M078 · Every star pulls every star

N-Body Star Cluster

Release a three-dimensional cloud of gravitating particles. Change its initial speed, mass spread and force softening; follow collapse, rebound and expansion while auditing 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 $\text{Ready}$

PLACE / RELEASE / COMPARE

Change the start. Follow the 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\,

Dimensionless units: gravity constant, total mass and the uncentred initial cloud radius are one. Initial positions sample a uniform ball, velocities sample an isotropic Gaussian; subtract mass-centre motion and rescale to the selected virial ratio. This is not an exact equilibrium distribution.

All particle pairs use Plummer-softened gravity and the matching potential. The virial ratio uses the force moment, which differs from the potential energy with softening. A ratio near one at one instant does not establish equilibrium.

Fixed-step velocity Verlet: standard step 0.0025, half step 0.00125, duration eight. Compare timestep runs and the displayed maximum energy drift. No collisions, stellar evolution, external tides, hard binaries or certified long-term relaxation.

Bright spheres are particle markers, not stellar surfaces; size and colour encode mass. Short trails follow eight fixed particle indices. The half-mass sphere is a geometric diagnostic, not a material shell. All records are model calculations.

Small, softened Newtonian cluster; not a stellar-evolution or hard-binary simulation.

Start with the essentials

Focus question: Does a cold cloud settle, rebound or escape?
One-sentence intuition: One instantaneous virial ratio cannot establish equilibrium.

Core mathematical model

Pair interaction

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

Force and potential use the same softening.

Virial is a force moment

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

With softening, the virial moment differs from the potential energy.

Finite-time identity

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

A turning radius is not a proof of equilibrium.

Conservation audit

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

Track numerical drift independently of the apparent motion.

Common difficulties

A model boundary

Typical misconceptionA bright particle is a resolved star.

Better mental modelMarkers encode particle mass; close stellar encounters require a different model.

Run the experiment

01
Release cold
Play the cold-collapse preset.
What to observe: The cloud contracts, then rebounds without an imposed restoring spring.
02
Add speed
Compare the warm and expanding clouds.
What to observe: The same initial geometry produces different outcomes.
03
Refine the force
Halve the timestep with the same cloud seed.
What to observe: Compare energy drift before interpreting detailed encounters.