1 00:00:07,745 --> 00:00:11,880 In 2009, two researchers ran a simple experiment. 2 00:00:11,880 --> 00:00:15,055 They took everything we know about our solar system 3 00:00:15,055 --> 00:00:21,107 and calculated where every planet would be up to 5 billion years in the future. 4 00:00:21,107 --> 00:00:25,107 To do so they ran over 2,000 numerical simulations 5 00:00:25,107 --> 00:00:29,829 with the same exact initial conditions except for one difference: 6 00:00:29,829 --> 00:00:35,136 the distance between Mercury and the Sun, modified by less than a millimeter 7 00:00:35,136 --> 00:00:37,796 from one simulation to the next. 8 00:00:37,796 --> 00:00:41,074 Shockingly, in about 1 percent of their simulations, 9 00:00:41,074 --> 00:00:46,420 Mercury’s orbit changed so drastically that it could plunge into the Sun 10 00:00:46,420 --> 00:00:48,780 or collide with Venus. 11 00:00:48,780 --> 00:00:49,500 Worse yet, 12 00:00:49,500 --> 00:00:54,983 in one simulation it destabilized the entire inner solar system. 13 00:00:54,983 --> 00:00:58,983 This was no error; the astonishing variety in results 14 00:00:58,983 --> 00:01:05,058 reveals the truth that our solar system may be much less stable than it seems. 15 00:01:05,058 --> 00:01:10,239 Astrophysicists refer to this astonishing property of gravitational systems 16 00:01:10,239 --> 00:01:12,419 as the n-body problem. 17 00:01:12,419 --> 00:01:15,239 While we have equations that can completely predict 18 00:01:15,239 --> 00:01:17,949 the motions of two gravitating masses, 19 00:01:17,949 --> 00:01:23,600 our analytical tools fall short when faced with more populated systems. 20 00:01:23,600 --> 00:01:28,861 It’s actually impossible to write down all the terms of a general formula 21 00:01:28,861 --> 00:01:34,771 that can exactly describe the motion of three or more gravitating objects. 22 00:01:34,771 --> 00:01:41,876 Why? The issue lies in how many unknown variables an n-body system contains. 23 00:01:41,876 --> 00:01:45,186 Thanks to Isaac Newton, we can write a set of equations 24 00:01:45,186 --> 00:01:49,186 to describe the gravitational force acting between bodies. 25 00:01:49,186 --> 00:01:53,863 However, when trying to find a general solution for the unknown variables 26 00:01:53,863 --> 00:01:55,153 in these equations, 27 00:01:55,153 --> 00:01:58,002 we’re faced with a mathematical constraint: 28 00:01:58,002 --> 00:02:01,833 for each unknown, there must be at least one equation 29 00:02:01,833 --> 00:02:04,043 that independently describes it. 30 00:02:04,043 --> 00:02:08,934 Initially, a two-body system appears to have more unknown variables 31 00:02:08,934 --> 00:02:12,724 for position and velocity than equations of motion. 32 00:02:12,724 --> 00:02:14,680 However, there’s a trick: 33 00:02:14,680 --> 00:02:18,915 consider the relative position and velocity of the two bodies 34 00:02:18,915 --> 00:02:22,625 with respect to the center of gravity of the system. 35 00:02:22,625 --> 00:02:27,353 This reduces the number of unknowns and leaves us with a solvable system. 36 00:02:27,353 --> 00:02:33,079 With three or more orbiting objects in the picture, everything gets messier. 37 00:02:33,079 --> 00:02:37,461 Even with the same mathematical trick of considering relative motions, 38 00:02:37,461 --> 00:02:42,088 we’re left with more unknowns than equations describing them. 39 00:02:42,088 --> 00:02:46,340 There are simply too many variables for this system of equations 40 00:02:46,340 --> 00:02:49,610 to be untangled into a general solution. 41 00:02:49,610 --> 00:02:53,520 But what does it actually look like for objects in our universe 42 00:02:53,520 --> 00:02:58,631 to move according to analytically unsolvable equations of motion? 43 00:02:58,631 --> 00:03:01,881 A system of three stars— like Alpha Centauri— 44 00:03:01,881 --> 00:03:05,359 could come crashing into one another or, more likely, 45 00:03:05,359 --> 00:03:10,471 some might get flung out of orbit after a long time of apparent stability. 46 00:03:10,471 --> 00:03:14,471 Other than a few highly improbable stable configurations, 47 00:03:14,471 --> 00:03:20,571 almost every possible case is unpredictable on long timescales. 48 00:03:20,571 --> 00:03:24,768 Each has an astronomically large range of potential outcomes, 49 00:03:24,768 --> 00:03:29,576 dependent on the tiniest of differences in position and velocity. 50 00:03:29,576 --> 00:03:33,742 This behaviour is known as chaotic by physicists, 51 00:03:33,742 --> 00:03:37,472 and is an important characteristic of n-body systems. 52 00:03:37,472 --> 00:03:42,201 Such a system is still deterministic— meaning there’s nothing random about it. 53 00:03:42,201 --> 00:03:45,791 If multiple systems start from the exact same conditions, 54 00:03:45,791 --> 00:03:48,241 they’ll always reach the same result. 55 00:03:48,241 --> 00:03:53,980 But give one a little shove at the start, and all bets are off. 56 00:03:53,980 --> 00:03:57,240 That’s clearly relevant for human space missions, 57 00:03:57,240 --> 00:04:02,489 when complicated orbits need to be calculated with great precision. 58 00:04:02,489 --> 00:04:06,489 Thankfully, continuous advancements in computer simulations 59 00:04:06,489 --> 00:04:09,379 offer a number of ways to avoid catastrophe. 60 00:04:09,379 --> 00:04:13,695 By approximating the solutions with increasingly powerful processors, 61 00:04:13,695 --> 00:04:19,565 we can more confidently predict the motion of n-body systems on long time-scales. 62 00:04:19,565 --> 00:04:22,755 And if one body in a group of three is so light 63 00:04:22,755 --> 00:04:25,885 it exerts no significant force on the other two, 64 00:04:25,885 --> 00:04:30,727 the system behaves, with very good approximation, as a two-body system. 65 00:04:30,727 --> 00:04:34,727 This approach is known as the “restricted three-body problem.” 66 00:04:34,727 --> 00:04:38,097 It proves extremely useful in describing, for example, 67 00:04:38,097 --> 00:04:41,607 an asteroid in the Earth-Sun gravitational field, 68 00:04:41,607 --> 00:04:46,700 or a small planet in the field of a black hole and a star. 69 00:04:46,700 --> 00:04:49,480 As for our solar system, you’ll be happy to hear 70 00:04:49,480 --> 00:04:52,650 that we can have reasonable confidence in its stability 71 00:04:52,650 --> 00:04:56,330 for at least the next several hundred million years. 72 00:04:56,330 --> 00:04:58,020 Though if another star, 73 00:04:58,020 --> 00:05:02,000 launched from across the galaxy, is on its way to us, 74 00:05:02,000 --> 00:05:03,850 all bets are off.