0:00:07.745,0:00:11.880 In 2009, two researchers ran a simple [br]experiment. 0:00:11.880,0:00:16.015 They took everything we know about our[br]solar system and calculated 0:00:16.015,0:00:21.107 where every planet would be up to 5 [br]billion years in the future. 0:00:21.107,0:00:25.107 To do so they ran over 2000 numerical [br]simulations 0:00:25.107,0:00:29.829 with the same exact initial conditions[br]except for one difference: 0:00:29.829,0:00:35.136 the distance between Mercury and the Sun,[br]modified by less than a millimeter 0:00:35.136,0:00:37.796 from one simulation to the next. 0:00:37.796,0:00:41.074 Shockingly, in about 1 percent of their [br]simulations, 0:00:41.074,0:00:46.420 Mercury’s orbit changed so drastically [br]that it could plunge into the Sun 0:00:46.420,0:00:48.780 or collide with Venus. 0:00:48.780,0:00:49.500 Worse yet, 0:00:49.500,0:00:54.983 in one simulation it destabilized[br]the entire inner solar system. 0:00:54.983,0:00:58.983 This was no error; the astonishing variety[br]in results 0:00:58.983,0:01:05.058 reveals the truth that our solar system [br]may be much less stable than it seems. 0:01:05.058,0:01:10.239 Astrophysicists refer to this astonishing[br]property of gravitational systems 0:01:10.239,0:01:12.419 as the n-body problem. 0:01:12.419,0:01:15.239 While we have equations that can [br]completely predict 0:01:15.239,0:01:17.949 the motions of two gravitating masses, 0:01:17.949,0:01:23.600 our analytical tools fall short when [br]faced with more populated systems. 0:01:23.600,0:01:28.861 It’s actually impossible to write down[br]all the terms of a general formula 0:01:28.861,0:01:34.771 that can exactly describe the motion[br]of three or more gravitating objects. 0:01:34.771,0:01:41.876 Why? The issue lies in how many unknown[br]variables an n-body system contains. 0:01:41.876,0:01:45.186 Thanks to Isaac Newton, we can write [br]a set of equations 0:01:45.186,0:01:49.186 to describe the gravitational force [br]acting between bodies. 0:01:49.186,0:01:53.863 However, when trying to find a general [br]solution for the unknown variables 0:01:53.863,0:01:55.153 in these equations, 0:01:55.153,0:01:59.592 we’re faced with a mathematical [br]constraint: for each unknown, 0:01:59.592,0:02:04.043 there must be at least one equation [br]that independently describes it. 0:02:04.043,0:02:08.934 Initially, a two-body system appears to[br]have more unknown variables 0:02:08.934,0:02:12.724 for position and velocity than [br]equations of motion. 0:02:12.724,0:02:14.680 However, there’s a trick: 0:02:14.680,0:02:18.915 consider the relative position and [br]velocity of the two bodies 0:02:18.915,0:02:22.625 with respect to the center of [br]gravity of the system. 0:02:22.625,0:02:27.353 This reduces the number of unknowns[br]and leaves us with a solvable system. 0:02:27.353,0:02:33.079 With three or more orbiting objects in the[br]picture, everything gets messier. 0:02:33.079,0:02:37.461 Even with the same mathematical trick [br]of considering relative motions, 0:02:37.461,0:02:42.088 we’re left with more unknowns than [br]equations describing them. 0:02:42.088,0:02:46.340 There are simply too many variables[br]for this system of equations 0:02:46.340,0:02:49.610 to be untangled into a general solution. 0:02:49.610,0:02:53.520 But what does it actually look like for [br]objects in our universe 0:02:53.520,0:02:58.631 to move according to analytically [br]unsolvable equations of motion? 0:02:58.631,0:03:00.451 A system of three stars–– 0:03:00.451,0:03:05.359 like Alpha Centauri could come crashing[br]into one another or, more likely, 0:03:05.359,0:03:10.471 some might get flung out of orbit [br]after a long time of apparent stability. 0:03:10.471,0:03:14.471 Other than a few highly improbable [br]stable configurations, 0:03:14.471,0:03:20.571 almost every possible case is [br]unpredictable on long timescales. 0:03:20.571,0:03:24.768 Each has an astronomically large range[br]of potential outcomes, 0:03:24.768,0:03:29.576 dependent on the tiniest of differences[br]in position and velocity. 0:03:29.576,0:03:33.742 This behaviour is known as chaotic [br]by physicists, 0:03:33.742,0:03:37.472 and is an important characteristic [br]of n-body systems. 0:03:37.472,0:03:42.201 Such a system is still deterministic—[br]meaning there’s nothing random about it. 0:03:42.201,0:03:45.791 If multiple systems start from the exact[br]same conditions, 0:03:45.791,0:03:48.241 they’ll always reach the same result. 0:03:48.241,0:03:53.980 But give one a little shove at the start,[br]and all bets are off. 0:03:53.980,0:03:57.240 That’s clearly relevant for human space[br]missions, 0:03:57.240,0:04:02.489 when complicated orbits need to [br]be calculated with great precision. 0:04:02.489,0:04:06.489 Thankfully, continuous advancements[br]in computer simulations 0:04:06.489,0:04:09.379 offer a number of ways[br]to avoid catastrophe. 0:04:09.379,0:04:13.695 By approximating the solutions with [br]increasingly powerful processors, 0:04:13.695,0:04:19.565 we can more confidently predict the motion[br]of n-body systems on long time-scales. 0:04:19.565,0:04:22.755 And if one body in a group of three[br]is so light 0:04:22.755,0:04:25.885 it exerts no significant force [br]on the other two, 0:04:25.885,0:04:30.727 the system behaves, with very good [br]approximation, as a two-body system. 0:04:30.727,0:04:34.727 This approach is known as the “restricted [br]three-body problem.” 0:04:34.727,0:04:38.097 It proves extremely useful in describing,[br]for example, 0:04:38.097,0:04:41.607 an asteroid in the Earth-Sun [br]gravitational field, 0:04:41.607,0:04:46.700 or a small planet in the field of a [br]black hole and a star. 0:04:46.700,0:04:49.480 As for our solar system, you’ll [br]be happy to hear 0:04:49.480,0:04:52.650 that we can have reasonable confidence[br]in its stability 0:04:52.650,0:04:56.330 for at least the next several [br]hundred million years. 0:04:56.330,0:05:02.000 Though if another star, launched from [br]across the galaxy, is on its way to us, 0:05:02.000,0:05:04.130 all bets are off.