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Two-body problem
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Reduces to one body with reduced mass.
Two masses orbit their common centre of mass. The relative motion is a one-body problem with reduced mass μ and total mass M in the force. Kepler's laws hold with M = m₁ + m₂.
Binary stars of equal mass orbit a point midway.
The central body also moves.
In more detail
The two-body problem describes two masses attracted by gravity, and it can be solved exactly. The motion reduces to a single body moving relative to the centre of mass.
Real-world uses
- Predicting the orbits of planets around the Sun.
- Modelling binary stars and double-planet systems.
- Astronomers model binary stars and planet-star pairs as two-body systems to predict their orbits.
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Frequently asked questions
Can three bodies be solved exactly?
No, in general the three-body problem has no closed-form solution.
Why is the two-body problem solvable?
Because the symmetry reduces it to a one-body problem in relative coordinates.
What is the advantage of the two-body approach?
It reduces the motion of two bodies to one effective body, which makes the equations much easier to solve.
Related topics in Gravitation & Orbits
Tidal locking · Roche limit · Lagrange points · Effective potential · Binary stars · Weighing the Sun and planets