What is the three-body problem in astrophysics?

The Netflix TV series based on the novel of the same name is based on a problem of classical mechanics

by Lorenzo Ciotti
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What is the three-body problem in astrophysics?
© NASA / Handout Getty Images

3 Body Problem was undoubtedly the most anticipated Netflix series of 2024. Created by David Benioff, D. B. Weiss, (former showrunners of Game of Thrones) and Alexander Woo, the series is the adaptation of the novel of the same name written by Liu Cixin, the first chapter of the Memory of the Earth's Past series.

But what is the three-body problem in astrophysics and astrodynamics?

It is actually a series or class of dynamics problems relating to classical mechanics, which consists in calculating - given the initial position, mass and velocity of three bodies subject to the influence of mutual gravitational attraction - the future evolution of the system they constitute.

Universe
Universe© NASA / Handout Getty Images
 

A bit of science, geometry and mathematics

There is a simplified three-body system. In the case of planets in reciprocal circular motion, one of which has negligible mass compared to the other two, there are five equilibrium points called Lagrangian points.

Three of these five points lie on the straight line of the two major bodies, one included in the segment having the two major bodies as endpoints and two external to it.

These positions are unstable. The remaining two points are located on the orbit of the intermediate mass planet and, compared to the latter, one point is 60° ahead and the other is 60° behind.

Therefore the imaginary segments that connect the two larger bodies to each other and form two equilateral triangles. For a high ratio between the masses of the two larger bodies.

Another simplified case is the three-body problem in Euler's formulation, in which a body is moving in the field produced by two other immobile masses. The problem is analytically soluble but requires the use of elliptic integrals.

It all started in 1687, when Isaac Newton formulated the formula for the gravitational attraction between two bodies. The British scientist described for the first time the principles of dynamics still used today to study the motion of bodies, but also the law of universal gravitation. The two-body problem.

If we add a third body to the system, the three-body problem occurs. Newton was unable to find a formula to describe what happens to a triple system over time. A system of this type is chaotic and therefore unpredictable for all practical purposes.

The situation becomes less compromised when one of the three bodies is much less massive than the other two: the restricted three-body problem.