Unstable Orbits and the Three-Body Problem

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If you start plugging in random masses and velocities, you’ll quickly discover that most configurations are wildly unstable. This isn’t a bug — it’s physics. Stable orbits are the exception, not the rule.

In a two-body system, stability is relatively easy to achieve: give the smaller body the right velocity at the right distance and it traces a clean ellipse forever. But the moment you add a third body, things get chaotic. The three-body problem has no general closed-form solution — small differences in initial conditions lead to dramatically different outcomes, including bodies being flung out of the system entirely.

A Chaotic Triple-Star System

Here’s an example: three equal-mass stars arranged in a triangle with slightly asymmetric velocities. It starts off looking like an interesting dance, but the asymmetry compounds and eventually one or more stars get ejected:

three_body <- create_system() |>
  add_body("Star A", mass = 1e30, x = 1e11, y = 0, vx = 0, vy = 15000) |>
  add_body("Star B", mass = 1e30, x = -5e10, y = 8.66e10, vx = -12990, vy = -7500) |>
  add_body("Star C", mass = 1e30, x = -5e10, y = -8.66e10, vx = 14000, vy = -8000) |>
  simulate_system(time_step = seconds_per_hour, duration = seconds_per_year * 3)

three_body |> plot_orbits()

The static plot is honestly hard to read — three crossed paths without a sense of when anything happens. Animating it makes the chaos legible. Watch the slingshot ejection unfold in real time:

animate_system(three_body, fps = 15, duration = 6)

This is actually what happens in real stellar dynamics — close three-body encounters in star clusters frequently eject one star at high velocity while the remaining two settle into a tighter binary. The process is called gravitational slingshot ejection.

Troubleshooting Unstable Simulations

If your simulations are producing messy, diverging trajectories, here are a few things to check before assuming something is wrong:

Velocity too high or too low. At a given distance \(r\) from a central mass \(M\), the circular orbit speed is \(v = \sqrt{GM/r}\). Deviating significantly from this produces eccentric orbits or escape trajectories.

Bodies too close together. Close encounters produce extreme accelerations that can blow up numerically. Try increasing softening or using a smaller time_step.

Three or more bodies. Chaos is the natural state of N-body systems. The stable examples in the documentation are carefully tuned — don’t expect random configurations to behave.

Time step too large. If bodies move a significant fraction of their orbital radius in a single step, the integrator can’t track the orbit accurately. Try halving time_step and see if the result changes.

The built-in constants and examples in orbitr are designed to give you stable starting points. From there you can tweak parameters and watch how the system responds — that’s where the real intuition for orbital mechanics comes from.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.