Gravity

Drop a few stars and let go. They fall toward each other, miss, swing around, and within seconds a cold scatter of points has organised itself into orbits, a binary, a slingshot, a disk. You did not arrange any of it. The inverse-square law did.

The concept

Every mass pulls every other, harder when closer, and the whole choreography of a galaxy is just that one rule applied to thousands of bodies at once. There is no central conductor. The structure, orbits, clusters, spiral arms, is what gravity does to a cloud given a little time and a little angular momentum.

The math

Each body feels the summed pull of all the others (Newton's law of gravitation):

$\mathbf{a}_i = G \sum_{j \neq i} m_j \, \frac{\mathbf{r}_j - \mathbf{r}_i}{\left(\lVert \mathbf{r}_j - \mathbf{r}_i \rVert^2 + \varepsilon^2\right)^{3/2}}$

The $\varepsilon$ is a softening length: it keeps the force finite when two bodies nearly coincide, so a close pass slingshots instead of exploding to infinity. Positions advance with a symplectic (velocity-Verlet) step, which conserves energy far better over long runs than naive Euler, so orbits stay closed instead of spiralling out from numerical drift.

Why it stays strange

The exact two-body orbit is a clean ellipse, solved for centuries. Add a third body and the system becomes chaotic, unsolvable in closed form, yet out of that chaos come the most ordered things in the sky. Gravity is the weakest force by an absurd margin, and given enough mass and patience it is the one that builds the universe's architecture.

Further reading

• Aarseth, Gravitational N-Body Simulations.
• Press et al., Numerical Recipes - symplectic integrators.

@book{aarseth2003nbody,
  author    = {Aarseth, Sverre J.},
  title     = {Gravitational N-Body Simulations: Tools and Algorithms},
  publisher = {Cambridge University Press},
  year      = {2003}
}