Three-body problem

Study deterministic chaos in celestial mechanics by simulating three suns in gravitational interaction with the FizziQ Web Orbits and Gravitation simulation.

Visão geral da atividade:

The student configures an original three-sun system in the Orbits and Gravitation simulation, vertically aligned with opposite and different speeds. The student runs the simulation and observes the complex trajectories that result from the mutual attraction of the three bodies. By modifying just one parameter very slightly (speed, mass or distance), the student discovers that the system evolves towards radically different fates: ejection of a body, collision, orbital capture or chaotic dance. The activity illustrates the sensitivity to initial conditions of the N-body problem, a phenomenon discovered by Henri Poincaré in 1889 and connected to the modern concept of deterministic chaos.

Nível:

Autor:

Middle school

FizziQ

Duração (minutos):

40-60

O que os alunos farão:

'- Configure a three-sun system in the FizziQ Web Orbits and Gravitation simulation
- Observe the chaotic evolution of the three trajectories under mutual gravitational attraction
- Discover sensitivity to initial conditions by modifying a parameter very slightly
- Identify the various possible behaviours (ejection, collision, capture, chaotic dance)
- Understand why the N-body problem has no general mathematical solution and requires numerical simulation

Conceitos científicos:

'- N-body problem
- Deterministic chaos
- Sensitivity to initial conditions
- Universal gravitation
- System stability
- Numerical integration
- Butterfly effect
- Celestial mechanics

Sensores:

'- FizziQ Web Orbits and Gravitation simulation

Materiais necessários:

'- Computer, tablet or smartphone with FizziQ Web
- FizziQ experiment notebook

Procedimento experimental:

Open the Orbits and Gravitation simulation in FizziQ Web (Experiment → Simulations → Orbits and gravitation).
Set the distance scale to 1,000,000 km/pixel (ruler icon) and the time scale to 12 hours per frame (speed icon). Cancel any centering by clicking the X button in the Centering area to observe motion in a fixed reference frame.
Configure body 1 as the first Sun: select "Sun" on the mass slider (333,000 M⊕), speed 20 km/s, initial angle 0° (motion to the right). Drag it into the upper half of the window to leave room for the others below. Choose a yellow colour.
Add a new body via the "+" tab and configure it as the second Sun: mass 333,000 M⊕, speed 20 km/s, initial angle 180° (motion to the left). Drag it 80 million km below body 1, watching the "distances" panel in the upper right. Choose an orange colour.
Add a third body as the third Sun: mass 333,000 M⊕, speed 40 km/s, initial angle 0° (motion to the right). Drag it 240 million km below body 1 (i.e. 160 million km below body 2). Choose a red colour.
Verify the configuration: three vertically aligned Suns, distances 80 and 160 million km between successive pairs, opposite speeds between body 1 and body 2, doubled speed for body 3.
Click the green Start button to launch the simulation. Watch the coloured trajectories carefully: the three bodies attract each other, their trajectories tangle, some bodies are ejected far away, others approach dangerously.
Let the simulation run for several minutes (about 200 to 500 simulated days). Observe that no orbit closes regularly: the three bodies follow complex, unpredictable trajectories. Save a screenshot using the IMG button.
Test 2 — Sensitivity to initial conditions: stop the simulation, modify body 1's speed very slightly (from 20 km/s to 21 km/s, only a 5% variation). All other parameters remain the same. Restart the simulation and observe that the trajectories obtained are completely different from the previous test.
Test 3 — Mass variation: replace body 1 with a "Red dwarf" (lower mass, around 100,000 M⊕). Restart the simulation and observe how the system evolves. Depending on the case, a body may be ejected to great distance or captured into orbit around another.
Test 4 — Distance variation: return to the initial configuration (3 Suns, 333,000 M⊕ each) but double the distance between bodies (160 and 480 million km). Restart and observe that interactions are slower and the system may seem more stable for the first dozens of days.
Test 5 — Adding bodies: return to the initial configuration and add a fourth body (e.g. an Earth-mass planet). Observe how this small body is tossed from one star to another without being able to settle into a stable orbit.
Fill in a 3-column summary table: Test, Modification made, Observed behaviour (collision, ejection, chaotic configuration, capture). Conclude that the three-body system displays a wide variety of unpredictable behaviours.

