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← Newton's three-body problem explained - Fabio Pacucci

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Showing Revision 3 created 07/23/2020 by lauren mcalpine .

  1. In 2009, two researchers ran
    a simple experiment.
  2. They took everything we know
    about our solar system
  3. and calculated where every planet would be
    up to 5 billion years in the future.
  4. To do so they ran over 2,000
    numerical simulations
  5. with the same exact initial conditions
    except for one difference:
  6. the distance between Mercury and the Sun,
    modified by less than a millimeter
  7. from one simulation to the next.
  8. Shockingly, in about 1 percent
    of their simulations,
  9. Mercury’s orbit changed so drastically
    that it could plunge into the Sun
  10. or collide with Venus.
  11. Worse yet,
  12. in one simulation it destabilized
    the entire inner solar system.
  13. This was no error;
    the astonishing variety in results
  14. reveals the truth that our solar system
    may be much less stable than it seems.
  15. Astrophysicists refer to this astonishing
    property of gravitational systems

  16. as the n-body problem.
  17. While we have equations
    that can completely predict
  18. the motions of two gravitating masses,
  19. our analytical tools fall short
    when faced with more populated systems.
  20. It’s actually impossible to write down
    all the terms of a general formula
  21. that can exactly describe the motion
    of three or more gravitating objects.
  22. Why? The issue lies in how many unknown
    variables an n-body system contains.

  23. Thanks to Isaac Newton,
    we can write a set of equations
  24. to describe the gravitational force
    acting between bodies.
  25. However, when trying to find a general
    solution for the unknown variables
  26. in these equations,
  27. we’re faced with
    a mathematical constraint:
  28. for each unknown,
    there must be at least one equation
  29. that independently describes it.
  30. Initially, a two-body system appears
    to have more unknown variables

  31. for position and velocity
    than equations of motion.
  32. However, there’s a trick:
  33. consider the relative position
    and velocity of the two bodies
  34. with respect to the center
    of gravity of the system.
  35. This reduces the number of unknowns
    and leaves us with a solvable system.
  36. With three or more orbiting objects
    in the picture, everything gets messier.

  37. Even with the same mathematical trick
    of considering relative motions,
  38. we’re left with more unknowns
    than equations describing them.
  39. There are simply too many variables
    for this system of equations
  40. to be untangled into a general solution.
  41. But what does it actually look like
    for objects in our universe

  42. to move according to analytically
    unsolvable equations of motion?
  43. A system of three stars—
    like Alpha Centauri—
  44. could come crashing
    into one another or, more likely,
  45. some might get flung out of orbit
    after a long time of apparent stability.
  46. Other than a few highly improbable
    stable configurations,
  47. almost every possible case
    is unpredictable on long timescales.
  48. Each has an astronomically large range
    of potential outcomes,
  49. dependent on the tiniest of differences
    in position and velocity.
  50. This behaviour is known
    as chaotic by physicists,
  51. and is an important characteristic
    of n-body systems.
  52. Such a system is still deterministic—
    meaning there’s nothing random about it.
  53. If multiple systems start
    from the exact same conditions,
  54. they’ll always reach the same result.
  55. But give one a little shove at the start,
    and all bets are off.
  56. That’s clearly relevant
    for human space missions,
  57. when complicated orbits need
    to be calculated with great precision.
  58. Thankfully, continuous advancements
    in computer simulations

  59. offer a number of ways
    to avoid catastrophe.
  60. By approximating the solutions
    with increasingly powerful processors,
  61. we can more confidently predict the motion
    of n-body systems on long time-scales.
  62. And if one body in a group
    of three is so light
  63. it exerts no significant force
    on the other two,
  64. the system behaves, with very good
    approximation, as a two-body system.
  65. This approach is known
    as the “restricted three-body problem.”
  66. It proves extremely useful
    in describing, for example,
  67. an asteroid in the Earth-Sun
    gravitational field,
  68. or a small planet in the field
    of a black hole and a star.
  69. As for our solar system,
    you’ll be happy to hear

  70. that we can have reasonable confidence
    in its stability
  71. for at least the next
    several hundred million years.
  72. Though if another star,
  73. launched from across the galaxy,
    is on its way to us,
  74. all bets are off.