N-Body Simulation

Numerical integration of gravitational systems with energy and time-step control.

Conceptual context

An N-body simulation follows several objects interacting through gravity. At each instant, the code knows $r_i$, $r_j$, masses, and velocities; it computes $F_ij$, then $a_i$, and finally updates positions and velocities. The challenge is not only obtaining plausible orbits but controlling how numerical integration alters $E$. The decisive parameter is $dt$. A large step can invent precession, ejections, or collisions; a smaller step increases computational cost. The central question is how $\epsilon _E$ decreases under temporal refinement and when the expected law stops applying.

🟢 Essential level

In a gravitational simulation, every body feels the attraction of all others. The computer repeats a cycle: it computes $r_ij$, obtains $F_ij$, sums $F_i$, determines $a_i$, and advances the system through a small $dt$. A smaller interval resolves orbital curvature and close encounters more accurately. It does not repair incorrect masses, inconsistent initial conditions, or an untreated singularity. Quality is monitored through $E$, which should remain nearly constant in an isolated system. $\epsilon _E$ compares current energy with $E_0$. A regular trend under decreasing $dt$ supports convergence. Growing drift, a plateau, or abrupt jumps indicate that the integrator, a close encounter, or the energy definition requires inspection.

Energy and orbital shape must be checked together because either diagnostic alone can hide a different numerical defect. A convincing orbit may accumulate phase error, and a dramatic ejection may disappear when the calculation is repeated at higher resolution. Runs must therefore be compared before accepting a physical conclusion, and the comparison should always report both the energy trend and the orbital shape side by side.

🔵 Formal level

Basic formulas

The gravitational interaction between two bodies follows:

Vector gravitational force

The total force on body i gives $a_i$ through:

Acceleration of one body

Leapfrog separates the update into staggered velocity and new position:

Leapfrog velocity update

Leapfrog position update

Global kinetic energy is:

Total kinetic energy

Potential energy must sum unique pairs:

Total gravitational potential energy

The mechanical total is:

Total mechanical energy

Real student questions

Does energy conservation mean the orbit is exact? No. Phase error can accumulate while energy stays bounded.

Why does Leapfrog use $v_half$? Staggering improves geometric structure and limits secular drift.

What if $E_0$ is nearly zero? Relative error becomes unstable; use another normalization scale.

Why do close encounters break convergence? They introduce a much shorter time scale and rapidly changing force.

Is adaptive stepping always better? It resolves encounters efficiently, but may lose symplectic advantages if designed carelessly.

Final synthesis

An N-body simulation is trustworthy when dynamics, units, and integration are coherent. Reducing $dt$ should decrease $\epsilon _E$ predictably until another source dominates. Convergence turns a computed orbit into numerical evidence.