timescales)

You’re not learning separate theories – you’re learning one framework that describes nature at every scale where statistics matters.

Part 1 Synthesis: The Statistical Lens

You’ve made the conceptual leap: stars are just particles at cluster scales. This isn’t analogy – it’s mathematical identity. The same phase space, the same Liouville’s theorem, the same statistical framework.

The key realizations:

  1. “Particle” is a role, not an identity: Anything numerous enough becomes a statistical ensemble
  2. Phase space unifies position and velocity: Complete description requires all 6 dimensions
  3. Liouville’s theorem constrains evolution: Phase space volume conservation is universal
  4. Scale invariance is real: Same mathematical framework from atoms to star clusters!

This abstraction power is why:

  • Your gravitational N-body code will simulate real star clusters
  • The virial theorem reveals masses of invisible matter
  • Statistical thinking connects astrophysics to machine learning

Ready to see how this abstraction leads to the fundamental equations of stellar dynamics? Part 2 awaits.