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  • 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, modified by less than a millimeter

  • from one simulation to the next.

  • Shockingly, in about 1 percent of their simulations,

  • Mercury's orbit changed so drastically that it could plunge into the Sun

  • or collide with Venus.

  • Worse yet,

  • in one simulation it destabilized the entire inner solar system.

  • This was no error; the astonishing variety in results

  • reveals the truth that our solar system may be much less stable than it seems.

  • Astrophysicists refer to this astonishing property of gravitational systems

  • as the n-body problem.

  • While we have equations that can completely predict

  • the motions of two gravitating masses,

  • our analytical tools fall short when faced with more populated systems.

  • It's actually impossible to write down all the terms of a general formula

  • that can exactly describe the motion of three or more gravitating objects.

  • Why? The issue lies in how many unknown variables an n-body system contains.

  • Thanks to Isaac Newton, we can write a set of equations

  • to describe the gravitational force acting between bodies.

  • However, when trying to find a general solution for the unknown variables

  • in these equations,

  • we're faced with a mathematical constraint:

  • for each unknown, there must be at least one equation

  • that independently describes it.

  • Initially, a two-body system appears to have more unknown variables

  • for position and velocity than equations of motion.

  • However, there's a trick:

  • consider the relative position and velocity of the two bodies

  • with respect to the center of gravity of the system.

  • This reduces the number of unknowns and leaves us with a solvable system.

  • With three or more orbiting objects in the picture, everything gets messier.

  • Even with the same mathematical trick of considering relative motions,

  • we're left with more unknowns than equations describing them.

  • There are simply too many variables for this system of equations

  • to be untangled into a general solution.

  • But what does it actually look like for objects in our universe

  • to move according to analytically unsolvable equations of motion?

  • A system of three starslike Alpha Centauri

  • could come crashing into one another or, more likely,

  • some might get flung out of orbit after a long time of apparent stability.

  • Other than a few highly improbable stable configurations,

  • almost every possible case is unpredictable on long timescales.

  • Each has an astronomically large range of potential outcomes,

  • dependent on the tiniest of differences in position and velocity.

  • This behaviour is known as chaotic by physicists,

  • and is an important characteristic of n-body systems.

  • Such a system is still deterministicmeaning there's nothing random about it.

  • If multiple systems start from the exact same conditions,

  • they'll always reach the same result.

  • But give one a little shove at the start, and all bets are off.

  • That's clearly relevant for human space missions,

  • when complicated orbits need to be calculated with great precision.

  • Thankfully, continuous advancements in computer simulations

  • offer a number of ways to avoid catastrophe.

  • By approximating the solutions with increasingly powerful processors,

  • we can more confidently predict the motion of n-body systems on long time-scales.

  • And if one body in a group of three is so light

  • it exerts no significant force on the other two,

  • the system behaves, with very good approximation, as a two-body system.

  • This approach is known as therestricted three-body problem.”

  • It proves extremely useful in describing, for example,

  • an asteroid in the Earth-Sun gravitational field,

  • or a small planet in the field of a black hole and a star.

  • As for our solar system, you'll be happy to hear

  • that we can have reasonable confidence in its stability

  • for at least the next several hundred million years.

  • Though if another star,

  • launched from across the galaxy, is on its way to us,

  • all bets are off.

In 2009, two researchers ran a simple experiment.

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B1 system body solar system solar gravitational motion

Newton’s three-body problem explained - Fabio Pacucci

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    林宜悉 posted on 2020/10/24
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