WEBVTT 00:00:07.745 --> 00:00:11.880 In 2009, two researchers ran a simple experiment. 00:00:11.880 --> 00:00:15.055 They took everything we know about our solar system 00:00:15.055 --> 00:00:21.107 and calculated where every planet would be up to 5 billion years in the future. 00:00:21.107 --> 00:00:25.107 To do so they ran over 2,000 numerical simulations 00:00:25.107 --> 00:00:29.829 with the same exact initial conditions except for one difference: 00:00:29.829 --> 00:00:35.136 the distance between Mercury and the Sun, modified by less than a millimeter 00:00:35.136 --> 00:00:37.796 from one simulation to the next. 00:00:37.796 --> 00:00:41.074 Shockingly, in about 1 percent of their simulations, 00:00:41.074 --> 00:00:46.420 Mercury’s orbit changed so drastically that it could plunge into the Sun 00:00:46.420 --> 00:00:48.780 or collide with Venus. 00:00:48.780 --> 00:00:49.500 Worse yet, 00:00:49.500 --> 00:00:54.983 in one simulation it destabilized the entire inner solar system. 00:00:54.983 --> 00:00:58.983 This was no error; the astonishing variety in results 00:00:58.983 --> 00:01:05.058 reveals the truth that our solar system may be much less stable than it seems. NOTE Paragraph 00:01:05.058 --> 00:01:10.239 Astrophysicists refer to this astonishing property of gravitational systems 00:01:10.239 --> 00:01:12.419 as the n-body problem. 00:01:12.419 --> 00:01:15.239 While we have equations that can completely predict 00:01:15.239 --> 00:01:17.949 the motions of two gravitating masses, 00:01:17.949 --> 00:01:23.600 our analytical tools fall short when faced with more populated systems. 00:01:23.600 --> 00:01:28.861 It’s actually impossible to write down all the terms of a general formula 00:01:28.861 --> 00:01:34.771 that can exactly describe the motion of three or more gravitating objects. NOTE Paragraph 00:01:34.771 --> 00:01:41.876 Why? The issue lies in how many unknown variables an n-body system contains. 00:01:41.876 --> 00:01:45.186 Thanks to Isaac Newton, we can write a set of equations 00:01:45.186 --> 00:01:49.186 to describe the gravitational force acting between bodies. 00:01:49.186 --> 00:01:53.863 However, when trying to find a general solution for the unknown variables 00:01:53.863 --> 00:01:55.153 in these equations, 00:01:55.153 --> 00:01:58.002 we’re faced with a mathematical constraint: 00:01:58.002 --> 00:02:01.833 for each unknown, there must be at least one equation 00:02:01.833 --> 00:02:04.043 that independently describes it. NOTE Paragraph 00:02:04.043 --> 00:02:08.934 Initially, a two-body system appears to have more unknown variables 00:02:08.934 --> 00:02:12.724 for position and velocity than equations of motion. 00:02:12.724 --> 00:02:14.680 However, there’s a trick: 00:02:14.680 --> 00:02:18.915 consider the relative position and velocity of the two bodies 00:02:18.915 --> 00:02:22.625 with respect to the center of gravity of the system. 00:02:22.625 --> 00:02:27.353 This reduces the number of unknowns and leaves us with a solvable system. NOTE Paragraph 00:02:27.353 --> 00:02:33.079 With three or more orbiting objects in the picture, everything gets messier. 00:02:33.079 --> 00:02:37.461 Even with the same mathematical trick of considering relative motions, 00:02:37.461 --> 00:02:42.088 we’re left with more unknowns than equations describing them. 00:02:42.088 --> 00:02:46.340 There are simply too many variables for this system of equations 00:02:46.340 --> 00:02:49.610 to be untangled into a general solution. NOTE Paragraph 00:02:49.610 --> 00:02:53.520 But what does it actually look like for objects in our universe 00:02:53.520 --> 00:02:58.631 to move according to analytically unsolvable equations of motion? 00:02:58.631 --> 00:03:01.881 A system of three stars— like Alpha Centauri— 00:03:01.881 --> 00:03:05.359 could come crashing into one another or, more likely, 00:03:05.359 --> 00:03:10.471 some might get flung out of orbit after a long time of apparent stability. 00:03:10.471 --> 00:03:14.471 Other than a few highly improbable stable configurations, 00:03:14.471 --> 00:03:20.571 almost every possible case is unpredictable on long timescales. 00:03:20.571 --> 00:03:24.768 Each has an astronomically large range of potential outcomes, 00:03:24.768 --> 00:03:29.576 dependent on the tiniest of differences in position and velocity. 00:03:29.576 --> 00:03:33.742 This behaviour is known as chaotic by physicists, 00:03:33.742 --> 00:03:37.472 and is an important characteristic of n-body systems. 00:03:37.472 --> 00:03:42.201 Such a system is still deterministic— meaning there’s nothing random about it. 00:03:42.201 --> 00:03:45.791 If multiple systems start from the exact same conditions, 00:03:45.791 --> 00:03:48.241 they’ll always reach the same result. 00:03:48.241 --> 00:03:53.980 But give one a little shove at the start, and all bets are off. 00:03:53.980 --> 00:03:57.240 That’s clearly relevant for human space missions, 00:03:57.240 --> 00:04:02.489 when complicated orbits need to be calculated with great precision. NOTE Paragraph 00:04:02.489 --> 00:04:06.489 Thankfully, continuous advancements in computer simulations 00:04:06.489 --> 00:04:09.379 offer a number of ways to avoid catastrophe. 00:04:09.379 --> 00:04:13.695 By approximating the solutions with increasingly powerful processors, 00:04:13.695 --> 00:04:19.565 we can more confidently predict the motion of n-body systems on long time-scales. 00:04:19.565 --> 00:04:22.755 And if one body in a group of three is so light 00:04:22.755 --> 00:04:25.885 it exerts no significant force on the other two, 00:04:25.885 --> 00:04:30.727 the system behaves, with very good approximation, as a two-body system. 00:04:30.727 --> 00:04:34.727 This approach is known as the “restricted three-body problem.” 00:04:34.727 --> 00:04:38.097 It proves extremely useful in describing, for example, 00:04:38.097 --> 00:04:41.607 an asteroid in the Earth-Sun gravitational field, 00:04:41.607 --> 00:04:46.700 or a small planet in the field of a black hole and a star. NOTE Paragraph 00:04:46.700 --> 00:04:49.480 As for our solar system, you’ll be happy to hear 00:04:49.480 --> 00:04:52.650 that we can have reasonable confidence in its stability 00:04:52.650 --> 00:04:56.330 for at least the next several hundred million years. 00:04:56.330 --> 00:04:58.020 Though if another star, 00:04:58.020 --> 00:05:02.000 launched from across the galaxy, is on its way to us, 00:05:02.000 --> 00:05:03.850 all bets are off.