Resultados esperados:

With the initial configuration (three Suns vertically aligned, speeds 20, 20 and 40 km/s with opposite directions), the three bodies follow intertwined trajectories. Bodies 1 and 2 first approach each other, while body 3, faster, is gradually pulled towards the central pair. Depending on the moment, the three bodies may form a moving triangular configuration, eject each other or undergo a collision (the simulation then stops). No orbit closes properly as in a two-body system: the motion is chaotic. Modifying just one parameter slightly (speed +5%, different mass, doubled distance) yields a completely different evolution — this is the hallmark of deterministic chaos. Doubling the initial distances slows the interactions and the system may seem stable initially, but the chaotic character reappears in the long run. Adding a fourth body (planet) shows how it is tossed between the three stars without being able to settle into a stable orbit.

Questões científicas:

'- Why is a three-body gravitational system fundamentally more complex than a two-body system?
- What does "sensitivity to initial conditions" mean and why does it make long-term prediction impossible?
- What happens when a body is ejected from the system? Where does its extra energy come from?
- Why does the FizziQ simulation use step-by-step calculation rather than a single formula?
- Could a habitable planet exist in a three-sun system? Why?

Explicações científicas:

The N-body problem consists of predicting the motion of several bodies attracting each other through gravity. For only two bodies (e.g. Earth around the Sun), Isaac Newton showed in 1687 that the trajectories are perfectly predictable ellipses, computable with simple formulas.

But starting from three bodies, the situation becomes extraordinarily more complex. Each body is simultaneously attracted by the other two, and the attraction varies constantly as positions change. The trajectories obtained are no longer simple curves but unpredictable tangles.

In 1889, the French mathematician Henri Poincaré showed that there is no general mathematical solution to the three-body problem. In other words, you cannot write a formula giving the position of the bodies at any future time: you must compute step by step, as the FizziQ simulation does.

The most surprising consequence is deterministic chaos: two configurations that differ only by a tiny variation at the start can evolve towards completely different trajectories after some time. This is what test 2 illustrates: changing just one body's speed from 20 to 21 km/s is enough to completely transform the system's fate.

This property is called sensitivity to initial conditions. It is also present in meteorology: the mathematician Edward Lorenz discovered it in 1963 and called it the butterfly effect by image — a variation as tiny as a flap of wings can, over the long term, change the climate at the other end of the planet.

Depending on initial conditions, several behaviours are possible:

- An ejection: one body gains enough energy to escape permanently to infinity, leaving the other two in orbit around each other.

- A collision: two bodies approach each other until they crash — the simulation then stops automatically.

- A capture: two bodies pair up in orbit while the third moves away without returning.

- A chaotic dance: the three bodies approach, move apart and exchange positions with no apparent logic, sometimes for a very long time.

This problem is very practical in astronomy: it must be solved numerically to compute the trajectories of space probes, potentially dangerous asteroids, or to study the very long-term stability of the solar system. Modern simulations use supercomputers to compute the motion of millions of bodies simultaneously.

The three-body problem has also inspired popular culture: the famous science fiction novel The Three-Body Problem by Liu Cixin imagines an extraterrestrial civilization living on a planet subjected to three suns, whose climate is radically unpredictable.

Atividades de extensão:

Perguntas frequentes:

Q: Why is it impossible to predict the system's long-term evolution?
A: Because a tiny measurement error on the bodies' initial position or velocity grows exponentially with time. After a certain time called the Lyapunov horizon, two configurations that were nearly identical at the start produce radically different results.

Q: Does the three-body problem exist in reality?
A: Yes, it is everywhere in astronomy. The Sun-Earth-Moon system is one example, and its exact study is already very complex. Triple star systems are also numerous in our galaxy.

Q: The simulation sometimes stops abruptly — is that normal?
A: Yes, the simulation automatically stops when two bodies collide (distance between their centres becoming less than the sum of their radii). This is expected behaviour for the three-body problem.

Q: If I rerun exactly the same configuration, will I get the same result?
A: Yes, provided the simulation is deterministic (same equations, same time steps, same values). The "chaos" of the three-body problem is not random: it is a deterministic amplification of small differences. That's why it is called deterministic chaos.

Q: Are there any known stable three-body configurations?
A: Yes, but very rare. The most famous is Lagrange's triangular configuration (three equal-mass bodies at the vertices of a rotating equilateral triangle). Another is the figure-eight orbit discovered in 1993, where three bodies follow a single inverted-8 trajectory.