Newton’s three-body problem explained - Fabio Pacucci | Summary and Q&A

TL;DR

The N-body problem states that it is impossible to write down a general formula that can accurately predict the motion of three or more gravitating objects. This leads to unpredictable orbits and the potential instability of our solar system.

Key Insights

• 🥺 A small modification in initial conditions can lead to drastically different outcomes in the orbits of celestial bodies, emphasizing the complex and unpredictable nature of n-body systems.
• 🙅 The N-body problem arises from the mathematical challenge of solving a system of equations with more unknowns than equations.
• 😀 The chaos exhibited by n-body systems is not random but deterministic, meaning that identical systems with the same initial conditions will always yield the same results.
• 💨 Computer simulations offer a way to approximate the motion of n-body systems and make long-term predictions.
• ❓ The restricted three-body problem provides a useful approach for analyzing systems where one body has negligible influence on the others.

Transcript

In 2009, two researchers ran a simple experiment. They took everything we know about our solar system and calculated where every planet would be up to 5 billion years in the future. To do so they ran over 2,000 numerical simulations with the same exact initial conditions except for one difference: the distance between Mercury and the Sun, modifi... Read More

Questions & Answers

Q: Why is it impossible to write down a general formula that accurately predicts the motion of three or more gravitating objects?

The N-body problem arises because the number of unknown variables in the system of equations describing gravitational forces is greater than the number of equations, making it mathematically infeasible to find a general solution.

Q: Can computer simulations help predict the motion of n-body systems?

Yes, continuous advancements in computer simulations allow for approximations of n-body systems, enabling more confident predictions of their motion over long timescales.

Q: What is the restricted three-body problem?

The restricted three-body problem refers to a situation where one body in a group of three exerts no significant force on the other two, making the system behave, with good approximation, as a two-body system. This approach is useful in describing scenarios like an asteroid in the Earth-Sun gravitational field.

Q: Is our solar system stable in the long term?

While there is reasonable confidence in the stability of our solar system for the next several hundred million years, the introduction of another star from across the galaxy could disrupt the system's stability.

Summary & Key Takeaways

• In a study, researchers found that even a small modification in the distance between Mercury and the Sun can lead to drastic changes in Mercury's orbit, potentially causing it to collide with Venus or plunge into the Sun.

• The N-body problem arises from the inability to find a general solution for the unknown variables in a system of equations that describe the gravitational force between multiple bodies.

• The unpredictable behavior of n-body systems, known as chaos, means that even slight differences in initial conditions can lead to a wide range of outcomes over long timescales.