M078 · Every star pulls every star

N-Body Star Cluster

What happens after a cold cloud collapses? It does not settle at once. The cloud falls inward, rebounds and partly expands again, while total energy and momentum stay fixed. A single moment when the virial ratio, the usual balance test between kinetic and potential energy, sits near one does not show the cluster has reached equilibrium; it may just be passing through.

Release a three-dimensional cloud of particles that all attract one another. Change their initial speeds, the spread of masses and the force softening, then follow the collapse, rebound and expansion while tracking total energy and momentum.

Model time $\text{Ready}$

Half-mass radius $\text{Ready}$

Instantaneous virial ratio $\text{Ready}$

Total energy $\text{Ready}$

Energy error / initial binding scale $\text{Ready}$

Total momentum magnitude $\text{Ready}$

Set the starting state and release it

Change the starting state and follow the motion under gravity

Initial virial ratio

Q_0

0.15\,

Particle count

N

48.00\,

Force softening

\epsilon

0.12\,

Maximum possible mass ratio

R_m

1.00\,

Reproducible cloud seed

s

3.00\,

Use half timestep

b_h

0.00\,

Units are dimensionless: the gravitational constant, the total mass and the radius of the initial cloud before centring are all one. Starting positions are drawn uniformly inside a ball and velocities from an isotropic Gaussian. The centre-of-mass motion is then removed and the velocities are rescaled to the chosen virial ratio. The result is not an exact equilibrium distribution.

Every pair of particles interacts through Plummer-softened gravity and the matching potential. The virial ratio uses the force moment, which no longer equals the potential energy once softening is on. A ratio near one at a single moment does not mean the cluster is in equilibrium.

The integrator is velocity Verlet with a fixed step: 0.0025 normally and 0.00125 at half step, over a duration of eight. Compare runs at the two steps and check the maximum energy drift shown. Collisions, stellar evolution, external tides and hard binaries are not modeled, and the run does not establish long-term relaxation.

The bright spheres mark particles; they are not star surfaces, and their size and colour show mass. Short trails follow the same eight particles throughout. The half-mass sphere is a geometric marker, not a physical shell. All records are model calculations.

Model assumptions

The cluster is small and uses softened Newtonian gravity. It does not simulate stellar evolution or hard binary stars.

Start with the essentials

Focus question: Does a cold cloud settle, rebound or escape?
One-sentence intuition: A virial ratio measured at one instant cannot show that the cluster is in equilibrium.

Core mathematical model

Softened gravity between each pair

U=-\sum_{i<j}\frac{Gm_im_j}{\sqrt{r_{ij}^2+\epsilon^2}}

Force and potential use the same softening.

The virial uses the force moment

W=\sum_i\mathbf r_i\cdot\mathbf F_i,\quad Q=\frac{2K}{|W|}

With softening, this force moment is not equal to the potential energy.

How the moment of inertia changes

\frac12\ddot I=2K+W,\quad I=\sum_i m_i r_i^2

When the radius stops shrinking and turns around, that alone does not prove equilibrium.

Energy and momentum conservation

E=K+U,\quad\mathbf P=\sum_i m_i\mathbf v_i

Track the numerical drift in these totals separately from how the motion looks.

Common difficulties

What a particle represents

Typical misconceptionEach bright dot is an individual, resolved star.

Better mental modelThe markers show particle mass. Close encounters between stars need a different model.

Run the experiment

01
Release a cold cloud
Choose Cold collapse and play.
What to observe: The cloud contracts and then rebounds, driven by gravity alone.
02
Give the stars more speed
Compare Warm cloud with Fast expansion.
What to observe: Clouds with the same initial shape end up behaving differently.
03
Halve the timestep
Keep the same cloud seed and turn on Use half timestep.
What to observe: Compare the energy drift of the two runs before reading anything into individual encounters.

Start with the essentials

Focus question: Why does the diagram change when the stars keep their age?
One-sentence intuition: A population record joins mass sampling, pairing, position and selected flux. Those steps can change a colour diagram without changing age.

Core mathematical model

Add light in one band

M_{b,\rm sys}=-2.5\log_{10}\sum_i10^{-0.4M_{b,i}}

Each band has its own zero point. Sum components in each band before taking a colour.

An imposed spatial snapshot

a_{\rm sys}=2\,\mathrm{pc}\left({M_{\rm birth,sys}\over0.5M_\odot}\right)^{-s}

This sets the sampled Plummer scale. It is not a dynamical evolution or a measured segregation clock.

Common difficulties

A narrow native mass interval

Typical misconceptionThe default covers every star born in a cluster.

Better mental modelThe licensed default samples98native H-burning nodes with IMF mass-cell weights only over0.851–1.262solar masses. Outside components are not invented.

Concentration does not measure time

Typical misconceptionA concentrated bright core gives the elapsed N-body time.

Better mental modelHere concentration is an imposed snapshot parameter. The original cluster mode retains its separate normalized dynamical clock.

Run the experiment

01
Keep the components, pair more of them
Choose more unresolved pairs, then move the binary fraction.
What to observe: System count and combined flux change. The component count remains1024; the initial component mass distribution is drawn before pairing.
02
Drag a real field boundary
Drag the green handle on the aperture ring.
What to observe: The cone selects the same system IDs in the scene, colour diagram and saved record.
03
Compare native ages locally
Use the tutorial recipe to prepare a verifiedPARSECorMISTmodel set, then import itsJSONfile.
What to observe: Only a verified twelve-node local model enables the age selector. The public3.95Gyrdefault remainsDR2; imported photometry isEDR3.

N-Body Star Cluster